# New revival phenomena for bidirectional dispersive hyperbolic equations

George Farmakis  George Farmakis
Heriot-Watt University & Maxwell Institute for Mathematical Sciences,
Edinburgh, UK<EMAIL_ADDRESS>,  Jing Kang  Jing Kang
Center for Nonlinear Studies and School of Mathematics, Northwest University,
Xi’an 710069, P.R. China<EMAIL_ADDRESS>,  Peter J. Olver  Peter J.
Olver
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
<EMAIL_ADDRESS>,  Changzheng Qu  Changzheng Qu
School of Mathematics and Statistics, Ningbo University, Ningbo 315211, P.R.
China<EMAIL_ADDRESS>and  Zihan Yin  Zihan Yin
School of Mathematics, Northwest University, Xi’an 710069, P.R. China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, the dispersive revival and fractalization phenomena for
bidirectional dispersive equations on a bounded interval subject to periodic
boundary conditions and discontinuous initial profiles are investigated.
Firstly, we study the periodic initial-boundary value problem of the linear
beam equation with step function initial data, and analyze the manifestation
of the revival phenomenon for the corresponding solution at rational times.
Next, we extend the investigation to periodic initial-boundary value problems
of more general bidirectional dispersive equations. We prove that, if the
initial functions are of bounded variation, the dynamical evolution of such
periodic problems depend essentially upon the large wave number asymptotics of
the associated dispersion relations. Integral polynomial or asymptotically
integral polynomial dispersion relations produce dispersive
revival/fractalization rational/irrational dichotomies, whereas those with
non-polynomial growth result in fractal profiles at all times. Finally,
numerical experiments, in the concrete case of the nonlinear beam equation,
are used to demonstrate how such effects persist into the nonlinear regime.

Key words and phrases: beam equation; revival; fractalization; Talbot effect;
dispersive equation. Mathematics Subject Classification (2020): 37K55, 35Q51

##  1\. Introduction

This paper is devoted to the study of the periodic initial-boundary value
problem for bidirectional dispersive partial differential equations. We prove
that, for linear equations, if the initial condition at time zero is a step
function or, more generally, a function of bounded variation, the time
evolution of the bidirectional dispersive equations subject to periodic
boundary conditions will exhibit new revival phenomena at rational times, of a
different form from that previously observed in unidirectional dispersive
evolution equations, whereas at irrational times the solution exhibits a
continuous, but non-differentiable fractal profile.

The term “revival” is based on the experimentally observed phenomenon of
quantum revival [3, 40], in which an electron that is initially concentrated
near a single location of its orbital shell is re-concentrated near a finite
number of orbital locations at certain times. A precursor of the revival
phenomenon was observed as far back as 1834 in a striking optical experiment,
[37], conducted in 1836 by William Henry Fox Talbot. This motivated the
pioneering work of Berry and his collaborators, [1, 2, 3], on what they called
the Talbot effect in the context of the linear free space Schrödinger
equation. Rigorous analytical results and estimates justifying the Talbot
effect can be found in the work of Kapitanski and Rodnianski, [26, 32],
Oskolkov, [30, 31], and Taylor, [38]. The Talbot effect governs, in the
quantum mechanical setting, the behavior of rough solutions subject to
periodic boundary conditions. The evolution of the rough initial profile, for
instance, a step function, also known as the Riemann problem [41], “quantizes”
into a dispersive revival profile at rational times, but “fractalizes” into a
continuous but nowhere differentiable profile having a specific fractal
dimension at irrational times.

In [7, 27], the same Talbot effect, which the authors called dispersive
quantization and fractalization, was shown to appear in general periodic
linear dispersive equations possessing an “integral polynomial” (a polynomial
with integer coefficients) dispersion relation, which included the
prototypical linearized Schrödinger and Korteweg-de Vries (KdV) equations.
Based on these investigations, one learns that a linear dispersive equation
admitting a polynomial dispersion relation and subject to periodic boundary
conditions will exhibit the revival phenomenon at each rational time, which
means that the fundamental solution, i.e., that induced by a delta function
initial condition, localizes into a finite linear combination of delta
functions. This has the remarkable consequence that the solution, to any
initial value problem, at rational times is a finite linear combination of
translates of the initial data and hence its value at any point on the
periodic domain depends only upon finitely many of the initial values. In
[28], the revival phenomenon for the linear free space Schrödinger equation
subject to pseudo-periodic boundary conditions was investigated, see also [4]
for the same model and for the quasi-periodic linear KdV equation. In [6], a
more general revival phenomenon, that produces dispersively quantized cusped
solutions of the periodic Riemann problem for three linear integro-
differential equations, including the Benjamin-Ono equation, the Intermediate
Long Wave equation and the Smith equation were studied. More recently, these
phenomena were shown to extend to multi-component dispersive equations, see
[42]. For a class of two-component linear systems of dispersive evolution
equations, the dispersive quantization conditions, which may yield quantized
structures for step-function initial value at rational times, are provided.

Inspired by these linear results, the phenomena of dispersive quantization and
fractalization for the periodic Riemann problem for nonlinear dispersive
evolution equations on periodic domains, including the integrable nonlinear
Schrödinger (NLS), KdV and modified KdV (mKdV) equations as well as non-
integrable versions with higher-order nonlinearities were studied numerically
in [8]. Erdo$\mathrm{\breve{g}}$an, Tzirakis and their collaborators
established rigorous results on the fractalization for the nonlinear equations
at a dense set of times. Quantifying the irrational time fractalization in
terms of the estimate on the fractal dimension, their results, on the one hand
extend the results of Oskolkov and Rodnianski to a class of nonlinear integer
polynomial dispersive equations subject to initial data of bounded variation,
and, on the other hand, confirm the numerical observations of fractalization
in [8]. Erdoǵan and Tzirakis studied the cubic NLS and KdV equations on a
periodic domain with initial data of bounded variation in [13] and [14],
respectively. Subsequently, together with Chousionis, they obtained some
results on the Minkowski dimension of the fractalization profiles for
dispersive linear partial differential equations with monomial dispersion
relation [10]. We refer the reader to the survey texts [12, 15] for irrational
time fractalization results. See also the recent survey [35].

To date, investigations have almost all concentrated on unidirectional
dispersive systems. In the present paper, we will show that the dispersive
revival/fractalization rational/irrational dichotomy extends to bidirectional
dispersive equations of the form

$u_{tt}=L[u],$ (1.1)

where $L$ is a scalar differential operator with constant coefficients.
Obviously, equation (1.1) is equivalent to the following two-component
evolutionary system

$u_{t}=v,\qquad v_{t}=L[u],$ (1.2)

which, however, does not satisfy the dispersive quantization conditions given
in [42]. As we describe below, in the bidirectional setting, if we set the
initial conditions equal to the same step function, then the solution of the
corresponding periodic Riemann problem will exhibit qualitative dispersive
quantization behaviour, of a different form than the standard piecewise
constant solutions admitted by the unidirectional systems, such as the linear
KdV and Schrödinger equations and their associated multi-component
generalizations. Interestingly, in the concrete case of the linear beam
equation

$u_{tt}+u_{xxxx}=0,$ (1.3)

these solutions at rational times $t^{\ast}=\pi p/q$ with $q>2$ appear to be
piecewise parabolic, non-constant between jump discontinuities, whereas at the
rational times $t_{k}^{0}=\pi(2k-1)/2$, the solution becomes a continuously
differentiable curve, with analytical expression (2.19).

Similar studies were initiated in one of the authors’ recent Ph.D. thesis
[16], which studies the revival property in bidirectional dispersive equations
(1.1) where the operator $L$ is an even-order poly-Laplacian, which includes
the linear wave equation and the beam equation (1.3), subject to periodic and
quasi-periodic boundary conditions. We should further mention that from a
general perspective, the form of the revival effect in the periodic
bidirectional problems considered here resembles this of the revival effect in
the free linear Schrödinger equation with Robin boundary conditions
$bu(t,0)=(1-b)u_{x}(t,\pi)$, where $b\in(0,1)$ is a parameter, see [4].
Indeed, in both cases the solution at rational times is given as the sum of
the revival of the initial condition and a more regular function, which can be
considered as a weak type of revival. Other models in the literature that
exhibit such weak revivals include the periodic cubic NLS and KdV equations
[13, 14], the periodic linear Schrödinger equation with periodic potential
[33, 9] and the linear Schrödinger equation subject to Dirichlet boundary
conditions [5].

The linear beam equation, which is studied in Section 2, is a typical example
of a model with the dispersion relation of the form $\omega(k)=\pm
k^{N},\;2\leq N\leq\mathbb{Z}^{+}$, namely when $N=2$. Although it is a
special case, it motivates the study of the more general case and illustrates
the idea and the method in the proof. More importantly, through the analysis
and derivation of its periodic initial-boundary value problem, we arrive at
some classical results for the Riemann zeta function. It implies that the
periodic initial-boundary value problems for such systems can provide an
alternative mechanism for establishing such classical identities.

The three concrete goals of the present paper are as follows. The first is to
investigate the new phenomenon of dispersive revival in greater detail, by
examining the periodic initial-boundary value problems for bidirectional
dispersive equations. We will provide an explicit characterization of the
solution profiles of the periodic Riemann problem for the linear beam
equation, leading to the general form of dispersive revival for bidirectional
periodic initial-boundary value problems with various dispersion relations,
including integral polynomial and non-polynomial. Our main results contain the
analytic description of the new phenomena of the dispersive revival, which can
be found in Section 2 for the linear beam equation, and Section 3 for general
bidirectional equations, respectively. In the particular case of monomial
dispersion relations, we present an alternative approach. Secondly, with the
aim to show that such effects can persist into the nonlinear regime, we
present numerical simulations, based on the Fourier spectral method, of the
periodic Riemann problem for the nonlinear beam equation in Section 4.
Numerical approximation supplies strong evidence that the dispersive
revival/fractalization rational/irrational dichotomy persists into the
nonlinear regime, whereas, when compared with the unidirectional systems, the
nonlinear terms induce greater variations of the curve profiles, including
their convexities. Finally, in the course of our analysis, we find that the
solutions at rational times of the periodic Riemann problem for bidirectional
dispersive equations with integral polynomial dispersion relations are closely
related to identities for the Riemann zeta function, which is of great
importance in analytic number theory. In summary, these new revival phenomena
warrant further investigation, both mathematically, and to develop its
profound applications.

##  2\. Revival for the linear beam equation

The starting point is the periodic initial-boundary value problem for the
linear beam equation on the interval $0\leq x\leq 2\pi$:

$\begin{cases}u_{tt}+u_{xxxx}=0,\\\ u(0,x)=f(x),\qquad\qquad
u_{t}(0,x)=g(x),\\\
{\displaystyle\partial_{x}^{j}u(t,0)=\partial_{x}^{j}u(t,2\pi)\atop\displaystyle\partial_{x}^{j}\partial_{t}u(t,0)=\partial_{x}^{j}\partial_{t}u(t,2\pi),}\qquad
j=0,\ 1,\ 2,\ 3.\end{cases}$ (2.1)

The linear beam equation is a bidirectional dispersive equation modeling small
vibrations of a thin elastic beam, with quadratic dispersion relation
$\omega(k)=\pm k^{2}$. We focus our attention on the initial data given by a
step function:

$f(x)=g(x)=\sigma(x)=\begin{cases}-1,\qquad&0\leq x<\pi,\\\ 1,\qquad&\pi\leq
x<2\pi,\end{cases}$ (2.2)

known as the _Riemann problem_. Without further mention, here and elsewhere
below, we assume that functions and distributions defined on $[\,0,\,2\pi\,]$
are extended $2\pi$-periodically to $\mathbb{R}$ in the usual way, when
required.

For the solution of (2.1) with initial data (2.2), we have the following
result.

######  Lemma 2.1.

The periodic initial-boundary value problem (2.1)-(2.2) has the following
solution

$u(t,x)=-\frac{4}{\pi}\left(\sum_{n=0}^{+\infty}\frac{\cos((2n+1)^{2}t)\sin((2n+1)x)}{2n+1}+\sum_{n=0}^{+\infty}\frac{\sin((2n+1)^{2}t)\sin((2n+1)x)}{(2n+1)^{3}}\right).$
(2.3)

Throughout the paper, a time $t>0$ will be designated as _rational_ if
$t/\pi\in\mathbb{Q}$, i.e., $t=t^{\ast}=\pi p/q$, with $p$ and $0\neq
q\in\mathbb{Z}^{+}$ having no common factors. Otherwise, if
$t/\pi\notin\mathbb{Q}$, the time is called _irrational_. To analyze the
qualitative behavior of solution (2.3) at the rational times, we invoke the
following Lemma.

######  Lemma 2.2.

Given $j,q\in\mathbb{Z^{+}}$ with $q\neq 0$, let $\sigma^{j,q}(x)$ be the box
function defined as

$\sigma^{j,q}(x)=\begin{cases}\,1,\qquad&\displaystyle\frac{\pi j}{q}\leq
x<\frac{\pi(j+1)}{q},\quad 0\leq j\leq 2q-1,\\\
\,0,\qquad&otherwise.\end{cases}$ (2.4)

Let $N,\,p,\,q\in\mathbb{Z^{+}},\,N\geq 2,\,q\neq 0$. Then, the following
formulae hold.

(i)

$-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\displaystyle\cos\left((2n+1)^{N}\frac{\pi
p}{q}\right)}{2n+1}\sin((2n+1)x)=\sum_{j=0}^{2q-1}a_{j}\left(\frac{p}{q}\right)\sigma^{j,q}(x),$
(2.5)

(ii) For each even $N$,

$-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\displaystyle\sin\left((2n+1)^{N}\frac{\pi
p}{q}\right)}{2n+1}\sin((2n+1)x)=\sum_{j=0}^{2q-1}b_{j}\left(\frac{p}{q}\right)\sigma^{j,q}(x),$
(2.6)

(iii) For each odd $N$,

$-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\displaystyle\sin\left(((2n+1)^{N}\frac{\pi
p}{q}\right)}{2n+1}\cos((2n+1)x)=\sum_{j=0}^{2q-1}\tilde{b}_{j}\left(\frac{p}{q}\right)\sigma^{j,q}(x),$
(2.7)

where $a_{j},b_{j},\tilde{b}_{j}\in\mathbb{R}$, $j=0,\ldots,2q-1$, are certain
constants which depend on $p$ and $q$.

To prove Lemma 2.2, we need the following theorem, which is based on Theorem
3.2 and Corollary 3.4 in [7], and underlies the dispersive quantization effect
for equations with “integral polynomial” dispersion relation.

######  Definition 2.3.

A polynomial $P(k)=c_{0}+c_{1}k+\cdots+c_{N}k^{N}$ is called an integral
polynomial if its coefficients are integers:
$c_{i}\in\mathbb{Z},i=0,\ldots,N$.

######  Theorem 2.4.

Suppose that the dispersion relation of the evolution equation $u_{t}=L[u]$ is
an integral polynomial:

$\omega(k)=P(k).$

Then at every rational time $t^{\ast}=\pi p/q$, with $p$ and $0\neq
q\in\mathbb{Z}$, the fundamental solution

$F(t,x)\sim\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}e^{\,{\rm
i}\,(kx+\omega(k)t)}$

is a linear combination of $q$ periodically extended delta functions
concentrated at the rational nodes $x_{j}=\pi j/q$ for $j\in\mathbb{Z}$.
Moreover, at $t^{\ast}=\pi p/q$, the solution profile to the periodic initial-
boundary value problem for $u_{t}=L[u]$ is a linear combination of $\leq 2q$
translates of its initial data $u(0,x)=f(x)$, i.e.,

$u\left(\frac{\pi
p}{q},x\right)=\sum_{j=0}^{2q-1}a_{j}\left(\frac{p}{q}\right)f\left(x-\frac{\pi
j}{q}\right).$

######  Remark 2.5.

Slightly more generally, the dispersion relation can be a nonzero multiple of
an integral polynomial. By suitably rescaling time, the stated result holds,
the only difference being which times are designated as rational or
irrational. A similar remark holds if one rescales space to consider the
equation on a different spatial interval.

###### Proof of Lemma 2.2.

According to Theorem 2.4, if the underlying equation admits a dispersion
relation $\omega(k)=\pm k^{N}$, and the initial data is the unit step function

$u(0,x)=\begin{cases}0,\qquad&0\leq x<\pi,\\\ 1,\qquad&\pi\leq
x<2\pi,\end{cases}$

with Fourier coefficients

$\displaystyle c_{k}=\begin{cases}\,\displaystyle\frac{1}{2},\qquad&k=0,\\\
\,0,\qquad&k\neq 0\;\mathrm{even},\\\ \,\displaystyle\frac{\,{\rm i}\,}{\pi
k},\qquad&k\;\mathrm{odd}.\end{cases}$

Then, at a rational time $t^{\ast}=\pi p/q$, the corresponding solution has
the Fourier series form

$u^{\pm}(t,x)=\sum_{k=-\infty}^{+\infty}c_{k}e^{\,{\rm i}\,(kx\pm k^{N}t)}$

and hence is constant on every subinterval $\pi j/q<x<\pi(j+1)/q$, for
$j=0,\ldots,2q-1$, namely

$u^{\pm}(t^{\ast},x)=\sum_{j=0}^{2q-1}\gamma_{j}^{\pm}\left(\frac{p}{q}\right)\sigma^{j,q}(x),$
(2.8)

for certain $\gamma_{j}^{\pm}\in\mathbb{C}$, $j=0,\ldots,2q-1$, dependent on
$p$ and $q$. We thus need to distinguish two cases:

Case 1. $N$ is even. It is easy to see that

$\displaystyle\begin{aligned}
u^{+}(t^{\ast},x)+u^{-}(t^{\ast},x)&=\sum_{k=-\infty}^{+\infty}c_{k}\left(e^{\,{\rm
i}\,(kx+k^{N}t^{\ast})}+e^{\,{\rm i}\,(kx-k^{N}t^{\ast})}\right)\\\
&=1-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\cos\left((2n+1)^{N}t^{\ast}\right)}{2n+1}\sin((2n+1)x),\\\
u^{+}(t^{\ast},x)-u^{-}(t^{\ast},x)&=\sum_{k=-\infty}^{+\infty}c_{k}\left(e^{\,{\rm
i}\,(kx+k^{N}t^{\ast})}-e^{\,{\rm i}\,(kx-k^{N}t^{\ast})}\right)\\\
&=-\frac{4\,{\rm
i}\,}{\pi}\sum_{n=0}^{+\infty}\frac{\sin\left((2n+1)^{N}t^{\ast}\right)}{2n+1}\sin((2n+1)x).\end{aligned}$

Case 2. $N$ is odd. Similar to the above, we have

$\displaystyle\begin{aligned}
u^{+}(t^{\ast},x)+u^{-}(t^{\ast},x)&=\sum_{k=-\infty}^{+\infty}c_{k}\left(e^{\,{\rm
i}\,(kx+k^{N}t^{\ast})}+e^{\,{\rm i}\,(kx-k^{N}t^{\ast})}\right)\\\
&=1-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\cos\left((2n+1)^{N}t^{\ast}\right)}{2n+1}\sin((2n+1)x),\\\
u^{+}(t^{\ast},x)-u^{-}(t^{\ast},x)&=\sum_{k=-\infty}^{+\infty}c_{k}\left(e^{\,{\rm
i}\,(kx+k^{N}t^{\ast})}-e^{\,{\rm i}\,(kx-k^{N}t^{\ast})}\right)\\\
&=-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\sin\left((2n+1)^{N}t^{\ast}\right)}{2n+1}\cos((2n+1)x).\end{aligned}$

Finally, these formulae, together with (2.8) yield (2.5)-(2.7), respectively,
proving the lemma.

∎

Denote the solution (2.3) as

$\displaystyle u(t,x):=\mathrm{I}(t,x)+\mathrm{II}(t,x).$ (2.9)

Note that (2.5) implies that the first summation in (2.9) evaluated at
rational times $t^{\ast}$ has the representation

$\mathrm{I}(t^{\ast},x)=\sum_{j=0}^{2q-1}a_{j}\left(\frac{p}{q}\right)\sigma^{j,q}(x),$
(2.10)

for certain constants $a_{0},\ldots,a_{2q-1}$ determined by (2.5). In
particular, if $q=2$, at the corresponding specific rational time
$t^{0}_{k}=\pi(2k-1)/2,\,k\in\mathbb{Z^{+}}$, it vanishes identically.

When it comes to $\mathrm{II}(t^{\ast},x)$, firstly, a direct computation
shows that for any $t>0$ we have that

$\displaystyle\begin{aligned}
&\frac{\sin((2n+1)^{2}t)\sin((2n+1)x)}{(2n+1)^{3}}\\\ &\hskip
50.58878pt=-\int_{0}^{x}\int_{0}^{y}\frac{\sin((2n+1)^{2}t)\sin((2n+1)z)}{2n+1}\;\mathrm{d}z\mathrm{d}y+\frac{\sin((2n+1)^{2}t)}{(2n+1)^{2}}x.\end{aligned}$
(2.11)

Next, again thanks to Lemma 2.1, we find that the first of the two components
in $\mathrm{II}(t,x)$ satisfies,

$\displaystyle\begin{aligned}
\mathrm{II}^{(1)}(t^{\ast},x)&=\frac{4}{\pi}\int_{0}^{x}\int_{0}^{y}\sum_{n=0}^{+\infty}\frac{\sin((2n+1)^{2}t^{\ast})\sin((2n+1)z)}{2n+1}\;\mathrm{d}z\mathrm{d}y\\\
&=-\int_{0}^{x}\int_{0}^{y}\sum_{j=0}^{2q-1}b_{j}\left(\frac{p}{q}\right)\sigma^{j,q}(z)\;\mathrm{d}z\mathrm{d}y,\end{aligned}$
(2.12)

for certain constants $b_{j},\,j=0,\ldots,2q-1$ determined by (2.6). We denote

$F(y)=\int_{0}^{y}\sum_{j=0}^{2q-1}b_{j}\sigma^{j,q}(z)\;\mathrm{d}z,\quad\mathrm{for}\quad
0\leq y\leq x,$

and set

$H(x)=\int_{0}^{x}F(y)\mathrm{d}y.$

It is easy to verify that

$F(y)=\begin{cases}b_{0}y,\quad\quad&0\leq y\leq\frac{\pi}{q},\\\
\displaystyle
b_{j}y+\frac{\pi}{q}\sum\limits_{m=0}^{j-1}b_{m}-\frac{\pi}{q}jb_{j},\quad&\displaystyle\frac{\pi}{q}j\leq
y\leq\frac{\pi}{q}(j+1),\quad j=1,\ldots,2q-1,\end{cases}$

and hence

$H(x)=\begin{cases}\frac{1}{2}b_{0}x^{2},\quad\quad&\displaystyle 0\leq
x\leq\frac{\pi}{q},\hbox{\vrule height=0.0pt,depth=13.0pt,width=0.0pt}\\\
\frac{1}{2}b_{1}x^{2}+\displaystyle\frac{\pi}{q}(b_{0}-b_{1})x+\frac{\pi^{2}}{2q^{2}}(b_{1}-b_{0}),\quad&\displaystyle\frac{\pi}{q}\leq
x\leq\frac{2\pi}{q},\hbox{\vrule height=0.0pt,depth=15.0pt,width=0.0pt}\\\
\frac{1}{2}b_{j}x^{2}+h_{1}x+h_{0},\quad&\displaystyle\frac{\pi}{q}j\leq
x\leq\frac{\pi}{q}(j+1),\quad j=1,\ldots,2q-1,\end{cases}$ (2.13)

where

$h_{1}=\frac{\pi}{q}\left(\sum_{m=0}^{j-1}b_{m}-mb_{m}\right)\quad\mathrm{and}\quad
h_{0}=\frac{\pi^{2}}{q^{2}}\left[\>\sum_{m=1}^{j-1}\left(\sum\limits_{i=0}^{m-1}b_{i}+\frac{b_{m}}{2}\right)+\frac{j^{2}}{2}b_{j}-j\sum_{m=0}^{j-1}b_{m}+\frac{b_{0}}{2}\>\right].$

This, when combined with (2.10) and (LABEL:eq-s2-l1), allows us to arrive at
the exact result for the solution (2.1) at the rational times, which is
summarized in the following theorem.

######  Theorem 2.6.

At a rational time $t^{\ast}=\pi p/q$, the solution to the periodic initial-
boundary value problem (2.1) for the linear beam equation with the step
function initial datum (2.2) takes the form

$u(t^{\ast},x)=\sum_{j=0}^{2q-1}a_{j}\left(\frac{p}{q}\right)\sigma^{j,q}(x)-H(x)+C(t^{\ast})x,$
(2.14)

where

$C(t^{\ast})=-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\sin((2n+1)^{2}t^{\ast})}{(2n+1)^{2}},$
(2.15)

$\sigma^{j,q}(x)$ is the box function defined in (2.4), $H(x)$ is the
piecewise quadratic function defined in (2.13), and $a_{j}$,
$j=0,\ldots,2q-1$, are certain constants determined by equation (2.5).

With the explicit expression (2.3) of the solution in hand, we now analyze its
qualitative behaviour. First of all, as a direct corollary of the estimate on
the solution of the linearized dispersion equation given by Oskolkov [31] and
Rodnianski [32] — see also [15] — one has, for almost all irrational
$t/(2\pi)$, the first summation
$\mathrm{I}\in\bigcap_{\epsilon>0}C^{\frac{1}{2}-\epsilon}$, which in turn
indicates that the second one
$\mathrm{II}\in\bigcap_{\epsilon>0}C^{\frac{5}{2}-\epsilon}$. We thus conclude
that, at the irrational times, the profile of the solution (2.1) is a
continuous fractal, with fractal dimension $D=3/2$. When it comes to the
rational times, note that in the expression of the solution (2.14), on the one
hand, $H(x)\in C^{1}$, on the other hand, $\mathrm{I}(t^{\ast},x)$ is a
piecewise constant with $\leq 2q$ discontinuities, apart from the specific
times $t^{0}_{k}=\pi(2k-1)/2,\,k\in\mathbb{Z^{+}}$, which will result in
$\mathrm{I}(t^{0}_{k},x)\equiv 0$. Therefore, we need to distinguish two
cases:

Case 1. $q=2$. At times $t^{0}_{k}=\pi(2k-1)/2,\,k\in\mathbb{Z^{+}}$,

$u(t^{0}_{k},x)=-H(x)+(-1)^{k}\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{1}{(2n+1)^{2}}x\in
C^{1}.$

Referring back to the relation (2.6) corresponding to $q=2,p=2k-1$, and $N=2$,
and solving for $b_{j}$ by making use of the inverse discrete Fourier
transform (IDFT) gives rise to

$b_{0}=b_{1}=(-1)^{k},\quad b_{2}=b_{3}=(-1)^{k-1}.$ (2.16)

Then, (2.13) readily leads to

$\displaystyle\begin{aligned}
H_{k}(x)&=(-1)^{k-1}\begin{cases}-\frac{1}{2}x^{2},\quad\quad&0\leq
x\leq\pi,\\\ \frac{1}{2}x^{2}-2\pi x+\pi^{2},\quad&\pi\leq x\leq
2\pi,\end{cases}\end{aligned}$ (2.17)

which, together with the fact that the solution (2.3) is $2\pi$-periodic,
i.e., $u(t^{0}_{k},0)=u(t^{0}_{k},2\pi)$, yields

$\sum_{n=0}^{+\infty}\frac{1}{(2n+1)^{2}}=\frac{\pi^{2}}{8}.$ (2.18)

It follows that, evaluated at each
$t^{0}_{k}=(2k-1)\pi/2,\,k\in\mathbb{Z^{+}}$, the solution can be written as

$\displaystyle\begin{aligned}
u(t^{0}_{k},x)=(-1)^{k-1}\begin{cases}\frac{1}{2}(x^{2}-\pi
x),\quad\quad&0\leq x\leq\pi,\\\ -\frac{1}{2}(x^{2}-3\pi
x+2\pi^{2}),\quad&\pi\leq x\leq 2\pi.\end{cases}\end{aligned}$ (2.19)

More interestingly, we find that the conclusion (2.18) agrees with the
classical result

$\zeta(2)=\sum_{n=0}^{+\infty}\frac{1}{n^{2}}=\frac{\pi^{2}}{6},$ (2.20)

where

$\zeta(s)=\sum_{n=0}^{+\infty}\frac{1}{n^{s}},\quad s>1$ (2.21)

is the Riemann zeta function. The above procedure provides an alternative
mechanism for establishing such classical identities, and the behavior of
these solutions at rational times has intriguing connections with such number-
theoretic exponential sums.

Case 2. $q\neq 2$. According to (2.14), at each rational time $t^{\ast}=\pi
p/q,\,q\neq 2$, the solution consists of a piecewise constant function, which
is constant on the intervals $\pi j/q<x<\pi(j+1)/q$ for $j=0,\ldots,2q-1$,
combined with a continuously differentiable function $-H(x)+C(t^{\ast})x$,
being composed of different parabolas defined on the intervals $\pi j/q\leq
x<\pi(j+1)/q,\ j=0,\ldots,2q-1$. Therefore, in the present case, the solution
profile is a discontinuous, piecewise parabolic curve.

For instance, let us take $t^{\ast}=\pi/3$ as an example. On the one hand,

$\mathrm{I}\left(\frac{\pi}{3},x\right)=-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\cos((2n+1)^{2}\frac{\pi}{3})}{2n+1}\sin((2n+1)x)=\sum_{j=0}^{5}a_{j}\left(\frac{1}{3}\right)\sigma^{j,q}(x),$

where, a direct computation through IDFT shows that

$a_{0}=a_{2}=a_{3}=a_{5}=0,\quad a_{1}=-1,\quad a_{4}=1.$

On the other hand, $b_{j}\,(0\leq j\leq 5)$ in $H(x)$ are obtained from the
relation

$-\frac{4}{\pi}\sum_{n=0}^{+\infty}\frac{\sin((2n+1)^{2}\frac{\pi}{3})}{2n+1}\sin((2n+1)x)=\sum_{j=0}^{5}b_{j}\left(\frac{1}{3}\right)\sigma^{j,q}(x).$

We have

$b_{0}=b_{2}=-b_{3}=-b_{5}=-\frac{\sqrt{3}}{3},\quad
b_{1}=-b_{4}=-\frac{2\sqrt{3}}{3}.$

Again, owing to the periodicity of the solution, one has

$\sum_{n=0}^{+\infty}\frac{\sin((2n+1)^{2}\frac{\pi}{3})}{(2n+1)^{2}}=-\frac{\pi}{4}C\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}\,\pi^{2}}{18}.$
(2.22)

Finally, inserting them into (2.13) and then (2.14) gives rise to the explicit
solution at time $t^{\ast}=\pi/3$, namely

$\displaystyle\begin{aligned}
u\left(\frac{\pi}{3},x\right)=\begin{cases}\frac{\sqrt{3}}{6}\left(x^{2}-\frac{4\pi}{3}x\right),\quad\quad&0\leq
x\leq\frac{\pi}{3},\\\ \frac{\sqrt{3}}{3}\left(x^{2}-\pi
x+\frac{\pi^{2}}{18}\right)-1,\quad\quad&\frac{\pi}{3}\leq
x\leq\frac{2\pi}{3},\\\
\frac{\sqrt{3}}{6}\left(x^{2}-\frac{2\pi}{3}x-\frac{\pi^{2}}{3}\right),\quad\quad&\frac{2\pi}{3}\leq
x\leq\pi,\\\
-\frac{\sqrt{3}}{6}\left(x^{2}-\frac{10\pi}{3}x+\frac{7\pi^{2}}{3}\right),\quad\quad&\pi\leq
x\leq\frac{4\pi}{3},\\\ -\frac{\sqrt{3}}{3}\left(x^{2}-3\pi
x+\frac{37\pi^{2}}{18}\right)+1,\quad\quad&\frac{4\pi}{3}\leq
x\leq\frac{5\pi}{3},\\\
-\frac{\sqrt{3}}{6}\left(x^{2}-\frac{8\pi}{3}x+\frac{4\pi^{2}}{3}\right),\quad\quad&\frac{5\pi}{3}\leq
x\leq 2\pi.\end{cases}\end{aligned}$ (2.23)

######  Remark 2.7.

Note that, at the rational points, the series (2.22) contains the odd terms of
Riemann’s non-differentiable function, which was introduced by Riemann in
1872; see [11] for details. The connection between the Riemann’s non-
differentiable function and solutions of the vortex filament equation with
polygonal initial data was recently established in [22, 23].

All in all, in the context of the linear beam equation, the evolution of the
periodic step function initial datum will take on three different qualitative
behaviors. At irrational times, it evolves into continuous but non-
differentiable fractal-like profile. At rational times $t^{\ast}=\pi
p/q\,(q\neq 2)$, the solution takes on a discontinuous, piecewise parabola
behavior. On the other hand, at each specific rational time
$t^{0}_{k}=\pi(2k-1)/2,\,k\in\mathbb{Z^{+}}$, the quantization effect
disappears entirely, and the solution instantly becomes a continuously
differentiable function, emerging at regular $\pi$-periodic intervals. We
conclude that the revival phenomena exhibited by the periodic evolution of the
linear beam equation differs from that arising in the linear KdV, the linear
Schrödinger, and other unidirectional linear dispersive evolution equations
studied previously.

(a) $t=\pi/2$

(b) $t=\pi/3$

(c) $t=\pi/5$

Figure 1. The solutions to the periodic initial-boundary value problem for the
linear beam equation at rational times.

(a) $t=0.1$

(b) $t=0.3$

(c) $t=0.5$

Figure 2. The solutions to the periodic initial-boundary value problem for the
linear beam equation at irrational times.

In Figures 1 and 2, we display the graphs of the solution at some
representative rational and irrational times, respectively. These figures are
plotted by straightforwardly applying the Fourier series representation of
$u(t,x)$ given by (2.3). We sum over 1001 terms111Summing over a larger number
of terms produces no appreciable difference in the solution profiles. to
obtain the numerical approximation of the solution. As illustrated in Figure
1(a), the solution is continuous at $\pi/2$. While, referring to Figure 1(b)
and Figure 1(c), it appears that, at $\pi/3$ and $\pi/5$, there exist a finite
number of jump discontinuities, and between which parabolic curves of
different form arise. Obviously, the plots in Figure 1, which obtained by
simply truncating the Fourier series (2.3), are entirely consistent with the
explicit expressions given by (2.19) and (2.23). On the other hand, Figure 2
shows that, at irrational times, the solution displays continuous, but nowhere
differentiable fractal-like profiles, as claimed above.

######  Remark 2.8.

It is worth mentioning that, in view of the series (2.10) and that in
$\mathrm{II}^{(1)}(t^{\ast},x)$, the distribution of the discontinuity points
in $H(x)$ depends on the value of $q$, especially on its parity. As studied in
[29], general speaking, the piecewise subintervals for these series are $\pi
j/q\leq x<\pi(j+1)/q,j=0,\ldots,2q-1$. However, if $q$ is even (for instance
$q=2$, whose corresponding solution is a representative example which can
manifest such characteristic), the solutions sometimes assume identical values
on adjacent subintervals, and so exhibits larger regions of constancy. See
also [29] for a number-theoretic characterization of these occurrences.

(a) $t=\pi/2$

(b) $t=\pi/3$

(c) $t=\pi/5$

(d) $t=0.1$

(e) $t=0.3$

(f) $t=0.5$

Figure 3. The solutions to the periodic initial-boundary value problem (2.1)
with initial data $f(x)=\tilde{\sigma}(x),\,g(x)=\sigma(x)$.

(a) $t=\pi/2$

(b) $t=\pi/3$

(c) $t=\pi/5$

(d) $t=0.1$

(e) $t=0.3$

(f) $t=0.5$

Figure 4. The solutions to the periodic initial-boundary value problem (2.1)
with initial data $f(x)=0,\,g(x)=\sigma(x)$.

(a) $t=\pi/2$

(b) $t=\pi/3$

(c) $t=\pi/5$

(d) $t=0.1$

(e) $t=0.3$

(f) $t=0.5$

Figure 5. The solutions to the periodic initial-boundary value problem (2.1)
with initial data $f(x)=\sigma(x),\,g(x)=0$.

Formula (2.3) provides the solution of (2.1) with the same step function
$f=g=\sigma(x)$ (2.2) as the initial data. Indeed, in view of Theorem 3.1
below, we find that the first and second terms in solution (2.3) are induced
by the initial data $u|_{t=0}=f(x)$ and $u_{t}|_{t=0}=g(x)$, respectively. It
is noticed that the emergence of such a dichotomy phenomenon is not only for
the case $f=g$. For instance, if we take the initial $g(x)=\sigma(x)$, while

$f(x)=\tilde{\sigma}(x)=\begin{cases}0,\qquad&0\leq x<\pi,\\\ 1,\qquad&\pi\leq
x<2\pi.\end{cases}$

Figure 3 suggests that the different steps functions will also evolve into
three different qualitative behaviors. Furthermore, in Figure 4 and Figure 5,
we display the graphs of solutions corresponding to $f(x)=0,\,g(x)=\sigma(x)$,
and $f(x)=\sigma(x),\,g(x)=0$, respectively. As demonstrated in Figure 4, if
$f(x)=0$, the dispersive quantization induced by $f(x)$ dispears entirely, and
then the solution will retain a $C^{1}$ profile all the time. On the other
hand, if $f(x)=\sigma(x),\,g(x)=0$, referring to Figure 5(b) and Figure 5(c),
the solutions take on dispersive quantization at the rational times. While, as
shown in Figure 5(a), the solution will vanish at $\pi/2$, since
$\mathrm{I}(\pi/2,x)\equiv 0$ as claimed above.

##  3\. Revival for bidirectional dispersive equations

In the preceding section, we concentrated on the periodic initial-boundary
problem for the linear beam equation. In this section, we will seek to extend
our analysis to bidirectional dispersive equations possessing more general
dispersion relations, as before, subject to general initial conditions and
periodic boundary conditions.

Let $L$ be a scalar, constant coefficient integro-differential operator with
real-valued Fourier transform $\mathaccent 866{L}(k)=-\varphi(k)$, where
$\varphi(k)>0$ is a real-valued function of $k$. The associated bidirectional
scalar equation

$u_{tt}=L[u]$ (3.1)

has real dispersion relation $\omega(k)=\pm\sqrt{\varphi(k)}$. We subject
(3.1) to periodic boundary conditions posed on the interval $0\leq x\leq
2\pi$, and initial conditions

$u(0,x)=f(x),\quad u_{t}(0,x)=g(x),$ (3.2)

where $f(x)$ and $g(x)$ are of bounded variation, and $g(x)$ is required to
satisfy $\int_{0}^{2\pi}g(x)\;\mathrm{d}x=0$.

As usual, the first step is to construct the (formal) solution as a Fourier
series

$u(t,x)\sim\sum_{k=-\infty}^{+\infty}a_{k}(t)e^{\,{\rm i}\,kx}.$

To this end, we first expand the initial data $f(x)$ and $g(x)$ in Fourier
series

$f(x)\sim\sum_{k=-\infty}^{+\infty}c_{k}e^{\,{\rm
i}\,kx},\quad\mathrm{where}\quad c_{k}=\mathaccent
866{f}(k)=\frac{1}{2\pi}\int_{0}^{2\pi}\,f(x)e^{-\,{\rm i}\,kx}\,\mathrm{d}x$

and

$g(x)\sim\sum_{k=-\infty}^{+\infty}d_{k}e^{\,{\rm
i}\,kx},\quad\mathrm{where}\quad d_{k}=\mathaccent
866{g}(k)=\frac{1}{2\pi}\int_{0}^{2\pi}\,g(x)e^{-\,{\rm i}\,kx}\,\mathrm{d}x.$

Next, an analogous analysis as used for the linear beam equation with step
function initial conditions, implies that the corresponding coefficients
$a_{k}(t)$ satisfy the following linear ODE

$a_{k}^{\prime\prime}(t)+\varphi(k)a_{k}(t)=0.$ (3.3)

Solving it yields

$a_{k}(t)=A_{k}e^{\,{\rm i}\,\sqrt{\varphi(k)}t}+B_{k}e^{-\,{\rm
i}\,\sqrt{\varphi(k)}t}.$

Finally, using the initial data again, we find that the solution to the
periodic initial-boundary value problem (3.1)-(3.2) is given by

$\displaystyle\begin{aligned} u(t,x)=\sum_{k}\mathaccent
866{f}(k)\cos\left(\sqrt{\varphi(k)}\>t\right)e^{\,{\rm i}\,kx}+\sum_{k\neq
0}\frac{\mathaccent
866{g}(k)}{\sqrt{\varphi(k)}}\sin\left(\sqrt{\varphi(k)}\>t\right)e^{\,{\rm
i}\,kx}.\end{aligned}$ (3.4)

With the Fourier series representation (3.4) in hand, we are now able to
analyze the qualitative behavior of the solution at rational times. We will
show that the dynamical evolution of equation (3.1) on periodic domains with
initial profiles (3.2) depends dramatically upon the asymptotics of the
dispersion relation at large wave number. In all cases considered here, the
large wave number asymptotics of the dispersion relation is given by a
positive power of the wave number:

$\sqrt{\varphi(k)}\sim|k|^{\alpha},\quad
2\leq\alpha\in\mathbb{R},\quad\mathrm{as}\quad|k|\rightarrow\infty.$ (3.5)

###  3.1. Monomial dispersion relation:

As the first step, we will study the special case of monomial dispersion
relation given by

$\omega(k)=\pm k^{N},\quad 2\leq N\in\mathbb{Z}^{+}.$ (3.6)

The main results for the corresponding solutions are summarized in Theorem 3.1
below. Hereafter, we define the operator $\partial_{x}^{-1}$ by the formula

$\partial_{x}^{-1}\,P(x)=\int_{2(k-1)\pi}^{x}\,P(y)\,\mathrm{d}y,\qquad
x\in[\,2(k-1)\pi,\,2k\pi\,].$ (3.7)

We further define its $M$-th order power $\partial_{x}^{-M}$ via the recursive
relation $\partial_{x}^{-M}=\partial_{x}^{-1}\partial_{x}^{-(M-1)}$, for
$M\geq 1$.

######  Theorem 3.1.

Suppose that equation (3.1) has the monomial dispersion relation (3.6), the
initial data $f(x)$ and $g(x)$ in (3.2) are of bounded variation, and $g(x)$
satisfies $\int_{0}^{2\pi}g(x)\;\mathrm{d}x=0$. Let
$G(x)=\partial_{x}^{-N}g(x)$. Then at each rational time $t^{\ast}=\pi p/q$,
the solution to the periodic initial-boundary value problem (3.1)-(3.2) takes
the form

$u(t^{\ast},x)=\sum_{j=0}^{2q-1}a_{j}\left(\frac{p}{q}\right)\,f\left(x-\frac{\pi
j}{q}\right)+\,{\rm
i}\,^{N}\sum_{j=0}^{2q-1}b_{j}\left(\frac{p}{q}\right)\,G\left(x-\frac{\pi
j}{q}\right)+\sum_{j=0}^{N-1}C_{j}x^{j},$ (3.8) $C_{j}=\frac{\,{\rm
i}\,^{j}}{j\,!}\sum_{k\neq 0}\frac{\mathaccent
866{g}(k)\sin(k^{N}t^{\ast})}{k^{N-j}},$ (3.9)

where the coefficients $a_{j},\,b_{j}\in\mathbb{C}$, $j=0,\ldots,2q-1$, are
constants depending on $p$ and $q$.

The proof of the theorem relies on the following lemma, which is a direct
corollary of Theorem 3.2 established in [7] and is a special case of Lemmas
7.5 and 7.6 in [16]. Thus, we omit the proof. Moreover, we remark that the
expression (3.8) for the exact solution is equivalent to (3.25) in the next
subsection, although not exactly the same in form.

######  Lemma 3.2.

Let $P(k)$ be an integral polynomial. Assume that $f(x)$ is of bounded
variation, and let $\mathaccent 866{f}(k)$ be the Fourier coefficient of
$f(x)$, i.e.,

$\mathaccent 866{f}(k)=\frac{1}{2\pi}\int_{0}^{2\pi}\,f(x)e^{-\,{\rm
i}\,kx}\,\mathrm{d}x.$

Given $t^{\ast}=\pi p/q$, with $p$ and $0\neq q\in\mathbb{Z}^{+}$, there exist
constants $a_{j}^{1},a_{j}^{2}\in\mathbb{C}$, $j=0,\ldots,2q-1$, depending on
$p$ and $q$, such the following two formulae hold:

$\sum_{k=-\infty}^{\infty}\mathaccent
866{f}(k)\cos\left(P(k)t^{\ast}\right)e^{\,{\rm
i}\,kx}=\sum_{j=0}^{2q-1}a_{j}^{1}\left(\frac{p}{q}\right)f\left(x-\frac{\pi
j}{q}\right),$ (3.10) $\sum_{k=-\infty}^{\infty}\mathaccent
866{f}(k)\sin\left(P(k)t^{\ast}\right)e^{\,{\rm
i}\,kx}=\sum_{j=0}^{2q-1}a_{j}^{2}\left(\frac{p}{q}\right)f\left(x-\frac{\pi
j}{q}\right).$ (3.11)

###### Proof of Theorem 3.1.

First of all, according to (3.4), under the assumption of the theorem, the
solution to the corresponding periodic initial-boundary problem has the form

$\displaystyle\begin{aligned} u(t,x)&=\sum_{k}\mathaccent
866{f}(k)\cos(k^{N}t)e^{\,{\rm i}\,kx}+\sum_{k\neq 0}\frac{\mathaccent
866{g}(k)\sin(k^{N}t)}{k^{N}}e^{\,{\rm
i}\,kx}:=\mathrm{I}(t,x)+\mathrm{II}(t,x).\end{aligned}$ (3.12)

Furthermore, since $f(x)$ and $g(x)$ are of bounded variation and $N\geq 2$,
the first summation in expression (3.12) is conditionally convergent, and the
second one is absolutely convergent.

At the rational times $t^{\ast}=\pi p/q$, by Lemma 3.2, the first summation is
a linear combination of translates of $f(x)$, i.e.,

$\mathrm{I}(t^{\ast},x)=\sum_{j=0}^{2q-1}a^{1}_{j}\left(\frac{p}{q}\right)\,f\left(x-\frac{\pi
j}{q}\right),$

for certain $a_{0}^{1},\ldots,a_{2q-1}^{1}\in\mathbb{C}$ determined by (3.10)
with $P(k)=k^{N}$.

Note that

$\partial_{x}^{-M}e^{\,{\rm i}\,kx}=\frac{1}{(\,{\rm i}\,k)^{M}}e^{\,{\rm
i}\,kx}-\sum_{j=0}^{M-1}\frac{1}{j!\>(\,{\rm i}\,k)^{M-j}}x^{j},\quad{\rm
for}\quad 0\leq x\leq 2\pi.$ (3.13)

It follows that, at the rational times $t^{\ast}=\pi p/q$, the second
summation satisfies

$\displaystyle\begin{aligned} \mathrm{II}(t^{\ast},x)&=\sum_{k}\,{\rm
i}\,^{N}\,\mathaccent 866{g}(k)\sin(k^{N}t^{\ast})\partial_{x}^{-N}e^{\,{\rm
i}\,kx}+\sum_{k\neq 0}\mathaccent
866{g}(k)\sin(k^{N}t^{\ast})\sum_{j=0}^{N-1}\frac{\,{\rm
i}\,^{j}}{j!\>k^{N-j}}x^{j}\\\
&:=\mathrm{II}^{(1)}(x)+\mathrm{II}^{(2)}(x).\end{aligned}$

Since $g(x)$ is of bounded variation, the series $C_{j}$ given in (3.9) is
convergent for each $j=0,\ldots,N-1$, then the second component
$\mathrm{II}^{(2)}(x)$ readily leads to the last term in (3.8). On the other
hand, in the case of $P(k)=k^{N}$, applying equation (3.11) to the delta
function $\delta(x)$ yields

$\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}\sin\left(k^{N}t^{\ast}\right)e^{\,{\rm
i}\,kx}=\sum_{j=0}^{2q-1}b_{j}\left(\frac{p}{q}\right)\delta\left(x-\frac{\pi
j}{q}\right),$

for some constants $b_{j}\in\mathbb{C},\;j=0,\ldots,2q-1.$ We thus deduce
$\mathrm{II}^{(1)}(x)$ as follows:

$\displaystyle\begin{aligned} \mathrm{II}^{(1)}(x)&=\frac{\,{\rm
i}\,^{N}}{2\pi}\sum_{k}\sin(k^{N}t^{\ast})\int_{0}^{2\pi}g(y)e^{-\,{\rm
i}\,ky}\partial_{x}^{-N}e^{\,{\rm i}\,kx}\,\mathrm{d}y\\\ &=\frac{\,{\rm
i}\,^{N}}{2\pi}\sum_{k}\sin(k^{N}t^{\ast})\int_{0}^{2\pi}e^{\,{\rm
i}\,ky}\partial_{x}^{-N}g(x-y)\,\mathrm{d}y\\\ &=\,{\rm
i}\,^{N}\int_{0}^{2\pi}G(x-y)\sum_{j=0}^{2q-1}b_{j}\delta\left(y-\frac{\pi
j}{q}\right)\,\mathrm{d}y=\,{\rm
i}\,^{N}\sum_{j=0}^{2q-1}b_{j}G\left(x-\frac{\pi j}{q}\right).\end{aligned}$

Summing $\mathrm{II}^{(1)}(x)$, $\mathrm{II}^{(2)}(x)$ and
$\mathrm{I}(t^{\ast},x)$ gives (3.8), which justifies the statement of the
theorem. ∎

In particular, if the initial data $f(x)$ and $g(x)$ are the step function
$\sigma(x)$ given in (2.2), the following corollary holds.

######  Corollary 3.3.

Let $\sigma^{j,q}(x)$ be the box function defined in (2.4). At a rational time
$t^{\ast}=\pi p/q$, the solution to the periodic initial-boundary value
problem (3.1)-(3.2) on the interval $0\leq x\leq 2\pi$, with initial data
$f(x)=g(x)=\sigma(x)$ given in (2.2) takes the form

$u(t^{\ast},x)=\sum_{j=0}^{2q-1}a_{j}\left(\frac{p}{q}\right)\sigma^{j,q}(x)+(-1)^{\left[\frac{N}{2}\right]}\partial_{x}^{-N}\sum_{j=0}^{2q-1}b_{j}\left(\frac{p}{q}\right)\sigma^{j,q}(x)+\sum_{j=0}^{\left[\frac{N}{2}\right]-1}D_{j}x^{2j+1},$
(3.14)

where

$D_{j}=\frac{(-1)^{j+1}4}{\pi(2j+1)!}\sum_{n=0}^{+\infty}\frac{\sin((2n+1)^{N}t^{\ast})}{(2n+1)^{N-2j}},\qquad
j=0,\ldots,\left[\frac{N}{2}\right]-1,$ (3.15)

and the coefficients $a_{j},j=0,\ldots,2q-1$ are determined by formula (2.5)
in Lemma 2.1, $b_{j},j=0,\ldots,2q-1$ satisfy (2.6) for even $N$, and (2.7)
for odd $N$, respectively.

###### Proof.

If $f(x)=g(x)=\sigma(x)$, the corresponding solution (3.4) reduces to

$\displaystyle\begin{aligned}
u(t^{\ast},x)&=-\frac{4}{\pi}\left[\>\sum_{n=0}^{+\infty}\frac{\cos((2n+1)^{N}t^{\ast})\sin((2n+1)x)}{2n+1}+\sum_{n=0}^{+\infty}\frac{\sin((2n+1)^{N}t^{\ast})\sin((2n+1)x)}{(2n+1)^{N+1}}\>\right].\end{aligned}$

Obviously, the first summation is exactly the first term in (3.14). As for the
second summation, a direct induction procedure shows that, if $N$ is even,

$\frac{\sin((2n+1)x)}{(2n+1)^{N+1}}=(-1)^{\frac{N}{2}}\partial_{x}^{-N}\frac{\sin((2n+1)x)}{2n+1}+\sum_{j=0}^{\frac{N}{2}-1}\frac{(-1)^{j}}{(2j+1)!(2n+1)^{N-2j}}x^{2j+1},$

whereas, if $N$ is odd,

$\frac{\sin((2n+1)x)}{(2n+1)^{N+1}}=(-1)^{\frac{N-1}{2}}\partial_{x}^{-N}\frac{\cos((2n+1)x)}{2n+1}+\sum_{j=0}^{\frac{N-1}{2}-1}\frac{(-1)^{j}}{(2j+1)!(2n+1)^{N-2j}}x^{2j+1}.$

Substituting into the second summation, and making use of formulae (2.6) and
(2.7) for even and odd $N$, respectively, verifies (3.14), proving the
corollary. ∎

More specifically, if the underlying equation is exactly the linear beam
equation in (2.1) with dispersion relation $\omega(k)=\pm k^{2}$. It follows
that the second term in (3.14) reduces to (2.12), which is nothing but $-H(x)$
in (2.14). Meanwhile, the third term is identical to $C(t^{\ast})x$ in (2.14).
This indicates that in this particular case, Corollary 3.3 is in accordance
with Theorem 2.6.

As in Section 2, let us now illustrate how, by Corollary 3.3, we can calculate
the value of the Riemann zeta function at $s=4$. We define

$H_{p,q}^{N}(x)=\partial_{x}^{-N}\sum_{j=0}^{2q-1}b_{j}\left(\frac{p}{q}\right)\sigma^{j,q}(x),$

where $b_{j}$ are determined by the formulae (2.6) for even $N$, or (2.7) for
odd $N$, respectively. Denote

$S_{l}^{N}(t)=\sum_{n=0}^{+\infty}\frac{\sin((2n+1)^{N}t)}{(2n+1)^{l}},\quad\quad\mathrm{for}\quad
l\in\mathbb{Z^{+}},\quad\mathrm{with}\quad l\geq 2,$ (3.16)

and let

$\Gamma_{N}=\begin{cases}\displaystyle\sum\limits_{k=1}^{\frac{N}{2}}\frac{(-1)^{k}(2\pi)^{N-2k+1}}{(N-2k+1)!}S_{2k}^{N}(t^{\ast}),\quad\quad&\mathrm{if}\;N\;\mathrm{even},\\\
\displaystyle\sum\limits_{k=1}^{\frac{N-1}{2}}\frac{(-1)^{k}(2\pi)^{N-2k}}{(N-2k)!}S_{2k+1}^{N}(t^{\ast}),\quad\quad&\mathrm{if}\;N\geq
3\;\mathrm{odd}.\end{cases}$ (3.17)

According to (3.14), we find a formula involving the sum $\Gamma_{N}$, which
along with the periodicity produces

$\Gamma_{N}=\frac{\pi}{4}H_{p,q}^{N}(2\pi).$ (3.18)

Note that, if $N$ is even, at the special rational times
$t^{\ast}_{l}=(2l-1)\pi/2,\,l\in\mathbb{Z^{+}}$,

$S_{N}^{N}\left(t^{\ast}_{l}\right)=(-1)^{l-1}\sum_{n=0}^{+\infty}\frac{1}{(2n+1)^{N}},$

while, if $N$ is odd, $S_{N}^{N}\left(t^{\ast}_{l}\right)$ is a alternating
series, namely,

$S_{N}^{N}\left(t^{\ast}_{l}\right)=(-1)^{l-1}\sum_{n=0}^{+\infty}\frac{(-1)^{n}}{(2n+1)^{N}}.$

Hereafter, we denote

$\sigma(N)=\sum_{n=0}^{+\infty}\frac{1}{(2n+1)^{N}},\quad\mathrm{for}\;\mathrm{even}\;N,\qquad\tau(N)=\sum_{n=0}^{+\infty}\frac{(-1)^{n}}{(2n+1)^{N}},\quad\mathrm{for}\;\mathrm{odd}\;N,$

respectively. Therefore, in the special rational times $t^{\ast}_{l}$ setting,
(3.18) establishes the recursion formulae for $\sigma(2k)$ and $\tau(2k+1)$
for each $k\in\mathbb{Z^{+}}$. More precisely, for even $N$

$\sigma(N)=\frac{(-1)^{\frac{N}{2}}}{8}\left(H_{1,2}^{N}(2\pi)-\frac{4}{\pi}\sum_{k=0}^{\frac{N}{2}-1}\frac{(-1)^{k}(2\pi)^{N-2k+1}}{(N-2k+1)!}\sigma(2k)\right),$

or, for odd $N$,

$\tau(N)=\frac{(-1)^{\frac{N-1}{2}}}{8}\left(H_{1,2}^{N}(2\pi)-\frac{4}{\pi}\sum_{k=0}^{\frac{N-1}{2}-1}\frac{(-1)^{k}(2\pi)^{N-2k}}{(N-2k)!}\tau(2k+1)\right),$

which are initiated by the series $\sigma(2)$ (2.18) for even $N$, or
$\tau(3)$ for odd $N$, respectively. As far as $\tau(3)$ is concerned, one can
verify from (2.7) for $N=3$ that

$\tilde{b}_{0}=-1,\quad\tilde{b}_{1}=\tilde{b}_{2}=1,\quad\tilde{b}_{3}=-1,$

which immediately yields

$\displaystyle\begin{aligned}
H_{1,2}^{3}(x)=\begin{cases}-\frac{1}{6}x^{3},\quad\quad&0\leq
x\leq\frac{\pi}{2},\\\ \frac{1}{6}\left(x^{3}-3\pi
x^{2}+\frac{3\pi^{2}}{2}x-\frac{\pi^{3}}{4}\right),\quad\quad&\frac{\pi}{2}\leq
x\leq\frac{3\pi}{2},\\\ -\frac{1}{6}\left(x^{3}-6\pi
x^{2}+12\pi^{2}x-\frac{13\pi^{3}}{2}\right),\quad\quad&\frac{3\pi}{2}\leq
x\leq 2\pi.\end{cases}\end{aligned}$

We thus arrive at

$\tau(3)=\sum_{n=0}^{+\infty}\frac{(-1)^{n}}{(2n+1)^{3}}=\frac{\pi^{3}}{32}.$

When it comes to $\zeta(4)$, we calculate from (2.16) that

$\displaystyle\begin{aligned}
H_{1,2}^{4}(x)=\begin{cases}-\frac{1}{24}x^{4},\quad\quad&0\leq x\leq\pi,\\\
\frac{1}{24}\left(x^{4}-8\pi
x^{3}+12\pi^{2}x^{2}-8\pi^{3}x+2\pi^{4}\right),\quad\quad&\pi\leq x\leq
2\pi.\end{cases}\end{aligned}$

Consequently,

$\sigma(4)=\sum_{n=0}^{+\infty}\frac{1}{(2n+1)^{4}}=\frac{1}{8}\left(H_{1,2}^{4}(2\pi)+\frac{4\pi^{2}}{3!}\bar{\zeta}(2)\right)=\frac{\pi^{4}}{96},$

which further yields the following classical result for the Riemann zeta
function at $s=4$:

$\zeta(4)=\sum_{n=0}^{+\infty}\frac{1}{n^{4}}=\frac{\pi^{4}}{90}.$

###  3.2. Monomial dispersion relation — second approach

We now briefly consider a different approach in the monomial case, which is
based on [16, Chapter 7] and derive an alternative representation of the
solution at rational times. Hence, the dispersion relation assumes the form
(3.6). Moreover, only for this subsection, we relax the condition on $g$ and
allow it to have non-zero mean over $[0,2\pi]$.

The solution to the periodic initial-boundary value problem (3.1)-(3.2) is
given by

$\displaystyle\begin{aligned} u(t,x)&=\sum_{k}\mathaccent
866{f}(k)\cos(k^{N}t)e^{\,{\rm
i}\,kx}+\frac{1}{2\pi}\int_{0}^{2\pi}g(y)\mathrm{d}y\ t+\sum_{k\neq
0}\frac{\mathaccent 866{g}(k)\sin(k^{N}t)}{k^{N}}e^{\,{\rm i}\,kx}\\\
&:=\mathrm{I}(t,x)+\langle g\rangle\ t+\mathrm{II}(t,x),\end{aligned}$ (3.19)

where $\langle g\rangle$ is the mean of $g$ and $\mathrm{I}(t,x)$,
$\mathrm{II}(t,x)$ correspond to the two Fourier series representations
respectively. In the following, we derive an alternative representation of the
term $\mathrm{II}(t,x)$. In particular, we will show that $\mathrm{II}(t,x)$
can be expressed as the time-evolution of the periodic convolution of the
function $g-\langle g\rangle$ with a polynomial of degree $N\geq 2$. As it is
known, see for example [24, Proposition 2.76], the convolution gains the
regularity of the most regular function between the two involved.
Consequently, we may deduce that at any time $t>0$, either rational or
irrational, the Fourier series representation of $\mathrm{II}(t,x)$ defines a
$2\pi$-periodic function of class $C^{N}(\mathbb{R})$.

Let us make all the above precise and define first the following family of
polynomials on $[0,2\pi]$.

######  Definition 3.4.

Let $N\geq 1$ be an integer. We denote by
$Q_{N}:[0,2\pi]\rightarrow\mathbb{C}$ the polynomial of degree $N$, defined
inductively by the formula

$Q_{N}(x)=\frac{(-\,{\rm
i}\,)^{N}x^{N}}{(-1)^{N-1}N!}-\sum_{\ell=1}^{N-1}\frac{(-1)^{\ell-N}}{(-\,{\rm
i}\,)^{\ell-N}}\frac{(2\pi)^{N-\ell}}{(N-\ell+1)!}Q_{\ell}(x).$ (3.20)

The crucial feature of the polynomial $Q_{N}$ is the form of its Fourier
coefficients, which are equal to $k^{-N}$. As we shall shortly see, this will
allow us to invoke the operation of the periodic convolution in the
representation of $\mathrm{II}(t,x)$.

######  Lemma 3.5.

Fix an integer $N\geq 1$ and consider the polynomial
$Q_{N}:[0,2\pi]\rightarrow\mathbb{C}$. Then, for $k\not=0$, $\mathaccent
866{Q_{N}}(k)=k^{-N}$.

###### Proof.

The proof follows by induction on $N$. It is easy to show that the statement
holds for $N=1$ and $N=2$. We assume that $\mathaccent
866{Q_{\ell}}(k)=k^{-\ell}$ for $\ell=1,2,\dots,N$, with $N\geq 3$, and
calculate the Fourier coefficients of $Q_{N+1}$.

Let $k\not=0$. Then, we have that

$\displaystyle\mathaccent 1371{Q_{N+1}}(k)$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}Q_{N+1}(y)e^{-\,{\rm i}\,ky}dy$
$\displaystyle=\frac{(-i)^{N+1}}{2\pi(-1)^{N}(N+1)!}\int_{0}^{2\pi}y^{N+1}e^{-\,{\rm
i}\,ky}dy-\sum_{\ell=1}^{N}\frac{(-1)^{\ell-N-1}}{(-\,{\rm
i}\,)^{\ell-N-1}}\frac{(2\pi)^{N+1-\ell}}{(N-\ell)!}\frac{1}{k^{\ell}}.$

However, a direct calculation shows that

$\int_{0}^{2\pi}y^{N+1}e^{-\,{\rm i}\,ky}dy=\frac{2\pi(-1)^{N}(N+1)!}{(-\,{\rm
i}\,)^{N+1}k^{N+1}}+(N+1)!\sum_{\ell=1}^{N}\frac{(-1)^{\ell-1}}{(-\,{\rm
i}\,)^{\ell}}\frac{(2\pi)^{N+2-\ell}}{(N-\ell)!}\frac{1}{k^{\ell}}.$ (3.21)

Substituting back for $\mathaccent 1371{Q_{N+1}}(k)$ we find that

$\displaystyle\mathaccent 1371{Q_{N+1}}(k)=\frac{1}{k^{N+1}},$

which concludes the proof. ∎

We now turn our attention to the second ingredient needed for the alternative
representation of $\mathrm{II}(t,x)$. Thus, we recall the definition of the
periodic convolution, see [36].

######  Definition 3.6.

Let $f$ and $g$ be $2\pi$-periodic on $\mathbb{R}$ and such that $f$, $g$ $\in
L^{1}(0,2\pi)$. Then the _$2\pi$ -periodic convolution_ of $f$ and $g$ is
defined by

$f\ast g(x)=\frac{1}{2\pi}\int_{0}^{2\pi}f(x-y)g(y)dy,\quad x\in[0,2\pi].$
(3.22)

From [36, Proposition 3.1], we know that $f\ast g$ defines a $2\pi$-periodic
continuous function whose Fourier coefficients are given by $\mathaccent
1371{f\ast g}(k)=\mathaccent 866{f}(k)\mathaccent 866{g}(k)$. Summarizing the
above, we arrive at the following lemma which identifies $\mathrm{II}(t,x)$
based on the convolution of $g-\langle g\rangle$ with $Q_{N}$.

######  Lemma 3.7.

Assume that $g$ is of bounded variation over $[0,2\pi]$. Fix integer $N\geq 2$
and consider the function

$v(x)=(g-\langle g\rangle)\ast Q_{N}(x),\quad x\in[0,2\pi].$ (3.23)

Then, at any fixed time $t\geq 0$, we have that

$\mathrm{II}(t,x)=\sum_{k=-\infty}^{\infty}\mathaccent
866{v}(k)\sin(k^{N}t)e^{\,{\rm i}\,kx}.$ (3.24)

###### Proof.

Let $\tilde{g}=g-\langle g\rangle$. Then, the Fourier coefficients of
$\tilde{g}$ are given by

$\mathaccent 866{\tilde{g}}(k)=\mathaccent 866{g}(k),\quad
k\not=0,\quad\mathaccent 866{\tilde{g}}(0)=0.$

Moreover, from Lemma 3.5, we know that for $k\not=0$, $\mathaccent
866{Q_{N}}(k)=k^{-N}$. Hence,

$\mathaccent 866{v}(k)=\mathaccent 866{\tilde{g}}(k)\,\mathaccent
866{Q_{N}}(k)=\begin{cases}0,\quad k=0,\\\ \frac{\mathaccent
866{g}(k)}{k^{N}},\quad k\not=0,\end{cases}$

which implies that for any $t>0$,

$\sum_{k=-\infty}^{\infty}\mathaccent 866{v}(k)\sin(k^{N}t)e^{\,{\rm
i}\,kx}=\sum_{k\not=0}\frac{\mathaccent 866{g}(k)}{k^{N}}\sin(k^{N}t)e^{\,{\rm
i}\,kx}=\mathrm{II}(t,x).$

∎

The validity of the revival effect at rational times $t^{\ast}=\pi p/q$
follows again by Lemma 3.2 applied directly on $\mathrm{I}(t^{\ast},x)$ and
$\mathrm{II}(t^{\ast},x)$ in conjunction with Lemma 3.7. This is the context
of the next theorem.

######  Theorem 3.8.

Suppose that equation (3.1) admits the monomial dispersion relation (3.6), the
initial data $f(x)$ and $g(x)$ in (3.2) are of bounded variation. Then, at
each rational time $t^{\ast}=\pi p/q$, the solutions to the periodic initial-
boundary value problem (3.1)-(3.2) take the form

$u(t^{\ast},x)=\sum_{j=0}^{2q-1}a_{j}\left(\frac{p}{q}\right)\,f\left(x-\frac{\pi
j}{q}\right)+\langle
g\rangle\,t^{\ast}+\sum_{j=0}^{2q-1}d_{j}\left(\frac{p}{q}\right)\,v\left(x-\frac{\pi
j}{q}\right),$ (3.25)

where $v(x)=(g-\langle g\rangle)\ast Q_{N}(x)$ and the coefficients
$a_{j},\,d_{j}\in\mathbb{C}$, $j=0,\ldots,2q-1$, are certain constants
depending on $p,q$.

As a consequence of Theorem 3.8, equivalently of Theorem 3.1, the solution at
rational times is, at least, piecewise continuous, given that $f(x)$ has
finitely many jump discontinuities. More specifically, the first term in
(3.25), $\mathrm{I}(t^{\ast},x)$, corresponds to the revival of the initial
function $f(x)$, whereas the third term, $\mathrm{II}(t^{\ast})$, is the
revival of the $2\pi$-periodic, $C^{N}(\mathbb{R})$ function $v(x)=(g-\langle
g\rangle)\ast Q_{N}(x)$ and thus, together with the constant term, $\langle
g\rangle t^{\ast}$, a $2\pi$-periodic, $C^{N}(\mathbb{R})$ function.
Therefore, the solution is given as the sum of the revival of the initial
condition $f(x)$ and a more regular function, which ensures the revival of the
initial jump discontinuities of $f(x)$.

###  3.3. Integral polynomial dispersion relation

This subsection is concerned with the case that $\sqrt{\phi(k)}$ is an
integral polynomial $P(k)$. The corresponding solution takes the form

$\displaystyle\begin{aligned} u(t,x)&=\sum_{k}\mathaccent
866{f}(k)\cos(P(k)t)e^{\,{\rm i}\,kx}+\sum_{k\neq 0}\frac{\mathaccent
866{g}(k)\sin(P(k)t)}{P(k)}e^{\,{\rm
i}\,kx}:=\mathrm{I}(t,x)+\mathrm{II}(t,x).\end{aligned}$ (3.26)

Firstly, using (3.10) again, we obtain that at each rational time
$t^{\ast}=\pi p/q$, the first term in (3.26) satisfies

$\mathrm{I}(t^{\ast},x)=\sum_{j=0}^{2q-1}a_{j}\left(\frac{p}{q}\right)\,f\left(x-\frac{\pi
j}{q}\right).$

Next, notice that

$\displaystyle\begin{aligned} \left|\>\mathrm{II}(t,x)-\sum_{k\neq
0}\frac{\mathaccent 866{g}(k)\sin(P(k)t)}{c_{N}k^{N}}e^{\,{\rm
i}\,kx}\>\right|&\leq\sum_{k\neq
0}\frac{|c_{N-1}k^{N-1}+\cdots+c_{1}k+c_{0}|}{|c_{N}P(k)k^{N}|}|\mathaccent
866{g}(k)|\\\ &\mathchar 13358\relax\sum_{k\neq
0}\frac{1}{k^{N+2}},\end{aligned}$ (3.27)

where the fact that $g(x)$ is of bounded variation has been used in the last
inequality and $c_{N}$ denotes the coefficient of the highest power of $P(k)$.
Since $N\geq 2$, the final series is absolutely convergent, whose sum is a
constant. Thus, the above estimate implies that the qualitative behavior of
the second term relies crucially on that of the series

$\sum_{k\neq 0}\frac{\mathaccent 866{g}(k)\sin(P(k)t)}{c_{N}k^{N}}e^{\,{\rm
i}\,kx}.$ (3.28)

While, as for (3.28), a direct generalization of the proof of Theorem 3.1
shows that, at each rational time $t^{\ast}=\pi p/q$,

$\displaystyle\begin{aligned} \sum_{k\neq 0}\frac{\mathaccent
866{g}(k)\sin(P(k)t^{\ast})}{c_{N}k^{N}}e^{\,{\rm i}\,kx}=\,{\rm
i}\,^{N}\sum_{j=0}^{2q-1}\bar{b}_{j}\left(\frac{p}{q}\right)\,G\left(x-\frac{\pi
j}{q}\right)+\sum_{j=0}^{N-1}\bar{C}_{j}x^{j},\end{aligned}$

where the coefficients $\bar{b}_{0},\ldots,\bar{b}_{2q-1}$ are determined by

$\sum_{k}\sin\left(P(k)t^{\ast}\right)e^{\,{\rm
i}\,kx}=\sum_{j=0}^{2q-1}\bar{b}_{j}\left(\frac{p}{q}\right)\delta\left(x-\frac{\pi
j}{q}\right),\quad\hbox{and}\quad\bar{C}_{j}=\sum_{k\neq 0}\frac{\mathaccent
866{g}(k)\sin(P(k)t^{\ast})}{c_{N}j!k^{N-j}}.$

We thus conclude that, at each rational time, the series (3.28) admits the
same discontinuities and revival structure as the second summation in solution
(3.12).

All in all, we may safely draw the conclusion that, in the present case, the
discontinuities of the solution will be determined by the initial data. For
instance, if the initial data are the step function $\sigma(x)$, as in (2.2),
by Corollary 3.3, $u(t,x)$ will be a $C^{N-1}$ curve at each
$t_{k}^{0}=(2k-1)\pi/2,\;k\in\mathbb{Z}^{+}$, and exhibit jump discontinuities
and revival profile at other rational times.

###  3.4. Non-polynomial dispersion relation

If $\sqrt{\varphi(k)}$ is not a polynomial, we distinguish two cases. The
first one assumes that, for large wave numbers, the dispersion relation is
asymptotically close to an integral polynomial $P(k)$. Hence, suppose

$\sqrt{\varphi(k)}\sim P(k)+O(k^{-1}),\quad\mathrm{as}\quad|k|\
\rightarrow\infty.$

Firstly, under the assumption that $f(x)$ is of bounded variation, the first
summation in (3.4) satisfies

$\displaystyle\begin{aligned} &\left|\ \sum_{k}\mathaccent
866{f}(k)\cos\left(\sqrt{\varphi(k)}\>t\right)e^{\,{\rm
i}\,kx}-\sum_{k}\mathaccent 866{f}(k)\cos(P(k)t)e^{\,{\rm i}\,kx}\ \right|\\\
&\hskip 108.405pt\leq\sum_{k}|\mathaccent
866{f}(k)|\,|\cos\left(\sqrt{\varphi(k)}\>t\right)-\cos(P(k)t)|\mathchar
13358\relax\sum_{k\neq 0}\frac{1}{k^{2}},\end{aligned}$ (3.29)

where the mean-value theorem has been used in the last inequality. Next, for
the second summation, one has

$\displaystyle\begin{aligned} &\left|\ \sum_{k\neq 0}\frac{\mathaccent
866{g}(k)}{\sqrt{\varphi(k)}}\sin\left(\sqrt{\varphi(k)}\>t\right)e^{\,{\rm
i}\,kx}-\sum_{k\neq 0}\frac{\mathaccent 866{g}(k)}{k^{N}}\sin(P(k)t)e^{\,{\rm
i}\,kx}\ \right|\\\ &\hskip 72.26999pt\leq\left|\ \sum_{k\neq
0}\frac{\mathaccent
866{g}(k)}{\sqrt{\varphi(k)}}\sin\left(\sqrt{\varphi(k)}\>t\right)e^{\,{\rm
i}\,kx}-\sum_{k\neq 0}\frac{\mathaccent
866{g}(k)}{P(k)}\sin\left(\sqrt{\varphi(k)}\>t\right)e^{\,{\rm i}\,kx}\
\right|\\\ &\hskip 108.405pt{}+\left|\ \sum_{k\neq 0}\frac{\mathaccent
866{g}(k)}{P(k)}\sin\left(\sqrt{\varphi(k)}\>t\right)e^{\,{\rm
i}\,kx}-\sum_{k\neq 0}\frac{\mathaccent 866{g}(k)}{P(k)}\sin(P(k)t)e^{\,{\rm
i}\,kx}\ \right|\\\ &\hskip 108.405pt{}+\left|\ \sum_{k\neq
0}\frac{\mathaccent 866{g}(k)}{P(k)}\sin(P(k)t)e^{\,{\rm i}\,kx}-\sum_{k\neq
0}\frac{\mathaccent 866{g}(k)}{k^{N}}\sin(P(k)t)e^{\,{\rm i}\,kx}\right|\\\
&\hskip
72.26999pt:=\mathrm{II}^{(1)}+\mathrm{II}^{(2)}+\mathrm{II}^{(3)}.\end{aligned}$

We directly estimate the above three terms as follows:

$\displaystyle\begin{aligned} \mathrm{II}^{(1)}&\leq\sum_{k\neq
0}\frac{|O(k^{-1})\mathaccent 866{g}(k)|}{|P(k)\varphi(k)|}\mathchar
13358\relax\sum_{k\neq 0}\frac{1}{k^{2N+2}},\\\
\mathrm{II}^{(2)}&\leq\sum_{k\neq
0}\frac{|\sin(\sqrt{\varphi(k)}t)-\sin(P(k)t)||\mathaccent
866{g}(k)|}{|P(k)|}\mathchar 13358\relax\sum_{k\neq 0}\frac{1}{k^{N+2}},\\\
\mathrm{II}^{(3)}&\leq\sum_{k\neq
0}\frac{|c_{N-1}k^{N-1}+\ldots+c_{0}||\mathaccent
866{g}(k)|}{|P(k)k^{N}|}\mathchar 13358\relax\sum_{k\neq
0}\frac{1}{k^{N+2}}.\end{aligned}$

We thus conclude that

$\displaystyle\begin{aligned} \left|\;\sum_{k\neq 0}\frac{\mathaccent
866{g}(k)}{\sqrt{\varphi(k)}}\sin(\sqrt{\varphi(k)}t)e^{\,{\rm
i}\,kx}-\sum_{k\neq 0}\frac{\mathaccent 866{g}(k)}{k^{N}}\sin(P(k)t)e^{\,{\rm
i}\,kx}\;\right|\mathchar 13358\relax\sum_{k\neq
0}\frac{1}{k^{N+2}},\end{aligned}$

which, together with the estimate (LABEL:non-poly-1) imply that, in the
present case, the solution $u(t,x)$ will exhibit the same asymptotic behavior
as the polynomial case. The times at which the solution (approximately)
exhibits revivals are densely embedded in the times at which it has a
continuous, fractal profile.

For example, the Boussinesq equation

$u_{tt}+\frac{1}{3}u_{xxxx}-u_{xx}+\frac{3}{2}\alpha(u^{2})_{xx}=0,$ (3.30)

has the linear dispersion relation $\omega(k)=\pm k\sqrt{\frac{1}{3}k^{2}+1}$,
and its leading order asymptotics is $\pm\frac{1}{\sqrt{3}}k^{2}$. The
solution of the periodic initial-boundary value problem for the linearization
of equation (3.30) subject to the step function initial data (2.2) at several
representative rational and irrational times are plotted in Figure 6. As
illustrated in these figures, the solutions exhibit the (approximately)
revival profile at rational times, and the overall jump discontinuities and
revival structure is very similar to that of the linear beam equation.

(a) $t=\pi/2$

(b) $t=\pi/3$

(c) $t=\pi/5$

(d) $t=0.1$

(e) $t=0.3$

(f) $t=0.5$

Figure 6. The solutions to the periodic initial-boundary value problem for the
linear Boussinesq equation.

On the other hand, if the equation admits a non-polynomial dispersion relation
with an non-integral asymptotic exponent, i.e.,

$\sqrt{\varphi(k)}\sim|k|^{\alpha},\quad
2\leq\alpha\notin\mathrm{Z},\;\mathrm{as}\quad|k|\rightarrow\infty,$

we still estimate the two summations in (3.4) separately. As far as the first
one is concerned, as studied in [7], its overall qualitative behavior is
entirely determined by the asymptotic exponent $\alpha$. In particular, when
$2\leq\alpha\notin\mathrm{Z}$ is not an integer, only fractal solution
profiles will be observed at every time. While, as for the second term,
observe that in the present situation,

$\displaystyle\begin{aligned} &\left|\ \sum_{k\neq 0}\frac{\mathaccent
866{g}(k)}{\sqrt{\varphi(k)}}\sin(\sqrt{\varphi(k)}t)e^{\,{\rm
i}\,kx}-\sum_{k\neq 0}\frac{\mathaccent
866{g}(k)}{|k|^{[\alpha]+1}}\sin(|k|^{\alpha}t)e^{\,{\rm i}\,kx}\ \right|\\\
&\hskip 108.405pt\leq\sum_{k\neq
0}\frac{|k|^{[\alpha]+1}-|k|^{\alpha}}{|k|^{\alpha}|k|^{[\alpha]+1}}|\mathaccent
866{g}(k)|\mathchar 13358\relax\sum_{k\neq
0}\frac{|k|^{\alpha^{\prime}}\ln|k|}{|k|^{\alpha}|k|^{[\alpha]+1}}|\mathaccent
866{g}(k)|,\end{aligned}$

for some $\alpha<\alpha^{\prime}<[\alpha]+1$. Note that if $g(x)$ is of
bounded variation, the estimate can only be obtained by using $\sum_{k\neq
0}{|k|^{-[\alpha]-1}}$. In view of this situation, we need to further require
that $g(x)$ satisfies

$\int\,e^{\,{\rm i}\,kx}\;\mathrm{d}g\sim
O(k^{(\alpha-\alpha^{\prime})-\delta})\quad{\rm for\ all}\quad\delta>0.$

Under this hypothesis, the above estimate is bounded by $\sum_{k\neq
0}{|k|^{-[\alpha]-2}}$, and hence the second term is completely determined by
the series

$\sum_{k\neq 0}\frac{\mathaccent
866{g}(k)}{|k|^{[\alpha]+1}}\sin(|k|^{\alpha}t)e^{\,{\rm i}\,kx},$

which, compared with the first term, will admit better regularity.

(a) $t=\pi/10$

(b) $t=\pi/5$

(c) $t=\pi/3$

(d) $t=0.1$

(e) $t=0.3$

(f) $t=0.5$

Figure 7. The solutions to the periodic Riemann problem for the linear
equation (3.31).

We conclude that, in the present case, the solutions will retain a fractal
profile at all times. Results confirming this are displayed in Figure 7, which
are the plots of the solutions to the periodic Riemann problem for the case of
three-halves dispersion relation $\omega(k)=\pm|k|^{\frac{3}{2}}$
corresponding to the equation

$u_{tt}=\mathcal{H}[u_{xxx}],$ (3.31)

where $\mathcal{H}$ denotes the periodic Hilbert transform,

$\mathcal{H}[f](x)=\frac{1}{\pi}\sum_{-\infty}^{+\infty}\,\int_{-\pi}^{\pi}\hskip-19.0pt\hbox{---}\hskip
10.0pt\frac{f(y)}{x-y+2\pi
k}\,\mathrm{d}y=\frac{1}{2\pi}\,\int_{-\pi}^{\pi}{\hskip-19.0pt\hbox{---}\hskip
10.0pt\cot\frac{x-y}{2}f(y)}\,\mathrm{d}y.$

##  4\. Numerical simulation of dispersive revival for nonlinear equations

In this section, we will explore the effect of periodicity on rough initial
data for nonlinear equations in the context of the nonlinear defocusing cubic
beam equation of the form

$u_{tt}+u_{xxxx}+\mu\,u+\varepsilon\,|u|^{2}u=0,$ (4.1)

which is motivated by the nonlinear Boussinesq equation, see [34] for details.
We will numerically approximate the solutions to the periodic initial-boundary
value problem for the beam equation (4.1) subject to periodic boundary
conditions on $[-\pi,\,\pi]$, with the same step function (2.2) as initial
data.

The goal of this section is to investigate to what extent revival and
fractalization phenomena persist into the nonlinear regime. A basic numerical
technique, the Fourier spectral method, will be employed to approximate the
solution to this initial-boundary value problem. As we will see, our numerical
studies strongly indicate that the dispersive revival phenomenon admitted by
the associated linearized equation will persist into the nonlinear regime.
However, some of the qualitative details — for instance, the convexity of the
curves between the jump discontinuities — will be affected by the
nonlinearity, in contrast to what was observed in the unidirectional case.

###  4.1.  The Fourier spectral method

Let us first summarize the basic ideas behind the Fourier spectral method for
approximating the solutions to nonlinear equations. One can refer to [19, 39]
for details of the method.

Formally, consider the initial value problem for a nonlinear evolution
equation

$u_{t}=K[u],\qquad u(0,x)=u_{0}(x),$ (4.2)

where $K$ is a differential operator in the spatial variable with no explicit
time dependence. Suppose $K$ can be written as $K=L+N$, in which $L$ is a
linear operator characterized by its Fourier transform $\mathaccent
866{Lu}(k)=\omega(k)\mathaccent 866{u}(k)$, while $N$ is a nonlinear operator.
We use $\mathcal{F}[\cdot]$ and $\mathcal{F}^{-1}[\cdot]$ to denote the
Fourier transform and the inverse Fourier transform of the indicated function,
respectively, so that the Fourier transform for equation in (4.2) takes the
form

$\mathaccent 866{u}_{t}=\omega(k)\mathaccent
866{u}+\mathcal{F}[\,N(\mathcal{F}^{-1}[\mathaccent 866{u}])\,].$

Firstly, periodicity and discretization of the spatial variable enables us to
apply the fast Fourier transform (FFT) based on, for instance, 512 space
nodes, and arrive at a system of ordinary differential equations (ODEs), which
we solve numerically. For simplicity, we adopt a uniform time step $0<\Delta
t\ll 1$, and seek to approximate the solution $\hat{u}(t_{n})$ at the
successive times $t_{n}=n\Delta t$ for $n=0,1,\ldots$. The classic fourth-
order Runge-Kutta method, which has a local truncation error of $O((\Delta
t)^{5})$, is adopted, and its iterative scheme is given by

$\mathaccent 866{u}(t_{n+1})=\mathaccent
866{u}(t_{n})+\frac{1}{6}(f_{k_{1}}+2f_{k_{2}}+2f_{k_{3}}+f_{k_{4}}),\quad
n=0,1,\ldots,\quad\mathaccent 866{u}(t_{0})=\mathaccent 866{u}_{0}(k),$

where

$\displaystyle\begin{aligned} f_{k_{1}}&=f(t_{n},\mathaccent
866{u}(t_{n})),&f_{k_{2}}&=f(t_{n}+\Delta t/2,\mathaccent 866{u}(t_{n})+\Delta
tf_{k_{1}}/2),\\\ f_{k_{3}}&=f(t_{n}+\Delta t/2,\mathaccent
866{u}(t_{n})+\Delta tf_{k_{2}}/2),&\qquad f_{k_{4}}&=f(t_{n}+\Delta
t,\mathaccent 866{u}(t_{n})+\Delta tf_{k_{3}})\end{aligned}$

with

$\displaystyle f(t,\mathaccent 866{u})=\omega(k)\mathaccent
866{u}+\mathcal{F}[\,N(\mathcal{F}^{-1}[\mathaccent 866{u}])\,].$

Accordingly, the approximate solution $u(t,x)$ can be obtained through the
inverse discrete Fourier transform.

Since the Runge-Kutta method is designed for first order systems of ordinary
differential equations, we convert our bidirectional second order in time
system (4.1) into a first order system by setting

$v=u_{t},$

an hence the beam equation (4.1) is mapped to the following evolutionary
system

$u_{t}=v,\qquad v_{t}=-u_{xxxx}-\mu\,u-\varepsilon\,|u|^{2}u.$ (4.3)

The Fourier transform for (4.3) takes the form

$\mathaccent 866{u}_{t}=\mathaccent 866{v},\qquad\mathaccent
866{v}_{t}=-(k)^{4}\mathaccent 866{u}-\mu\,\mathaccent
866{u}-\varepsilon\,\mathcal{F}[\,|\mathcal{F}^{-1}[\mathaccent
866{u}]|^{2}\mathcal{F}^{-1}[\mathaccent 866{u}]\,].$ (4.4)

Using the classic fourth-order Runge-Kutta method to solve the resulting
system (4.4), and then taking the inverse discrete Fourier transform, one can
obtain the numerical solution to the periodic initial-boundary value problem
for the nonlinear beam equation (4.1).

###  4.2.  Numerical Results

Figure 8 and Figure 9 display some results from our numerical approximations
of the solutions to the nonlinear beam equation (4.1) with periodic boundary
conditions and initial conditions (2.2) at some representative rational and
irrational times. Comparing the graphs in these two figures with the graphs
corresponding to the same times in Figure 1 and Figure 2, we find that, at
each irrational time, all sets of plots are fairly similar to those from the
associated linear beam equation, and the solution still takes a continuous,
non-differentiable profile. When it comes to the rational times, the same jump
discontinuities consistency for nonlinear and linear equations emerges as
well. Meanwhile, closer inspection will reveal some differences. The most
noticeable is that, the shape of the curves between jump discontinuities will
change with time evolution. More precisely, the graphs corresponding to
$t=\pi/5$ show that the differences of the solution profile between the linear
and nonlinear equations is slight, except that, in the nonlinear case, the
curves between the jumps become closer to constants. Further, as the power $p$
decreases with increasing time, the variation in the shape of the curves from
linear to nonlinear becomes greater and greater. As illustrated in the graphs
corresponding to $t=\pi/3$ and $t=\pi/2$, the convexity of the curves has
completely changed. These differences of the qualitative behavior of the
solutions exhibit the effect of the nonlinearity. Furthermore, in order to
better understand the effect of the nonlinearity, we perform further numerical
experiments for smaller values of coefficients $\varepsilon$ and $\mu$ in
equation (4.1). Referring to Figure 10, it appears that solution at $t=\pi/3$
tends to the linear profile as $\varepsilon$ tends to zero. Meanwhile, the
shape of the curves between jump discontinuities will change as $\varepsilon$
increases, the most noticeable variation being the changes in convexity. More
unexpected phenomena appear when $t=\pi/2$. We find the variation of the
profile of the solution will be affected not only by the nonlinear term but
also by the linear term involving $u$. The plots displayed in Figure 11,
corresponding to some representative coefficients $\varepsilon$ and $\mu$,
suggest that the solution profile, including its convexity and the values of
its peak and trough will be affected by the combination of both coefficients
$\varepsilon$ and $\mu$.

(a) $t=\pi/2$

(b) $t=\pi/3$

(c) $t=\pi/5$

Figure 8. The solutions to the periodic initial-boundary value problem for the
beam equation with $\mu=\varepsilon=1$ at rational times.

(a) $t=0.1$

(b) $t=0.3$

(c) $t=0.5$

Figure 9. The solutions to the periodic initial-boundary value problem for the
beam equation with $\mu=\varepsilon=1$ at irrational times.

(a) $\mu=1,\,\varepsilon=0.01$

(b) $\mu=1,\,\varepsilon=0.5$

(c) $\mu=1,\,\varepsilon=0.9$

Figure 10. The solutions to the periodic initial-boundary value problem for
the beam equation at $t=\pi/3$.

(a) $\mu=0.001,\,\varepsilon=0.001$

(b) $\mu=0.001,\,\varepsilon=1$

(c) $\mu=1,\,\varepsilon=0.001$

Figure 11. The solutions to the periodic initial-boundary value problem for
the beam equation at $t=\pi/2$.

Recall that the numerical experiments to the periodic initial-boundary value
problem for the KdV equation, the NLS equation and the multi-component KdV
system have been previously analyzed in [8, 42], which show that, in the
unidirectional regime, the effect of the nonlinear flow can be regarded as a
perturbation of the linearized flow. When it comes to the bidirectional
dispersive equations, our numerical simulation strongly indicates that, the
dichotomy of revival/fractalization at rational/irrational times in
linearization will persist into the nonlinear regime, and the finite “revival”
nature of the solutions at rational times is not affected by the nonlinearity,
however, the influence of the nonlinearity on the qualitative behavior of the
solutions is much greater than in the unidirectional setting. Motivated by
this observation, formulation of theorems and rigorous proofs concerning this
novel revival phenomenon in the nonlinear bidirectional regime, specially for
the nonlinear beam and Boussinesq equations, is eminently worth further study.

Acknowledgments. Part of Farmakis’ research was conducted during his Ph.D
studies which were supported by the Maxwell Institute Graduate School in
Analysis and its Applications, a Centre for Doctoral Training funded by EPSRC
(grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University
and the University of Edinburgh. Kang’s research was supported by NSFC
(Grants-11631007 and 11871395 and Basic Science Program of Shaanxi Province
(Grant-2019JC-28). Qu’s research was supported by NSFC (Grants-11971251,
11631007 and 12111530003). Yin’s research was supported by the NSFC
(Grant-11631007).

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