# A two-component nonlinear variational wave system

Peder Aursand  Aker BP ASA  and  Anders Nordli  Engineering Science and Safety
UiT
UiT The Arctic University of Norway
Tromsø
Norway<EMAIL_ADDRESS>
###### Abstract.

We derive a novel two-component generalization of the nonlinear variational
wave equation as a model for the director field of a nematic liquid crystal
with a variable order parameter. The two-component nonlinear variational wave
equation admits solutions locally in time. We show that a particular long time
asymptotic expansion around a constant state in a moving frame satisfy the
two-component Hunter–Saxton system.

##  1\. Introduction

The nonlinear variational wave equation
$\psi_{tt}-c(\psi)(c(\psi)\psi_{x})_{x}=0$ was derived by Saxton [24] as a
model of the director field of a nematic liquid crystal. The nonlinear
variational wave equation has recieved wide attention [2, 5, 11, 13] due to
mathematical challenges in the form of wavebreaking in finite time, and
distinct ways to extend the solution to a global weak solution. In this
context wavebreaking means that either the time or the space derivative
becomes unbounded at certain points, while the solution remains Hölder
continuous.

Furthermore, Hunter and Saxton derived the equation
$u_{tx}+(uu_{x})_{x}=\frac{1}{2}u_{x}^{2}$ from the nonlinear variational wave
equation [14] as an asymptotic equation for small perturbations in the long
time regime in a moving frame. The Hunter–Saxton equation share many of the
features of the nonlinear variational wave equation such as wavebreaking,
conservative and dissipative weak solutions, and Hölder continuity [1, 3, 4,
6, 9, 16, 17, 26, 27]. In addition it exhibited novel features such as
complete integrability and interpretation as a geodesic flow [15, 20]. It also
proved easier to work with due to there being only one family of
characteristics and existence of explicit solutions.

A two-component generalization for the Hunter–Saxton equation was derived from
two-component Camassa–Holm [8], and independently form the Gurevich–Zybin
system [23]. The two-component generalization is similar to the Hunter–Saxton
equation since wavebreaking is also possible, there are both conservative and
dissipative weak solutions, and $u$ is Hölder continuous [12, 22, 25]. However
when the second variable is nonzero almost everywhere initially, there will be
no wave breaking [22], and in that sense the introduction of a second variable
regularizes the equation. The two-component Hunter–Saxton system has, however,
not been shown to be related to the theory of nematic liquid crystals, and one
of the aims of this paper is to establish that the two-component Hunter–Saxton
system can indeed be derived from the theory of nematic liquid crystals, and
that the new variable is related to the order parameter.

In the Ericksen–Leslie theory of nematic liquid crystals the configuration is
described by a director field $\mathbf{n}$ which gives the local orientation
of the rods, and an order parameter field $s$ which gives the local degree of
orientation [10, 21]. When the nonlinear variational wave equation was first
derived the local degree of orientation was assumed constant, so that a scalar
equation was obtained [24]. Here we will follow a similar route, but account
for variable degree of orientation. Thus, in Section 2, we will arrive at a
novel two-component system of nonlinear wave equations. Moreover, we show that
a similar asymptotic expansion to the one by Hunter and Saxton [14] yields the
two-component Hunter–Saxton equation. In Section 3 existence of global
solutions is shown in the case of constant wave speed, and local solutions is
shown to exist in general. The proofs rely on fixed point iterations and
standard semigroup theory for evolution equations [18].

##  2\. Derivation of the equations

In Ericksen–Leslie theory a nematic liquid crystal in a domain $\Omega$ is
desribed by a director field
$\mathbf{n}:[0,T]\times\Omega\rightarrow\mathbb{R}\mathbb{P}^{2}$ and an order
parameter $s:[0,T]\times\Omega\rightarrow(-\frac{1}{2},1)$ [10, 21]. The
Lagrangian of the system is then [10] given by

(2.1) $\mathcal{L}=-(s\mathbf{n})_{t}^{2}+W_{2}(s,\nabla
s,\mathbf{n},\nabla\mathbf{n})+W_{0}(s),$

with the potential energy term $W_{2}$

$\displaystyle W_{2}(s,\nabla s,\mathbf{n},\nabla\mathbf{n})$
$\displaystyle=\left(K_{1}+L_{1}s\right)s^{2}\left(\nabla\cdot\mathbf{n}\right)^{2}+\left(K_{2}+L_{2}s\right)s^{2}\left(\mathbf{n}\cdot\nabla\times\mathbf{n}\right)^{2}$
$\displaystyle\quad+\left(K_{3}+L_{3}s\right)s^{2}\left|\mathbf{n}\times\nabla\times\mathbf{n}\right|^{2}$
$\displaystyle\quad+\left((K_{2}+K_{4})+(L_{2}+L_{4})s\right)s^{2}\left[\mathrm{tr}\>\nabla\mathbf{n}^{2}-(\nabla\cdot\mathbf{n})^{2}\right]$
$\displaystyle\quad+\left(\kappa_{1}+\lambda_{1}s\right)\left|\nabla
s\right|^{2}+\left(\kappa_{2}+\lambda_{2}s\right)\left(\nabla
s\cdot\mathbf{n}\right)^{2}$ (2.2)
$\displaystyle\quad+\left(\kappa_{3}+\lambda_{3}s\right)\left(\nabla\cdot\mathbf{n}\right)\left(\nabla
s\cdot\mathbf{n}\right)+\left(\kappa_{4}+\lambda_{4}s\right)\nabla
s\cdot\left((\mathbf{n}\cdot\nabla)\mathbf{n}\right),$

and some function $W_{0}$ satisfying

(2.3) $\lim_{s\downarrow-\frac{1}{2}}W_{0}(s)=+\infty,\quad\lim_{s\uparrow
1}W_{0}(s)=+\infty,\text{ and }\lim_{s\rightarrow
0}\frac{W_{0}(s)}{s^{2}}\in\mathbb{R}.$

We are interested in the case where $\mathbf{n}=(\cos\psi,\sin\psi,0)$, and
$s$ and $\psi$ depends on $t$ and $x$ only. Then the expression for $W_{2}$
can be written

$\displaystyle W_{2}$
$\displaystyle=s^{2}\psi_{x}^{2}\left((K_{1}+L_{1}s)\sin^{2}\psi+(K_{3}+L_{3}s)\cos^{2}\psi\right)$
$\displaystyle\quad+s_{x}^{2}\left((\kappa_{1}+\lambda_{1}s)\sin^{2}\psi+(\tilde{\kappa}_{2}+\tilde{\lambda}_{2}s)\cos^{2}\psi\right)$
(2.4) $\displaystyle\quad-s_{x}\psi_{x}\sin
2\psi\left(\frac{\kappa_{3}+\kappa_{4}}{2}+\frac{\lambda_{3}+\lambda_{4}}{2}s\right),$

where $\tilde{\kappa}_{2}=\kappa_{1}+\kappa_{2}$ and
$\tilde{\lambda}_{2}=\lambda_{1}+\lambda_{2}$. We simplify the problem by
considering the case where

(2.5a) $\displaystyle\kappa_{3}$
$\displaystyle=\kappa_{4}=\lambda_{3}=\lambda_{4}=0,$ (2.5b) $\displaystyle
L_{1}$ $\displaystyle=L_{3}=\lambda_{1}=\tilde{\lambda}_{2}=0,$ (2.5c)
$\displaystyle K_{1}$ $\displaystyle=\kappa_{1},$ (2.5d) $\displaystyle K_{3}$
$\displaystyle=\tilde{\kappa}_{2}.$

Then with the definition $c(\psi)^{2}=K_{1}\sin^{2}\psi+K_{3}\cos^{2}\psi$, we
get the Lagrangian density

$\displaystyle\mathcal{L}^{2NVW}$
$\displaystyle=-\frac{1}{2}s_{t}^{2}-\frac{1}{2}s^{2}\psi_{t}^{2}+\frac{1}{2}s^{2}c(\psi)^{2}\psi_{x}^{2}+\frac{1}{2}c(\psi)^{2}s_{x}^{2}+W_{0}(s)$
(2.6)
$\displaystyle=-\frac{1}{2}(s\mathbf{n})_{t}^{2}+\frac{1}{2}c(\mathbf{n})^{2}(s\mathbf{n})_{x}^{2}+W_{0}(s).$

######  Definition 2.1.

We define the two-component nonlinear variational wave system to be

(2.7a) $\displaystyle
s^{2}\left(\psi_{tt}-c(\psi)\left(c(\psi)\psi_{x}\right)_{x}\right)+2s\left(\psi_{t}s_{t}-c(\psi)^{2}\psi_{x}s_{x}\right)+c(\psi)c^{\prime}(\psi)s_{x}^{2}$
$\displaystyle=0,$ (2.7b) $\displaystyle
s_{tt}-c(\psi)\left(c(\psi)s_{x}\right)_{x}-c(\psi)c^{\prime}(\psi)\psi_{x}s_{x}-s\left(\psi_{t}^{2}-c(\psi)^{2}\psi_{x}^{2}\right)+W_{0}^{\prime}(s)$
$\displaystyle=0,$ (2.7c) $\displaystyle c(\psi)^{2}-K_{1}\sin^{2}\psi-
K_{3}\cos^{2}\psi$ $\displaystyle=0,$ (2.7d) $\displaystyle W_{0}$
$\displaystyle\in C^{4}\left(\left(-\frac{1}{2},1\right)\right).$

######  Remark 2.2.

The system (2.7) is the Euler–Lagrange equations for (2).

One can define a conserved energy for classical solutions.

######  Proposition 2.3.

Define the energy density

(2.8)
$\mathcal{E}=\frac{1}{2}\left(s^{2}(\psi_{t}^{2}+c(\psi)^{2}\psi_{x}^{2})+(s_{t}^{2}+c(\psi)^{2}s_{x}^{2})\right)+W_{0}(s).$

and the energy density flux

(2.9) $\mathcal{F}=\left(s^{2}\psi_{t}\psi_{x}+s_{t}s_{x}\right).$

Then for classical solutions of (2.7) the energy satisfy the equations

(2.10a) $\displaystyle\mathcal{E}_{t}-\left(c(\psi)^{2}\mathcal{F}\right)_{x}$
$\displaystyle=0,$ (2.10b)
$\displaystyle\mathcal{F}_{t}-\left(\mathcal{E}-2W_{0}(s)\right)_{x}$
$\displaystyle=0.$

We will now derive the two-component Hunter–Saxton system from (2). To follow
the work of Hunter and Saxton [14] we introduce
$\psi(t,x)=\psi_{0}+\varepsilon u(\varepsilon t,x-c_{0}t)$ and
$s=s_{0}+\varepsilon r(\varepsilon t,x-c_{0}t)$ with
$c_{0}^{2}=K_{1}\sin^{2}\psi_{0}+K_{3}\cos^{2}\psi_{0}$. Then expansion of (2)
in powers of $\varepsilon$ gives

$\displaystyle\mathcal{L}^{\varepsilon}$
$\displaystyle=W_{0}(s_{0})+\varepsilon
W_{0}^{\prime}(s_{0})r+\varepsilon^{2}\frac{1}{2}W_{0}^{\prime\prime}(s_{0})r^{2}$
(2.11)
$\displaystyle\quad+\varepsilon^{3}\bigg{(}c_{0}r_{t}r_{x}+s_{0}^{2}c_{0}u_{t}u_{x}+s_{0}^{2}\left(cc^{\prime}\right)_{0}uu_{x}^{2}+\left(cc^{\prime}\right)_{0}ur_{x}^{2}+\frac{1}{6}W_{0}^{\prime\prime\prime}(s_{0})r^{3}\bigg{)}+O(\varepsilon^{4}).$

If we select $s_{0}$ such that
$W_{0}^{\prime}(s_{0})=W_{0}^{\prime\prime}(s_{0})=W^{\prime\prime\prime}(s_{0})=0$,
the third order terms in the above Lagrangian gives (possibly by rescaling)

(2.12) $\mathcal{L}^{2HS}=u_{t}u_{x}+uu_{x}^{2}+r_{t}r_{x}+ur_{x}^{2}.$

######  Theorem 2.4.

The two-component Hunter–Saxton system

(2.13a) $\displaystyle(u_{t}+uu_{x})_{x}$
$\displaystyle=\frac{1}{2}u_{x}^{2}+\frac{1}{2}\rho^{2},$ (2.13b)
$\displaystyle\rho_{t}+(u\rho)_{x}$ $\displaystyle=0,$

is the Euler–Lagrange equations for the Lagrangian density (2.12) with
$\rho=r_{x}$.

##  3\. The two-component nonlinear variational wave system

We will first consider the case $K_{1}=K_{3}$, that is the wave speed $c$ is
independent of $\psi$. We will assume $s\geq 0$ such that we can introduce the
complex variable $\zeta=se^{i\psi}$, and (2.7) reduces to

(3.1)
$\zeta_{tt}-c^{2}\zeta_{xx}+\frac{W_{0}^{\prime}\left(|\zeta|\right)}{|\zeta|}\zeta=0,$

which is a defocusing nonlinear wave equation. Recall that the physical
interpretation of $\psi$ is an angle, and thus any solutions $\psi$ and
$\psi+n\cdot 2\pi$ should be considered equal from an application point of
view. We have conservation laws for energy
$\mathcal{E}=\frac{1}{2}|\zeta_{t}|^{2}+\frac{1}{2}c^{2}|\zeta_{x}|^{2}+W_{0}(|\zeta|)$
and energy flux
$\mathcal{F}=\frac{1}{2}\left(\bar{\zeta}_{t}\zeta_{x}+\zeta_{t}\bar{\zeta}_{x}\right)$
as follows

(3.2a) $\displaystyle\frac{\partial}{\partial
t}\mathcal{E}-c^{2}\frac{\partial}{\partial x}\mathcal{F}$ $\displaystyle=0,$
(3.2b) $\displaystyle\frac{\partial}{\partial
t}\mathcal{F}-\frac{\partial}{\partial
x}\left(\mathcal{E}-2W_{0}(|\zeta|)\right)$ $\displaystyle=0.$

Hence both $\int_{\mathbb{R}}\mathcal{E}\>\mathrm{d}x$ and
$\int_{\mathbb{R}}\mathcal{F}\>\mathrm{d}x$ are conserved for any classical
solution with bounded energy. To analyze solutions we will need some
conditions on the function $W_{0}$. We will assume that
$W_{0}(s)\rightarrow\infty$ rapidly enough as $s\rightarrow 1$ to be able to
bound $\|W_{0}(|\zeta|)\|_{\infty}$ in terms of the total energy $E$. In
addition we will assume that $W_{0}$ is non-negative and that $W_{0}$ is well
behaved close to $s=0$, and also close to zeros of $W_{0}$. We will also relax
(2.3) slightly to allow for $s=0$ to be a local maximum or minimum instead of
a global minimum, in accordance with [10].

######  Definition 3.1.

We will assume that $W_{0}\in C^{4}([0,1))$ and non-negative, and that there
exists a finite number of $s^{*}$ such that $W_{0}(s^{*})=0$, and for all
zeros $s^{*}$,

(3.3) $\displaystyle\lim_{s\rightarrow 0}\frac{W_{0}(s)-W_{0}(0)}{s^{2}}$
$\displaystyle=\frac{1}{2}W_{0}^{\prime\prime}(0)\in\mathbb{R},$ (3.4)
$\displaystyle\lim_{s\rightarrow s^{*}}\frac{W_{0}(s)}{(s-s^{*})^{2}}$
$\displaystyle=\frac{1}{2}W_{0}^{\prime\prime}(s^{*})\in[0,\infty).$

Moreover, we let $W_{0}$ satisfy

(3.5) $\int_{s}^{1}W_{0}(u)(1-u)\>du=\infty,$

and that there exists $\tilde{s}\in[0,1)$ such that $W_{0}(s)>0$ and
$W_{0}^{\prime}(s)>0$ for all $s\in(\tilde{s},1)$.

We define the function spaces for the solutions in the next definition.

######  Definition 3.2.

Let $\zeta^{*}\in\mathbb{C}$, $|\zeta^{*}|<1$, and

(3.6) $X_{\zeta^{*}}=\\{(\zeta,\sigma)\mid\zeta\in
W^{1,\infty}(\mathbb{R}),\zeta-\zeta^{*}\in H^{1}(\mathbb{R}),\sigma\in
L^{\infty}(\mathbb{R})\cap L^{2}(\mathbb{R})\\},$

and define

(3.7) $\displaystyle\mathcal{E}(\zeta,\sigma)$
$\displaystyle=\frac{1}{2}|\sigma|^{2}+\frac{1}{2}|\zeta_{x}|^{2}+W_{0}(|\zeta|),$
(3.8) $\displaystyle\mathcal{F}(\zeta,\sigma)$
$\displaystyle=\frac{1}{2}\sigma\bar{\zeta}_{x}+\frac{1}{2}\bar{\sigma}\zeta_{x},$

and note that both $\mathcal{E}$ and $\mathcal{F}$ are real valued. Let now

(3.9) $X_{E}=\\{(\zeta,\sigma)\in X_{\zeta^{*}}\mid|\mathcal{E}\|_{L^{1}}\leq
E\\},$

and denote

(3.10) $\displaystyle\mathcal{D}_{T,E}$ $\displaystyle=\left\\{\zeta\in
C([0,T],W^{1,\infty}(\mathbb{R},\mathbb{C}))\cap
C^{1}([0,T],L^{\infty}(\mathbb{R},\mathbb{C})\mid(\zeta(t),\zeta_{t}(t))\in
X_{E}\right\\}.$

In the case $T<\infty$ we equip $\mathcal{D}_{T,E}$ with the metric
$d_{\mathcal{D}_{E,T}}$ induced from

$\displaystyle\|\zeta\|$
$\displaystyle=\sup_{t\in[0,T]}\big{(}\|\zeta(t)\|_{W^{1,\infty}(\mathbb{R})}+\|\zeta(t)-\zeta^{*}\|_{H^{1}(\mathbb{R})}$
(3.11)
$\displaystyle\quad+\|\zeta_{t}(t))\|_{L^{\infty}(\mathbb{R})}+\|\zeta_{t}(t))\|_{L^{2}(\mathbb{R})}+\|W_{0}(|\zeta(t)|)\|_{L^{1}(\mathbb{R})}\big{)}.$

For bounded energy we can now a priori put constraints on possible solutions.

######  Proposition 3.3.

Let $W_{0}$ satisfy the conditions of Definition 3.1. Then there exist
positive constants $c_{E},C_{E}$, depending on $W_{0}$ and $E$ only, such that
for any $(\zeta,\sigma)\in X_{E}$ we have

$\displaystyle\|W_{0}(|\zeta|)\|_{L^{\infty}(\mathbb{R})}$ $\displaystyle\leq
C_{E},$ $\displaystyle\|\zeta\|_{L^{\infty}(\mathbb{R})}$ $\displaystyle\leq
c_{E}<1.$

Moreover for $s\in[0,c_{E}]$ there is a positive constant $k_{E}$ such that

(3.12) $W_{0}^{\prime}(s)^{2}\leq k_{E}W_{0}(s).$

Furthermore, we can define

(3.13) $\displaystyle L_{E}$
$\displaystyle=\sup_{s\in[0,c_{E}]}|W_{0}^{\prime}(s)|,$ (3.14) $\displaystyle
L_{E}^{\prime}$
$\displaystyle=\sup_{s\in[0,c_{E}]}|\frac{W_{0}^{\prime}(s)}{s}|,$ (3.15)
$\displaystyle L_{E}^{\prime\prime}$
$\displaystyle=\sup_{s\in[0,c_{E}]}|W_{0}^{\prime\prime}(s)|.$

###### Proof.

Since
$\frac{1}{2}\|\sigma\|_{2}^{2}+\frac{1}{2}c^{2}\|\zeta_{x}\|_{2}^{2}+\|W_{0}(|\zeta|)\|_{1}\leq
E$ we have that
$|\zeta(x_{1})-\zeta(x_{2})|\leq\frac{\sqrt{2E_{1}}}{c}\sqrt{|x_{1}-x_{2}|}$
where $E_{1}=E-\frac{1}{2}\|\sigma\|_{2}^{2}-\|W_{0}(|\sigma|)\|_{1}\geq 0$.
Since $\zeta$ is continuous we have that there exists $\hat{x}$ such that
$|\zeta(\hat{x})|=\|\zeta\|_{L^{\infty}(\mathbb{R})}\leq 1$. Then we have that

(3.16)
$|\zeta(x)|\geq\begin{cases}\|\zeta\|_{\infty}-\frac{\sqrt{2E_{1}}}{c}\sqrt{|x-\hat{x}|},&\hat{x}-\frac{\|\zeta\|_{\infty}^{2}c^{2}}{2E_{1}}<x<\hat{x}+\frac{\|\zeta\|_{\infty}^{2}c^{2}}{2E_{1}},\\\
0,&\text{ else}.\end{cases}$

In particular, we have that

$\displaystyle\int_{\mathbb{R}}W_{0}(|\zeta(x)|)\>\mathrm{d}x$
$\displaystyle\geq\int_{\hat{x}-\frac{c^{2}}{2E_{1}}(\|\zeta\|_{\infty}-\tilde{s})^{2}}^{\hat{x}+\frac{c^{2}}{2E_{1}}(\|\zeta\|_{\infty}-\tilde{s})^{2}}W_{0}(|\zeta(x)|)\>\mathrm{d}x$
$\displaystyle\geq\int_{\hat{x}-\frac{c^{2}}{2E_{1}}(\|\zeta\|_{\infty}-\tilde{s})^{2}}^{\hat{x}+\frac{c^{2}}{2E_{1}}(\|\zeta\|_{\infty}-\tilde{s})^{2}}W_{0}\left(\|\zeta\|_{\infty}-\frac{\sqrt{2E_{1}}}{c}\sqrt{|x-\hat{x}|})\right)\>\mathrm{d}x$
(3.17)
$\displaystyle=\int_{\tilde{s}}^{\|\zeta\|_{\infty}}W_{0}(u)\left(\|\zeta\|_{\infty}-u\right)\>\mathrm{d}u$

By the monotone convergence theorem we have that

(3.18) $\lim_{S\rightarrow
1^{-}}\int_{s}^{S}W_{0}(u)\left(S-u\right)\>\mathrm{d}u=\int_{s}^{1}W_{0}(u)\left(1-u\right)\>\mathrm{d}u=\infty.$

Hence we have the inequality

(3.19)
$\int_{\tilde{s}}^{\|\zeta\|_{\infty}}W_{0}(u)\left(\|\zeta\|_{\infty}-u\right)\>\mathrm{d}u\leq
E-E_{1},$

which proves the existence of $c_{E},C_{E}$ such that
$\|\zeta\|_{\infty}=c_{E}<1$ and
$\|W_{0}(|\zeta|)\|_{\infty}=W_{0}(\|\zeta\|_{\infty})=C_{E}<\infty$. The
constant can be chosen to depend on $W_{0}$ and $E$ only.

To prove that $W_{0}^{\prime 2}\leq k_{E}W_{0}$, note that whenever $W_{0}$
tends to zero that

(3.20) $\lim_{s\rightarrow
s^{*}}\frac{W_{0}^{\prime}(s)^{2}}{W_{0}(s)}=2W_{0}^{\prime\prime}(s^{*})\in[0,\infty).$

In between zeros of $W_{0}$ the fraction $\frac{W_{0}^{\prime 2}}{W_{0}}$ is
continuous. Since there is only a finite number of intervals where $W_{0}$ is
zero and $[0,c_{E}]$ is bounded, $\frac{W_{0}^{\prime 2}}{W_{0}}$ has to be
bounded as well by a constant dependent on $c_{E}$. Since $W_{0}$ is non-
negative we get the desired inequality. ∎

######  Remark 3.4.

Note that $\zeta_{1}$ and $\zeta_{2}$ in $X_{E}$ we have that

$\displaystyle\int_{\mathbb{R}}W_{0}(|\zeta_{1}(x)|)\>\mathrm{d}x$
$\displaystyle=\int_{\mathbb{R}}\frac{1}{2}L_{E}^{\prime\prime}|\zeta_{1}-\zeta^{*}|^{2}\>\mathrm{d}x$
(3.21)
$\displaystyle\leq\frac{1}{2}L_{E}^{\prime\prime}\|\zeta_{1}-\zeta^{*}\|_{2}^{2},$

and similarly, since
$W_{0}(|\zeta|)=\int_{|\zeta^{*}|}^{|\zeta|}W_{0}^{\prime\prime}(u)(|\zeta|-u)\>\mathrm{d}u$,

$\displaystyle\int_{\mathbb{R}}|W_{0}(|\zeta_{1}|)-W_{0}(|\zeta_{2}|)|\>\mathrm{d}x$
$\displaystyle=\int_{\mathbb{R}}\Bigg{|}\int_{|\zeta^{*}|}^{|\zeta_{1}|}W_{0}^{\prime\prime}(u)\left(|\zeta_{1}|-|\zeta_{2}|\right)\>\mathrm{d}u$
$\displaystyle\qquad+\int_{|\zeta_{2}|}^{|\zeta_{1}|}W_{0}^{\prime\prime}(u)\left(|\zeta_{2}|-u\right)\>\mathrm{d}u\Bigg{|}\>\mathrm{d}x$
(3.22) $\displaystyle\leq
L_{E}^{\prime\prime}\|\zeta_{1}-\zeta_{2}\|_{2}^{2}+L_{E}^{\prime\prime}\|\zeta_{1}-\zeta^{*}\|_{2}\|\zeta_{1}-\zeta_{2}\|_{2}.$

######  Theorem 3.5.

Let $(\zeta_{0},\zeta_{t0})\in X_{E}$, and assume that $W_{0}$ satisfies the
conditions in Definition 3.1. Then there exists a unique global strong energy
conserving solution $(\zeta,\zeta_{t})\subseteq\mathcal{D}_{\infty,E}$ of
(3.1) depending continuously on the initial data $(\zeta_{0},\zeta_{t0})\in
X_{E}$.

###### Proof.

The idea of the proof is to show that the solution in strong form

$\displaystyle\zeta(t,x)$
$\displaystyle=\frac{1}{2}\left(\zeta_{0}(x-ct)+\zeta_{0}(x+ct)\right)+\frac{1}{2c}\int_{x-ct}^{x+ct}\zeta_{t0}(y)\>\mathrm{d}y$
(3.23)
$\displaystyle\quad-\frac{1}{2c}\int_{0}^{t}\int_{x-c(t-s)}^{x+c(t-s)}\frac{W_{0}^{\prime}(|\zeta(s,y)|)}{|\zeta(s,y)|}\zeta(s,y)\>\mathrm{d}y\mathrm{d}s,$

is the unique fixed point of a contraction on $\mathcal{D}_{T,E}$, as is done
for Picard iteration for a to be specified $T$. To be able to prove that
energy is conserved by strong solutions we will at first assume that the
initial data in $X_{E}$ is smooth, $(\zeta_{0},\zeta_{t0})\in
C^{2}(\mathbb{R})\times C^{1}(\mathbb{R})$. We will then prove that the strong
solution with smooth initial data preserve energy, and that the solution
operator taking initial data in $X_{E}$ to solutions in $\mathcal{D}_{T,E}$ is
continuous and can thus be extended to all of $X_{E}$. We prove that the
energy is still conserved for these solutions. Finally we note that the
existence time $T$ depends on $W_{0}$ and $E$ only, and thus that the
existence time can be extended indefinitely.

We will now assume that $0<E<E^{\prime}$ and that $(\zeta_{0},\zeta_{t0})\in
X_{E}$ and $\zeta_{0}\in C^{2}(\mathbb{R}),\zeta_{t0}\in C^{1}(\mathbb{R})$,
and let $(\hat{\zeta},\hat{\zeta}_{t})\in\mathcal{D}_{T,E^{\prime}}\cap
C^{2}([0,\infty)\times\mathbb{R})$ with
$(\hat{\zeta}(0),\hat{\zeta}_{t}(0))=(\zeta_{0},\zeta_{t0})$. Then let
$(\zeta,\zeta_{t})\in C([0,T],X_{E^{\prime}})$, $\zeta\in
C^{2}([0,\infty)\times\mathbb{R})$, be the classical solution of

(3.24)
$\zeta_{tt}-c^{2}\zeta_{xx}=-\frac{W^{\prime}_{0}(|\hat{\zeta}|)}{|\hat{\zeta}|}\hat{\zeta},\quad(\zeta,\zeta_{t})|_{t=0}=(\zeta_{0},\zeta_{t0}).$

Then by Duhamel’s principle we have

$\displaystyle\zeta(t,x)$
$\displaystyle=\frac{1}{2}\left(\zeta_{0}(x-ct)+\zeta_{0}(x+ct)\right)+\frac{1}{2c}\int_{x-ct}^{x+ct}\zeta_{t0}(y)\>\mathrm{d}y$
(3.25a)
$\displaystyle\quad-\frac{1}{2c}\int_{0}^{t}\int_{x-c(t-s)}^{x+c(t-s)}\frac{W_{0}^{\prime}(|\hat{\zeta}(s,y)|)}{|\hat{\zeta}(s,y)|}\hat{\zeta}(s,y)\>\mathrm{d}y\>\mathrm{d}s,$
$\displaystyle\zeta_{t}(t,x)$
$\displaystyle=\frac{c}{2}\left(\zeta_{x0}(x+ct)-\zeta_{x0}(x-ct)\right)+\frac{1}{2}\left(\zeta_{t0}(x+ct)+\zeta_{t0}(x-ct)\right)\>\mathrm{d}y$
$\displaystyle\quad-\frac{1}{2}\int_{0}^{t}\frac{W_{0}^{\prime}(|\hat{\zeta}(s,x-c(t-s))|)}{|\hat{\zeta}(s,x-c(t-s))|}\hat{\zeta}(s,x-c(t-s))$
(3.25b)
$\displaystyle\qquad\qquad+\frac{W_{0}^{\prime}(|\hat{\zeta}(s,x+c(t-s))|)}{|\hat{\zeta}(s,x+c(t-s))|}\hat{\zeta}(s,x+c(t-s))\>\mathrm{d}s.$

To prove that the strong solution (3.25) is a member of
$\mathcal{D}_{T,E^{\prime}}$ we will need norm estimates on the solutions. For
$(\zeta,\zeta_{t})\in\mathcal{D}_{T,E}$ we have that

(3.26)
$\displaystyle\left|\frac{1}{2c}\int_{x-c(t-s)}^{x+c(t-s)}\frac{W_{0}^{\prime}(|\zeta(s,y)|)}{|\zeta(s,y)|}\zeta(s,y)\>\mathrm{d}y\right|$
$\displaystyle\leq L_{E}(t-s),$ (3.27)
$\displaystyle\left|\frac{\partial}{\partial
x}\frac{1}{2c}\int_{x-c(t-s)}^{x+c(t-s)}\frac{W_{0}^{\prime}(|\zeta(s,y)|)}{|\zeta(s,y)|}\zeta(s,y)\>\mathrm{d}y\right|$
$\displaystyle\leq\frac{1}{c}L_{E}.$

The integrals can be interpreted as convolutions

(3.28) $\int_{x-ct}^{x+ct}f(y)\>\mathrm{d}y=\mathbf{1}_{[-ct,ct]}*f(x),$

and then by Young’s inequality we get for any $p\in[1,\infty]$,

(3.29) $\left\|\int_{x-ct}^{x+ct}f(y)\>\mathrm{d}y\right\|_{p}\leq
2ct\|f\|_{p}.$

Thus, the solution $(\zeta,\zeta_{t})$ satisfies for $p=2,\infty$ the
estimates

(3.30) $\displaystyle\|\zeta(t)-\zeta^{*}\|_{p}$
$\displaystyle\leq\|\zeta_{0}-\zeta^{*}\|_{p}+t\|\zeta_{t0}\|_{p}+\frac{1}{2}t^{2}\max\\{L_{E^{\prime}},\sqrt{k_{E^{\prime}}E^{\prime}}\\},$
(3.31) $\displaystyle\|\zeta_{x}(t)\|_{p}$
$\displaystyle\leq\|\zeta_{x0}\|_{p}+\frac{1}{c}\|\zeta_{t0}\|_{p}+\frac{2}{c}t\max\\{L_{E^{\prime}},\sqrt{k_{E^{\prime}}E^{\prime}}\\},$
(3.32) $\displaystyle\|\zeta_{t}(t)\|_{p}$ $\displaystyle\leq
c\|\zeta_{x0}\|_{p}+\|\zeta_{t0}\|_{p}+\frac{1}{2}t^{2}\max\\{L_{E^{\prime}},\sqrt{k_{E^{\prime}}E^{\prime}}\\}$

We need to show that the energy is bounded. We have that

$\displaystyle\frac{\partial}{\partial
t}\left(\frac{1}{2}|\zeta_{t}|^{2}+\frac{1}{2}c^{2}|\zeta_{x}|^{2}+W_{0}(|\zeta|)\right)-\frac{\partial}{\partial
x}\frac{1}{2}c^{2}\left(\bar{\zeta}_{t}\zeta_{x}+\zeta_{t}\bar{\zeta}_{x}\right)=$
(3.33)
$\displaystyle\quad\frac{1}{2}\zeta_{t}\left(\frac{W_{0}^{\prime}(|\zeta|)}{|\zeta|}\bar{\zeta}-\frac{W_{0}^{\prime}(|\hat{\zeta}|)}{|\hat{\zeta}|}\bar{\hat{\zeta}}\right)+\frac{1}{2}\bar{\zeta}_{t}\left(\frac{W_{0}^{\prime}(|\zeta|)}{|\zeta|}\zeta-\frac{W_{0}^{\prime}(|\hat{\zeta}|)}{|\hat{\zeta}|}\hat{\zeta}\right).$

Integrating in space and using the above estimates gives, whenever
$E(t)<E^{\prime}$,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}E(t)$
$\displaystyle\leq\frac{1}{2}\int_{\mathbb{R}}\left|\zeta_{t}\right|\left|\frac{W_{0}^{\prime}(|\zeta|)}{|\zeta|}\bar{\zeta}-\frac{W_{0}^{\prime}(|\hat{\zeta}|)}{|\hat{\zeta}|}\bar{\hat{\zeta}}\right|\>\mathrm{d}x$
$\displaystyle\quad+\frac{1}{2}\int_{\mathbb{R}}\left|\bar{\zeta}_{t}\right|\left|\frac{W_{0}^{\prime}(|\zeta|)}{|\zeta|}\zeta-\frac{W_{0}^{\prime}(|\hat{\zeta}|)}{|\hat{\zeta}|}\hat{\zeta}\right|\>\mathrm{d}x$
$\displaystyle\leq\|\zeta_{t}(t)\|_{2}\left(\|W_{0}^{\prime}(|\zeta(t)|)\|_{2}+\|W_{0}^{\prime}(|\hat{\zeta}(t)|)\|_{2}\right)$
$\displaystyle\leq\sqrt{E(t)}\left(\sqrt{k_{E^{\prime}}\|W_{0}(|\zeta(t)|)\|_{1}}+\sqrt{k_{E^{\prime}}\|W_{0}(|\hat{\zeta}(t)|)\|_{1}}\right)$
(3.34) $\displaystyle\leq 2\sqrt{k_{E^{\prime}}}E^{\prime}$

which yields

(3.35) $E(t)\leq E+2\sqrt{k_{E^{\prime}}}E^{\prime}t.$

Thus for
$T\leq\frac{1}{2\sqrt{k_{E^{\prime}}}}\left(1-\frac{E}{E^{\prime}}\right)$ we
have that the solution map $\Phi:\hat{\zeta}\mapsto\zeta$ of (3.24) maps
smooth functions in $\mathcal{D}_{T,E^{\prime}}$ to smooth functions in
$\mathcal{D}_{T,E^{\prime}}$.

To prove that $\Phi$ is a contraction we will need further estimates on the
integrals in (3.25). For $\zeta_{1},\zeta_{2}\in\mathbb{C}$ with
$0<|\zeta_{1}|<|\zeta_{2}|<1$ we have

$\displaystyle\left|\frac{W^{\prime}(|\zeta_{1}|)}{|\zeta_{1}|}\zeta_{1}-\frac{W^{\prime}(|\zeta_{2}|)}{|\zeta_{2}|}\zeta_{2}\right|$
$\displaystyle\leq\left|\frac{W^{\prime}(|\zeta_{1}|)}{|\zeta_{1}|}\left(\zeta_{1}-\frac{|\zeta_{1}|}{|\zeta_{2}|}\zeta_{2}\right)+\frac{\zeta_{2}}{|\zeta_{2}|}\left(W^{\prime}(|\zeta_{1}|)-W^{\prime}(|\zeta_{2}|)\right)\right|$
(3.36)
$\displaystyle\leq\left|\frac{W^{\prime}(|\zeta_{1}|)}{|\zeta_{1}|}\right||\zeta_{1}-\zeta_{2}|+\sup_{\zeta\in(\zeta_{1},\zeta_{2})}\left|W^{\prime\prime}(\zeta)\right|\left|\zeta_{1}-\zeta_{2}\right|$

Then for $(\zeta_{1},\zeta_{1t}),(\zeta_{2},\zeta_{2t})\in\mathcal{D}_{T,E}$
we get

(3.37)
$\left|\frac{W^{\prime}(|\zeta_{1}(t,x)|)}{|\zeta_{1}(t,x)|}\zeta_{1}(t,x)-\frac{W^{\prime}(|\zeta_{2}(t,x)|)}{|\zeta_{2}(t,x)|}\zeta_{2}(t,x)\right|\leq(L_{E}^{\prime}+L_{E}^{\prime\prime})|\zeta_{1}(t,x)-\zeta_{2}(t,x)|.$

Hence

$\displaystyle\left|\frac{1}{2c}\int_{x-c(t-s)}^{x+c(t-s)}\frac{W_{0}^{\prime}(|\zeta_{1}(s,y)|)}{|\zeta_{1}(s,y)|}\zeta_{1}(s,y)-\frac{W_{0}^{\prime}(|\zeta_{2}(s,y)|)}{|\zeta_{2}(s,y)|}\zeta_{2}(s,y)\>\mathrm{d}y\right|$
(3.38)
$\displaystyle\qquad\leq(t-s)(L_{E}^{\prime}+L_{E}^{\prime\prime})\|\zeta_{1}(s)-\zeta_{2}(s)\|_{L^{\infty}(\mathbb{R})},$
$\displaystyle\left|\frac{\partial}{\partial
x}\frac{1}{2c}\int_{x-c(t-s)}^{x+c(t-s)}\frac{W^{\prime}(|\zeta_{1}(s,y)|)}{|\zeta_{1}(s,y)|}\zeta_{1}(s,y)-\frac{W^{\prime}(|\zeta_{2}(s,y)|)}{|\zeta_{2}(s,y)|}\zeta_{2}(s,y)\>\mathrm{d}y\right|$
(3.39) $\displaystyle\qquad\leq
2(L_{E}^{\prime}+L_{E}^{\prime\prime})\|\zeta_{1}(s)-\zeta_{2}(s)\|_{L^{\infty}(\mathbb{R})}.$

We will need $L^{2}(\mathbb{R})$-estimates in $x$ as well. For
$(\zeta_{1},\zeta_{1t}),(\zeta_{2},\zeta_{2t})\in\mathcal{D}_{T,E}$ we get
from (3.37) and (3.29)

$\displaystyle\quad\left\|\int_{x-c(t-s)}^{x+c(t-s)}\frac{W_{0}^{\prime}(|\zeta_{1}(s,y)|)}{|\zeta_{1}(s,y)|}\zeta_{1}(s,y)-\frac{W_{0}^{\prime}(|\zeta_{2}(s,y)|)}{|\zeta_{2}(s,y)|}\zeta_{2}(s,y)\>\mathrm{d}y\right\|_{2}$
(3.40)
$\displaystyle\leq(L_{E}^{\prime}+L_{E}^{\prime\prime})2c(t-s)\|\zeta_{1}(s,\cdot)-\zeta_{2}(s,\cdot)\|_{2}.$

To prove that $\Phi$ is a contraction let
$\hat{\zeta}_{1},\hat{\zeta}_{2}\in\mathcal{D}_{T,E^{\prime}}\cap
C^{2}([0,T]\times\mathbb{R})$ with coinciding initial data
$(\zeta_{0},\zeta_{t0})$, and denote
$\zeta_{1}=\Phi(\hat{\zeta}_{1}),\zeta_{2}=\Phi(\hat{\zeta}_{2})$. Then by
(3), (3), (3), and (3.25), we have for $p=2,\infty$,

(3.41)
$\displaystyle\sup_{t\in[0,T]}\|\zeta_{1}(t)-\zeta_{2}(t)\|_{L^{p}(\mathbb{R})}$
$\displaystyle\leq\left(L_{E^{\prime}}^{\prime}+L_{E^{\prime}}^{\prime\prime}\right)\int_{0}^{t}(t-s)\|\hat{\zeta}_{1}(s)-\hat{\zeta}_{2}(s)\|_{p}\>\mathrm{d}s,$
(3.42)
$\displaystyle\sup_{t\in[0,T]}\|\zeta_{1,t}(t)-\zeta_{2,t}(t)\|_{L^{p}(\mathbb{R})}$
$\displaystyle\leq\left(L_{E^{\prime}}^{\prime}+L_{E^{\prime}}^{\prime\prime}\right)\int_{0}^{t}\|\hat{\zeta}_{1}(s)-\hat{\zeta}_{2}(s)\|_{p}\>\mathrm{d}s,$
(3.43)
$\displaystyle\sup_{t\in[0,T]}\|\zeta_{1,x}(t)-\zeta_{2,x}(t)\|_{L^{p}(\mathbb{R})}$
$\displaystyle\leq\frac{1}{c}\left(L_{E^{\prime}}^{\prime}+L_{E^{\prime}}^{\prime\prime}\right)\int_{0}^{t}\|\hat{\zeta}_{1}(s)-\hat{\zeta}_{2}(s)\|_{p}\>\mathrm{d}s.$

We have that

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(W_{0}(|\zeta_{1}|)-W_{0}(|\zeta_{2}|)\right)$
$\displaystyle=W_{0}^{\prime}(|\zeta_{1}|)\frac{\bar{\zeta}_{1}\zeta_{1,t}+\zeta_{1}\bar{\zeta}_{1,t}}{2|\zeta_{1}|}$
$\displaystyle\quad-
W_{0}^{\prime}(|\zeta_{2}|)\frac{\bar{\zeta}_{2}\zeta_{2,t}+\zeta_{2}\bar{\zeta}_{2,t}}{2|\zeta_{2}|}$
$\displaystyle=\left(W_{0}^{\prime}(|\zeta_{1}|)-W_{0}^{\prime}(|\zeta_{2}|)\right)\frac{\bar{\zeta}_{1}\zeta_{1,t}+\zeta_{1}\bar{\zeta}_{1,t}}{2|\zeta_{1}|}$
(3.44)
$\displaystyle\quad+W_{0}^{\prime}(|\zeta_{2}|)\left(\frac{\bar{\zeta}_{1}\zeta_{1,t}+\zeta_{1}\bar{\zeta}_{1,t}}{2|\zeta_{1}|}-\frac{\bar{\zeta}_{2}\zeta_{2,t}+\zeta_{2}\bar{\zeta}_{2,t}}{2|\zeta_{2}|}\right),$

Further note that

$\displaystyle\left(\frac{\bar{\zeta}_{1}\zeta_{1,t}+\zeta_{1}\bar{\zeta}_{1,t}}{2|\zeta_{1}|}-\frac{\bar{\zeta}_{2}\zeta_{2,t}+\zeta_{2}\bar{\zeta}_{2,t}}{2|\zeta_{2}|}\right)$
$\displaystyle=\frac{1}{|\zeta_{2}|}\bigg{[}\frac{1}{2}\left(|\zeta_{2}|-|\zeta_{1}|\right)(\frac{\bar{\zeta}_{1}}{|\zeta_{1}|}\zeta_{1,t}+\frac{\bar{\zeta}_{1}}{|\zeta_{1}|}\bar{\zeta}_{1,t})$
$\displaystyle\quad+\frac{1}{2}\big{(}(\bar{\zeta}_{1}-\bar{\zeta}_{2})\zeta_{1,t}+\bar{\zeta}_{2}(\zeta_{1,t}-\zeta_{2,t})$
(3.45)
$\displaystyle\quad+(\zeta_{1}-\zeta_{2})\bar{\zeta}_{1,t}+\zeta_{2}(\bar{\zeta}_{1,t}-\bar{\zeta}_{2,t})\big{)}\bigg{]},$

and hence

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left\|W_{0}(|\zeta_{1}|)-W_{0}(|\zeta_{2}|)\right\|_{1}$
$\displaystyle\leq\int_{\mathbb{R}}\left|W_{0}^{\prime}(|\zeta_{1}|)-W_{0}^{\prime}(|\zeta_{2}|)\right|\left|\frac{\bar{\zeta}_{1}\zeta_{1,t}+\zeta_{1}\bar{\zeta}_{1,t}}{2|\zeta_{1}|}\right|\>\mathrm{d}x$
$\displaystyle\quad+2\int_{\mathbb{R}}\left|\frac{W_{0}^{\prime}(|\zeta_{2}|)}{|\zeta_{2}|}\right|\left|\zeta_{1}-\zeta_{2}\right|\left|\zeta_{1,t}\right|\>\mathrm{d}x$
$\displaystyle\quad+\int_{\mathbb{R}}\left|\frac{W_{0}^{\prime}(|\zeta_{2}|)}{|\zeta_{2}|}\right|\left|\zeta_{1,t}-\zeta_{2,t}\right||\zeta_{2}|\>\mathrm{d}x$
(3.46)
$\displaystyle\leq\left(L_{E^{\prime}}^{\prime\prime}+2L_{E^{\prime}}^{\prime}\right)\sqrt{E^{\prime}}\|\zeta_{1}-\zeta_{2}\|_{2}+\sqrt{k_{E^{\prime}}E^{\prime}}\|\zeta_{1,t}-\zeta_{2,t}\|_{2}.$

Hence for $T$ small enough $\Phi$ is a contraction on the subspace of
$\mathcal{D}_{T,E^{\prime}}$ consisting of $\zeta\in
C^{2}([0,T]\times\mathbb{R})$. Thus there is a unique fixed point in
$\mathcal{D}_{T,E^{\prime}}$ which is a strong solution in the sense of (3).
The limit may not be $C^{2}$, however. We want to prove that energy is
actually conserved. From the construction of the metric on
$\mathcal{D}_{T,E^{\prime}}$ we see that the energy converges. A more careful
investigation of (3) reveals that

$\displaystyle E(t)-E(0)$
$\displaystyle=\frac{1}{2}\int_{0}^{t}\int_{\mathbb{R}}\frac{W_{0}^{\prime}(|\hat{\zeta}(s,y)|)}{|\hat{\zeta}(s,y)|}\bar{\hat{\zeta}}(s,y)(\hat{\zeta}_{t}(s,y)-\zeta_{t}(s,y))$
(3.47)
$\displaystyle\qquad+\frac{W_{0}^{\prime}(|\hat{\zeta}(s,y)|)}{|\hat{\zeta}(s,y)|}\hat{\zeta}(s,y)(\bar{\hat{\zeta}}_{t}(s,y)-\bar{\zeta}_{t}(s,y))\>\mathrm{d}y\mathrm{d}s.$

Thus the limit conserves energy since the right hand side of (3) tends to zero
as $\zeta$ tends to the fixed point. Then we know that the solution
$(\zeta,\zeta_{t})$ is actually in
$\mathcal{D}_{T,E}\subsetneq\mathcal{D}_{T,E^{\prime}}$, and hence,
$(\zeta(T),\zeta_{t}(T))\in X_{E}$.

We will now show that we can relax the smoothness conditions on the initial
values. From the strong formulation of the equation (3) and estimate (3.37) we
get, from the Grönwall inequality, the estimates

(3.48) $\displaystyle\|\zeta_{1}(t)-\zeta_{2}(t)\|_{p}$
$\displaystyle\leq\left(\|\zeta_{1,0}-\zeta_{2,0}\|_{p}+t\|\zeta_{1,t,0}-\zeta_{2,t,0}\|_{p}\right)e^{\frac{1}{2}(L_{E}^{\prime}+L_{E}^{\prime\prime})t^{2}},$
$\displaystyle\|\zeta_{1,t}(t)-\zeta_{2,t}(t)\|_{p}$
$\displaystyle\leq\|\zeta_{1,t,0}-\zeta_{2,t,0}\|_{p}+c\|\zeta_{1,x,0}-\zeta_{2,x,0}\|_{p}$
(3.49)
$\displaystyle\quad+(L_{E}^{\prime}+L_{E}^{\prime\prime})t\left(\|\zeta_{1,0}-\zeta_{2,0}\|_{p}+t\|\zeta_{1,t,0}-\zeta_{2,t,0}\|_{p}\right)e^{\frac{1}{2}(L_{E}^{\prime}+L_{E}^{\prime\prime})t^{2}},$
$\displaystyle\|\zeta_{1,x}(t)-\zeta_{2,x}(t)\|_{p}$
$\displaystyle\leq\frac{1}{c}\|\zeta_{1,t,0}-\zeta_{2,t,0}\|_{p}+\|\zeta_{1,x,0}-\zeta_{2,x,0}\|_{p}$
(3.50)
$\displaystyle\quad+\frac{L_{E}^{\prime}+L_{E}^{\prime\prime}}{c}t\left(\|\zeta_{1,0}-\zeta_{2,0}\|_{p}+t\|\zeta_{1,t,0}-\zeta_{2,t,0}\|_{p}\right)e^{\frac{1}{2}(L_{E}^{\prime}+L_{E}^{\prime\prime})t^{2}},$

for $p=2,\infty$. Thus the strong solution depends continuously on the initial
data in the metric induced from
$\|(\zeta_{0},\zeta_{t0})\|=\|\zeta_{0}-\zeta^{*}\|_{H^{1}(\mathbb{R})}+\|\zeta_{0}\|_{W^{1,\infty}(\mathbb{R})}\|+\|\zeta_{t0}\|_{L^{2}(\mathbb{R})}+\|\zeta_{t0}\|_{L^{\infty}(\mathbb{R})}+\|W_{0}(|\zeta_{0}|)\|_{L^{1}(\mathbb{R})}$.
Hence we can extend the solution operator to all initial data in $X_{E}$. The
argument can now be repeated to yield a solution in $[0,nT]$ for any
$n\in\mathbb{N}$, and hence there exists a global strong solution that
conserve energy. ∎

In the case where the initial data is smooth, $\zeta_{0}\in C^{2}(\mathbb{R})$
and $\zeta_{t0}\in C^{1}(\mathbb{R})$, the strong solution can be upgraded to
a classical solution $\zeta\in C^{2}([0,\infty)\times\mathbb{R})$.

######  Corollary 3.6.

For initial data $(\zeta_{0},\zeta_{t0})\in X_{E}\cap C^{2}(\mathbb{R})\times
C^{1}(\mathbb{R})$ the strong solution $\zeta$ in Theorem 3.5 belongs to
$C^{2}([0,\infty)\times\mathbb{R})$ and satisfies the equation (3.1)
pointwise.

###### Proof.

We now assume that $\zeta$ is a strong solution given implicitly by (3), and
that $\zeta\in C([0,\infty),W^{1,\infty}(\mathbb{R}))\cap
C^{1}([0,\infty),L^{\infty}(\mathbb{R}))$. Note that we have

(3.51)
$W_{0}(s)=W_{0}(0)+\frac{1}{2}W_{0}^{(2)}(0)s^{2}+\frac{1}{6}W_{0}^{(3)}(0)s^{3}+\int_{0}^{s}\frac{1}{6}W_{0}^{(4)}(u)(s-u)^{3}\>\mathrm{d}u,\\\
$

and thus

(3.52)
$\frac{\mathrm{d}}{\mathrm{d}s}\frac{W_{0}^{\prime}(s)}{s}=\frac{1}{2}W_{0}^{(3)}(0)-\frac{1}{s^{2}}\int_{0}^{s}\frac{1}{2}W_{0}^{(4)}(u)(s-u)^{2}\>\mathrm{d}u+\frac{1}{s}\int_{0}^{s}W_{0}^{(4)}(u)(s-u)\>\mathrm{d}u.$

One can then readily verify that

(3.53)
$\displaystyle\left|\frac{\mathrm{d}}{\mathrm{d}s}\frac{W_{0}^{\prime}(s)}{s}(s)\right|$
$\displaystyle\leq\frac{1}{2}\left|W_{0}^{(3)}(0)\right|+\frac{3}{2}\left\|W_{0}^{(4)}\right\|_{L^{\infty}([0,s])}s,$
(3.54)
$\displaystyle\left|\frac{\mathrm{d}}{\mathrm{d}s}\frac{W_{0}^{\prime}(s)}{s}(s_{1})-\frac{\mathrm{d}}{\mathrm{d}s}\frac{W_{0}^{\prime}(s)}{s}(s_{2})\right|$
$\displaystyle\leq\frac{5}{2}\left\|W_{0}^{(4)}\right\|_{L^{\infty}([0,\max\\{s_{1},s_{2}\\}])}\left|s_{1}-s_{2}\right|.$

Thus $\zeta\mapsto\frac{W_{0}^{\prime}(|\zeta|)}{|\zeta|}\zeta$ is $C^{1}$ in
the open unit disk in two dimensions. Direct computation then yields that for

(3.55)
$I(t,x)=\int_{0}^{t}\int_{x-c(t-\tau)}^{x+c(t-\tau)}\frac{W_{0}^{\prime}(|\zeta(\tau,y)|)}{|\zeta(\tau,y)|}\zeta(\tau,y)\>\mathrm{d}y\mathrm{d}\tau$

the first derivatives are

$\displaystyle\frac{\partial}{\partial t}I(t,x)$
$\displaystyle=c\int_{0}^{t}\frac{W_{0}^{\prime}(|\zeta(\tau,x+c(t-\tau))|)}{|\zeta(\tau,x+c(t-\tau))|}\zeta(\tau,x+c(t-\tau))$
(3.56)
$\displaystyle\qquad+\frac{W_{0}^{\prime}(|\zeta(\tau,x-c(t-\tau))|)}{|\zeta(\tau,x-c(t-\tau))|}\zeta(\tau,x-c(t-\tau))\>\mathrm{d}\tau,$
$\displaystyle\frac{\partial}{\partial x}I(t,x)$
$\displaystyle=\int_{0}^{t}\frac{W_{0}^{\prime}(|\zeta(\tau,x+c(t-\tau))|)}{|\zeta(\tau,x+c(t-\tau))|}\zeta(\tau,x+c(t-\tau))$
(3.57)
$\displaystyle\qquad-\frac{W_{0}^{\prime}(|\zeta(\tau,x-c(t-\tau))|)}{|\zeta(\tau,x-c(t-\tau))|}\zeta(\tau,x-c(t-\tau))\>\mathrm{d}\tau,$

which are continuous since $\zeta$ is continuous by assumption. Hence
$\zeta\in C^{1}([0,\infty)\times\mathbb{R})$. Similarly the second derivatives
are

$\displaystyle\frac{\partial^{2}}{\partial t^{2}}I(t,x)$
$\displaystyle=2c\frac{W_{0}^{\prime}(|\zeta(t,x)|)}{|\zeta(t,x)|}\zeta(t,x)$
$\displaystyle\quad+c^{2}\int_{0}^{t}\frac{\partial}{\partial
x}\frac{W_{0}^{\prime}(|\zeta(\tau,x+c(t-\tau))|)}{|\zeta(\tau,x+c(t-\tau))|}\zeta(\tau,x+c(t-\tau))$
(3.58) $\displaystyle\qquad-\frac{\partial}{\partial
x}\frac{W_{0}^{\prime}(|\zeta(\tau,x-c(t-\tau))|)}{|\zeta(\tau,x-c(t-\tau))|}\zeta(\tau,x-c(t-\tau))\>\mathrm{d}\tau,$
$\displaystyle\frac{\partial^{2}}{\partial t\partial x}I(t,x)$
$\displaystyle=c\int_{0}^{t}\frac{\partial}{\partial
x}\frac{W_{0}^{\prime}(|\zeta(\tau,x+c(t-\tau))|)}{|\zeta(\tau,x+c(t-\tau))|}\zeta(\tau,x+c(t-\tau))$
(3.59) $\displaystyle\qquad+\frac{\partial}{\partial
x}\frac{W_{0}^{\prime}(|\zeta(\tau,x-c(t-\tau))|)}{|\zeta(\tau,x-c(t-\tau))|}\zeta(\tau,x-c(t-\tau))\>\mathrm{d}\tau,$
$\displaystyle\frac{\partial^{2}}{\partial x^{2}}I(t,x)$
$\displaystyle=\int_{0}^{t}\frac{\partial}{\partial
x}\frac{W_{0}^{\prime}(|\zeta(\tau,x+c(t-\tau))|)}{|\zeta(\tau,x+c(t-\tau))|}\zeta(\tau,x+c(t-\tau))$
(3.60) $\displaystyle\qquad-\frac{\partial}{\partial
x}\frac{W_{0}^{\prime}(|\zeta(\tau,x-c(t-\tau))|)}{|\zeta(\tau,x-c(t-\tau))|}\zeta(\tau,x-c(t-\tau))\>\mathrm{d}\tau,$

which are continuous. Thus $\zeta$ given by

(3.61)
$\zeta(t,x)=\frac{1}{2}\left(\zeta(t,x+ct)+\zeta(t,x-ct)\right)+\frac{1}{2c}\int_{x-ct}^{x+ct}\zeta_{t0}(y)\>\mathrm{d}y-\frac{1}{2c}I(t,x),$

is a sum of $C^{2}([0,\infty)\times\mathbb{R})$ functions and is thus
$C^{2}([0,\infty)\times\mathbb{R})$. We get

$\displaystyle\frac{\partial^{2}}{\partial
t^{2}}\zeta(t,x)-c^{2}\frac{\partial^{2}}{\partial t^{2}}\zeta(t,x)$
$\displaystyle=-\frac{\partial^{2}}{\partial
t^{2}}\frac{1}{2c}I(t,x)+c^{2}\frac{\partial^{2}}{\partial
x^{2}}\frac{1}{2c}I(t,x)$ (3.62)
$\displaystyle=-\frac{W_{0}^{\prime}(|\zeta(t,x)|)}{|\zeta(t,x)|}\zeta(t,x)$

pointwise. ∎

We will only give a local existence theorem for (2.7). The main difficulty for
(2.7) compared to (3.1) is that (2.7) is quasilinear, that is the terms
$c(\psi)^{2}\psi_{xx}$ and $c(\psi)^{2}s_{xx}$ are nonlinear. The method used
in the proof of Theorem 3.5 relied heavily on (3.1) being semilinear.

######  Theorem 3.7.

There exists a unique short time solution of (2.7) for smooth initial data
$(\psi_{0},\psi_{t0},s_{0},s_{t0})$, where $s_{0}$ is bounded away from $0$
and the energy is finite.

###### Proof.

Local existence, uniqueness, and continuous dependence on initial data follows
from the standard approach taken to semigroups of nonlinear evolution
equations. Here we will diverge from the standard semigroup approach by
considering linear operators on Banach spaces, and instead solve the linear
equations by characteristics.

In particular, inspired by [19], the system (2.7) can be rewritten as a
quasilinear symmetric hyperbolic system and the results in [18] applied.
Indeed, introduce the variables

(3.63a) $\displaystyle\phi$ $\displaystyle=\psi_{t},$ (3.63b) $\displaystyle
v$ $\displaystyle=s_{t},$ (3.63c) $\displaystyle\omega$
$\displaystyle=c(\psi)\psi_{x},$ (3.63d) $\displaystyle r$
$\displaystyle=c(\psi)s_{x},$

the system (2.7) can be written as a quasilinear symmetric hyperbolic system

(3.64a) $\displaystyle\psi_{t}$ $\displaystyle=\phi,$ (3.64b) $\displaystyle
s_{t}$ $\displaystyle=v,$ (3.64c) $\displaystyle\phi_{t}-c(\psi)\omega_{x}$
$\displaystyle=-\frac{2}{s}\left(\phi v-\omega
r\right)+\frac{c^{\prime}(\psi)}{c(\psi)}r^{2},$ (3.64d) $\displaystyle
v_{t}-c(\psi)r_{x}$
$\displaystyle=s\left(\phi^{2}-\omega^{2}\right)+\frac{c^{\prime}(\psi)}{c(\psi)}\omega
r-W_{0}^{\prime}(s),$ (3.64e) $\displaystyle\omega_{t}-c(\psi)\phi_{x}$
$\displaystyle=\frac{c^{\prime}(\psi)}{c(\psi)}\phi\omega,$ (3.64f)
$\displaystyle r_{t}-c(\psi)v_{x}$
$\displaystyle=\frac{c^{\prime}(\psi)}{c(\psi)}\phi r.$

We introduce the Banach space $X=W^{1,\infty}(\mathbb{R},\mathbb{R}^{6})$, and
let

$\displaystyle X_{E,L}=\bigg{\\{}$ $\displaystyle
U=(\psi,s,\phi,v,\omega,r)\in X\mid\|U\|_{W^{1,\infty}}\leq
L,\left\|\frac{1}{s}\right\|_{L^{\infty}(\mathbb{R})}\leq L,$ (3.65)
$\displaystyle\int_{\mathbb{R}}\frac{1}{2}s^{2}\left(\phi^{2}+\omega^{2}\right)+\frac{1}{2}\left(v^{2}+r^{2}\right)+W_{0}(s)\>\mathrm{d}x\leq
E\bigg{\\}}.$

Note that for elements of $X_{E_{L}}$ the variable $s$ is of definite sign
since $s$ is continuous and $\frac{1}{s}$ is bounded. We define our solution
space $\mathcal{D}_{T,E,L}$ as follows

(3.66) $\mathcal{D}_{T,E,L}=C([0,T],X_{E,L})\cap
C^{1}([0,T],L^{\infty}(\mathbb{R},\mathbb{R}^{6})).$

We equip $\mathcal{D}_{T,E,L}$ with the metric

(3.67) $d_{\mathcal{D}_{T,E,L}}(U_{1},U_{2})=\sup_{0\leq t\leq
T}\|U_{1}(t)-U_{2}(t)\|_{W^{1,\infty}(\mathbb{R},\mathbb{R}^{6})}+\sup_{0\leq
t\leq T}\|U_{1,t}(t)-U_{2,t}(t)\|_{L^{\infty}(\mathbb{R},\mathbb{R}^{6})},$

which renders $\mathcal{D}_{T,E,L}$ a complete metric space. Formally, we can
formulate (3.64) as

(3.68) $U_{t}-c(\psi)AU_{x}=F(U),$

where $A$ is a constant symmetric hyperbolic matrix with eigenvalues $-1,0,1$.
We want to show existence of short time solutions from a fix point argument.
To be able to make a contraction we will further restrict our space for
approximate solutions to

(3.69) $\mathcal{D}_{T,E,L}^{\mathrm{lin}}=\left\\{U\in\mathcal{D}_{T,E,L}\mid
U(t)\in
W^{2,\infty}(\mathbb{R},\mathbb{R}^{6}),\|U_{xx}(t)\|_{L^{\infty}(\mathbb{R},\mathbb{R}^{6})}\leq
L\right\\}.$

Let $0<E<E^{\prime},0<L<L^{\prime}$ and $U_{0}\in X_{E,L}$ with
$\|U_{0xx}\|_{\infty}\leq L$ be the initial data, then for any
$\hat{U}\in\mathcal{D}_{T,E^{\prime},L^{\prime}}^{\mathrm{lin}}$ with
$\hat{U}(0)=U_{0}$, the linear system of transport equations

(3.70) $U_{t}-c(\hat{\psi})AU_{x}=F(\hat{U})$

can be solved by characteristics. Define the backward characteristics
$x_{\pm}$ at time $\tau$ from point $(t,x)$ by

(3.71) $\frac{\mathrm{d}}{\mathrm{d}\tau}x_{\pm}(\tau;t,x)=\mp
c\bigg{(}\hat{\psi}(\tau,x_{\pm}(\tau;t,x))\bigg{)},$

and note that $x_{\pm}(\tau;t,x)=x_{\pm}(\tau;s,x_{\pm}(s;t,x))$ and
$x_{\pm}(t;t,x))=x$. Then, by taking the derivative with respect to $s$, we
get the following

(3.72) $\frac{\partial}{\partial t}x_{\pm}(\tau;t,x)\mp
c(\hat{\psi}(\tau,x_{\pm}(\tau;t,x)))\frac{\partial}{\partial
x}x_{\pm}(\tau;t,x)=0.$

Thus

(3.73) $\displaystyle|x_{\pm}(\tau_{1};t,x)-x_{\pm}(\tau_{2};t,x)|$
$\displaystyle\leq\|c\|_{\infty}|\tau_{2}-\tau_{1}|,$ (3.74)
$\displaystyle|x_{\pm}(\tau;t,x_{1})-x_{\pm}(\tau;t,x_{2})|$
$\displaystyle\leq\mathrm{exp}\left\\{\left\|\frac{1}{c}\right\|_{\infty}L^{\prime}|t-\tau|\right\\}|x_{1}-x_{2}|,$
(3.75) $\displaystyle|x_{+}(\tau;t,x)-x_{-}(\tau;t,x)|$ $\displaystyle\leq
2\|c\|_{\infty}|t-\tau|,$ (3.76) $\displaystyle\left|\frac{\partial}{\partial
t}x_{\pm}(\tau;t,x)\right|$
$\displaystyle\leq\|c\|_{\infty}\>\mathrm{exp}\left\\{\left\|\frac{1}{c}\right\|_{\infty}L^{\prime}|t-\tau|\right\\}.$

We need to show that the Duhamel operator
$DF(\hat{U})(t)=\int_{0}^{t}T_{\tau}(t-\tau)F(\hat{U}(\tau))\>\mathrm{d}\tau$,
where $T_{\tau}$ is the solution operator of
$U_{t}(t)-c(\psi(t+\tau))AU_{x}(t)=0$, satisfies $DF(\hat{U})\in
C([0,\infty),W^{2,\infty}(\mathbb{R},\mathbb{R}^{6}))$, and
$\|DF(\hat{U}_{1})(t)-DF(\hat{U}_{2})(t)\|_{W^{1,\infty}(\mathbb{R})}\leq
C(t,E^{\prime},L^{\prime})\|\hat{U}_{1}-\hat{U}_{2}\|_{W^{1,\infty}(\mathbb{R})}$
for
$\hat{U}_{1},\hat{U}_{2}\in\mathcal{D}_{T,E^{\prime},L^{\prime}}^{\mathrm{lin}}$.
Since

(3.77) $\frac{\partial}{\partial
x}\frac{c^{\prime}(\psi)}{c(\psi)}=\frac{c^{\prime\prime}(\psi)c(\psi)-c^{\prime}(\psi)^{2}}{c(\psi)^{2}}\psi_{x},$

and $F$ is a smooth function, we have that for
$\hat{U}\in\mathcal{D}_{T,E^{\prime},L^{\prime}}^{\mathrm{lin}}$ there is

(3.78) $\|F(\hat{U}(t))\|_{W^{2,\infty}(\mathbb{R})}\leq
C_{E^{\prime},L^{\prime}}.$

Hence the Duhamel operator satisfies
$\|DF(\hat{U})(t)\|_{W^{2,\infty}(\mathbb{R})}\leq
C_{E^{\prime},L^{\prime}}t$. Note that

(3.79)
$\frac{\partial}{\partial\tau}\left(x_{\pm}^{1}(\tau;t,x_{1})-x_{\pm}^{2}(\tau;t,x_{2})\right)=\mp\left(c(\hat{\psi}_{1}(\tau,x_{\pm}^{1}(\tau;t,x_{1})))-c(\hat{\psi}_{2}(\tau,x_{\pm}^{2}(\tau;t,x_{2})))\right),$

and thus

$\displaystyle|x_{\pm}^{1}(\tau;t,x_{1})-x_{\pm}^{2}(\tau;t,x_{2})|$
$\displaystyle\leq|x_{1}-x_{2}|e^{\|c^{\prime}\|_{\infty}L^{\prime}(t-\tau)}$
(3.80)
$\displaystyle\quad+\frac{1}{L^{\prime}}\left(e^{\|c^{\prime}\|_{\infty}L^{\prime}(t-\tau)}-1\right)\|\hat{\psi}_{1}-\hat{\psi}_{2}\|_{L^{\infty}([0,t],L^{\infty}(\mathbb{R}))},$
(3.81) $\displaystyle\left|\frac{\partial}{\partial
x}x_{\pm}^{1}(\tau;t,x)-\frac{\partial}{\partial
x}x_{\pm}^{2}(\tau;t,x)\right|$ $\displaystyle\leq
C_{L^{\prime}}\mathrm{exp}\left\\{k_{L^{\prime}}(t-\tau)\right\\}\sup_{0\leq\tau\leq
t}\|\hat{\psi}_{1}-\hat{\psi}_{2}\|_{W^{1,\infty}(\mathbb{R})},$

where $C_{L^{\prime}}$ and $k_{L^{\prime}}$ depends on $c$ and $L^{\prime}$
only. Thus, since $F$ is a smooth function and
$\|\hat{U}_{1}\|_{W^{2,\infty}(\mathbb{R})},\|\hat{U}_{2}\|_{W^{2,\infty}(\mathbb{R})}\leq
L^{\prime}$,

$\displaystyle\left|\int_{0}^{t}F_{j}(\hat{U}_{1})(\tau,x^{1}_{\pm}(\tau;t,x))-F_{j}(\hat{U}_{2})(\tau,x^{2}_{\pm}(\tau;t,x))\>\mathrm{d}\tau\right|$
$\displaystyle\leq\left|\int_{0}^{t}F_{j}(\hat{U}_{1})(\tau,x^{1}_{\pm}(\tau;t,x))-F_{j}(\hat{U}_{2})(\tau,x^{1}_{\pm}(\tau;t,x))\>\mathrm{d}\tau\right|$
$\displaystyle\quad+\left|\int_{0}^{t}F_{j}(\hat{U}_{2})(\tau,x^{1}_{\pm}(\tau;t,x))-F_{j}(\hat{U}_{2})(\tau,x^{2}_{\pm}(\tau;t,x))\>\mathrm{d}\tau\right|$
$\displaystyle\leq\left|\int_{0}^{t}F_{j}(\hat{U}_{1})(\tau,x^{1}_{\pm}(\tau;t,x))-F_{j}(\hat{U}_{2})(\tau,x^{1}_{\pm}(\tau;t,x))\>\mathrm{d}\tau\right|$
$\displaystyle\quad+t\bar{C}(L^{\prime},E^{\prime})\|\hat{\psi}_{1}-\hat{\psi}_{2}\|_{L^{\infty}([0,t],L^{\infty}(\mathbb{R})}$
(3.82) $\displaystyle\leq
t\tilde{C}(L^{\prime},E^{\prime})\|\hat{U}_{1}-\hat{U}_{2}\|_{L^{\infty}([0,t],L^{\infty}(\mathbb{R},\mathbb{R}^{6})}$

To compute $\frac{\partial}{\partial
x}\left(DF(\hat{U}_{1})(t)-DF(\hat{U}_{2})(t)\right)$ we estimate

$\displaystyle\quad\left|\frac{\partial}{\partial
x}\int_{0}^{t}F_{j}(\hat{U}_{1})(\tau,x_{\pm}^{1}(\tau;t,x))-F_{j}(\hat{U}_{2})(\tau,x_{\pm}^{2}(\tau;t,x))\>\mathrm{d}\tau\right|$
$\displaystyle\leq\left|\int_{0}^{t}\left(\nabla_{U}F_{j}(\hat{U}_{1})\frac{\partial\hat{U}_{1}}{\partial
x}-\nabla_{U}F_{j}(\hat{U}_{2})\frac{\partial\hat{U}_{2}}{\partial
x}\right)(\tau,x_{\pm}^{1}(\tau;t,x))\frac{\partial}{\partial
x}x_{\pm}^{1}(\tau;t,x)\>\mathrm{d}\tau\right|$
$\displaystyle\quad+\left|\int_{0}^{t}\nabla_{U}F_{j}(\hat{U}_{2})\frac{\partial\hat{U}_{2}}{\partial
x}(\tau,x_{\pm}^{1}(\tau;t,x))\frac{\partial}{\partial
x}\left(x_{\pm}^{1}(\tau;t,x)-x_{\pm}^{2}(\tau;t,x)\right)\>\mathrm{d}\tau\right|$
$\displaystyle\quad+\bigg{|}\int_{0}^{t}\left(\nabla_{U}F_{j}(\hat{U}_{2})\frac{\partial\hat{U}_{2}}{\partial
x}(\tau,x_{\pm}^{1}(\tau;t,x))-\nabla_{U}F_{j}(\hat{U}_{2})\frac{\partial\hat{U}_{2}}{\partial
x}(\tau,x_{\pm}^{2}(\tau;t,x))\right)$
$\displaystyle\qquad\qquad\times\frac{\partial}{\partial
x}x_{\pm}^{2}(\tau;t,x)\>\mathrm{d}\tau\bigg{|}$ $\displaystyle\leq
C_{E^{\prime},L^{\prime}}\left(\mathrm{exp}\left\\{\left\|\frac{1}{c}\right\|_{\infty}L^{\prime}t\right\\}-1\right)\sup_{0\leq\tau\leq
t}\|\hat{U}_{1}(\tau)-\hat{U}_{2}(\tau)\|_{W^{1,\infty}(\mathbb{R})}$
$\displaystyle\quad+C_{E^{\prime},L^{\prime}}\>(\mathrm{exp}\left\\{K_{L^{\prime}}t\right\\}-1)\sup_{0\leq\tau\leq
t}\|\hat{\psi}_{1}(\tau)-\hat{\psi}_{2}(\tau)\|_{W^{1,\infty}(\mathbb{R})}$
(3.83)
$\displaystyle\quad+C_{E^{\prime},L^{\prime}}\>(\mathrm{exp}\left\\{K_{L^{\prime}}t\right\\}-1)\sup_{0\leq\tau\leq
t}\|\hat{\psi}_{1}(\tau)-\hat{\psi}_{2}(\tau)\|_{L^{\infty}(\mathbb{R})}.$

That is, we have that

(3.84) $\|U_{1}(t)-U_{2}(t)\|_{W^{1,\infty}(\mathbb{R})}\leq
C_{L^{\prime},E^{\prime}}(\mathrm{exp}\\{k_{L^{\prime}}t\\}-1)\|\hat{U}_{1}-\hat{U}_{2}\|_{W^{1,\infty}(\mathbb{R})},$

and hence for $T$ small enough we have that

(3.85) $\sup_{0\leq t\leq
T}\|U_{1}(t)-U_{2}(t)\|_{W^{1,\infty}(\mathbb{R})}<\sup_{0\leq t\leq
T}\|\hat{U}_{1}(t)-\hat{U}_{2}(t)\|_{W^{1,\infty}(\mathbb{R})}.$

Since $U$ is continuously differentiable, we can do similar estimates for
$\sup_{0\leq t\leq T}\|U_{1,t}(t)-U_{2,t}(t)\|_{L^{\infty}(\mathbb{R})}$
directly from the linear system. We need that the energy of the solution is
bounded by $E^{\prime}$. Recall that
$E(t)=\int_{\mathbb{R}}\mathcal{E}(t)\>\mathrm{d}x=\int_{\mathbb{R}}s(t)^{2}(\phi(t)^{2}+\omega(t)^{2})+v(t)^{2}+r(t)^{2}+W_{0}(s(t))\>\mathrm{d}x$,
and thus

$\displaystyle\frac{\partial}{\partial t}\mathcal{E}-\frac{\partial}{\partial
x}c(\hat{\psi})\left(s^{2}\phi\omega+vr\right)$
$\displaystyle=s\phi^{2}\hat{v}+v\hat{s}\hat{\phi}^{2}-\frac{2}{\hat{s}}s^{2}\phi\hat{\phi}\hat{v}$
$\displaystyle\quad+s\omega^{2}\hat{v}-v\hat{s}\hat{\omega}^{2}+W_{0}^{\prime}(s)\hat{v}-W_{0}^{\prime}(\hat{s})v$
$\displaystyle\quad+\frac{c^{\prime}(\hat{\psi})}{c(\hat{\psi})}\left(s^{2}\omega\hat{\omega}(\hat{\phi}-\phi)+\hat{r}(\hat{\phi}r-\phi\hat{r})+v(\hat{\omega}\hat{r}-\omega
r)\right)$ (3.86)
$\displaystyle\quad+\frac{c^{\prime}(\hat{\psi})}{c(\hat{\psi})}2s\phi\left(\frac{s}{\hat{s}}\hat{\omega}\hat{r}-\omega
c(\hat{\psi})s_{x}\right).$

Integration yields

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}E(t)$ $\displaystyle\leq
2{L^{\prime}}^{2}E(t)+2{L^{\prime}}^{2}E^{\prime}+2{L^{\prime}}^{2}\sqrt{E^{\prime}}\sqrt{E(t)}+\|W_{0}^{\prime}(s)\hat{v}\|_{1}+\|W_{0}^{\prime}(\hat{s})v\|_{1}$
$\displaystyle\quad+K\left({L^{\prime}}^{3}\sqrt{E(t)}\sqrt{E^{\prime}}+4{L^{\prime}}\sqrt{E(t)}\sqrt{E^{\prime}}\right)$
(3.87)
$\displaystyle\quad+K\left(2{L^{\prime}}^{3}\sqrt{E(t)}\sqrt{E^{\prime}}+2E(t)\|c\|_{\infty}L^{\prime}\right)$

By Definition 3.1 we have that

$\displaystyle\|W_{0}^{\prime}(s)\hat{v}\|_{1}$
$\displaystyle=\left\|W_{0}^{\prime}(s)\right\|_{2}\left\|\hat{v}\right\|_{2}$
$\displaystyle\leq\sqrt{k_{E(t)}\|W_{0}(s)\|_{1}}\sqrt{E^{\prime}}$ (3.88)
$\displaystyle\leq\sqrt{k_{E^{\prime}}E(t)}\sqrt{E^{\prime}},$

and hence for $t$ small enough, depending on $E^{\prime}$, $L^{\prime}$,
$W_{0}$, and $c$ only, $E(t)\leq E^{\prime}$.

Hence the sequence $U_{n}$, where $U_{n+1}$ is the solution to the linear
equation with $U_{n}$ inserted, is a Cauchy sequence in
$\mathcal{D}_{T,E^{\prime},L^{\prime}}$, and hence there is a solution of the
equations. Moreover, the limit will satisfy
$\mathcal{E}_{t}-(c(\psi)^{2}\mathcal{F})_{x}=0$, and thus conserves energy. ∎

## References

  * [1] A. Bressan and A. Constantin, Global solutions of the Hunter–Saxton equation. SIAM J. Math. Anal., 37,996–1026 (2005).
  * [2] A. Bressan, G. Chen, and Q. Zhang, Unique conservative solutions to a variational wave equation. Arch. Ration. Mech. Anal., 217, 1069–1101 (2015).
  * [3] A. Bressan, H. Holden, and X. Raynaud, Lipschitz metric for the Hunter–Saxton equation. J. Math. Pures Appl., 94, 68–92 (2010).
  * [4] A. Bressan, P. Zhang, and Y. Zheng, Asymptotic variational wave equations. Arch. Ration. Mech. Anal., 183, 163–185 (2007).
  * [5] A. Bressan and Y. Zheng, Conservative solutions to a nonlinear variational wave equation. Commun. Math. Phys., 266, 471–497 (2006).
  * [6] J. Carrillo, K. Grunert, and H. Holden, A Lipschitz metric for the Hunter–Saxton equation. Comm. Partial Differential Equations, 44, 309–334 (2019).
  * [7] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 71, 1661–1664 (1993).
  * [8] H.-H. Dai and M. V. Pavlov, Transformations for the Camassa–Holm equation, its high-frequency limit and the Sinh–Gordon equation. J. Phys. Soc. Japan, 67 3655–3657 (1998).
  * [9] C. M. Dafermos, Generalized characteristics and the Hunter–Saxton equation. J. Hyperbolic Differ. Equ., 8, 159–168, (2011).
  * [10] Ericksen, Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal.,113, 97–120 (1991).
  * [11] R. T. Glassey, J. K. Hunter, and Y. Zheng, Singularities in a nonlinear variational wave equation. J. Diff. Eqs., 129, 49–78 (1996).
  * [12] K. Grunert and A. Nordli, Existence and Lipschitz stability for $\alpha$-dissipative solutions of the two-component Hunter–Saxton system. J. Hyperbolic Differ. Equ., 15, 559–597 (2018).
  * [13] H. Holden and X. Raynaud, Global semigroup of conservative solutions of the nonlinear variational wave equation. Arch. Ration. Mech. Anal., 201, 871–964 (2011).
  * [14] J. K. Hunter and R. Saxton, Dynamics of director fields. SIAM J. Appl. Math., 56, 1498–1521 (1991).
  * [15] J. K. Hunter and Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation. Phys. D., 79, 361–386 (1994).
  * [16] J. K. Hunter and Y. Zheng, On a nonlinear hyperbolic variational equation. I. Global existence of weak solutions. Arch. Ration. Mech. Anal., 129, 305–353 (1995).
  * [17] J. K. Hunter and Y. Zheng, On a nonlinear hyperbolic variational equation. II. The zero-viscosity and dispersion limits. Arch. Ration. Mech. Anal., 129, 355–383 (1995).
  * [18] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal., 58, 181–205 (1975).
  * [19] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations. Lecture Notes in Mathematics, 448, 25–70 (1975).
  * [20] B. Khesin and G. Misiołek, Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math., 176, 116–144 (2003).
  * [21] F. Leslie, Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal., 28, 265–283 (1968).
  * [22] A. Nordli, A Lipschitz metric for conservative solutions of the two-component Hunter–Saxton system. Methods Appl. Anal., 23, 215–232 (2016).
  * [23] M. V. Pavlov, The Gurevich–Zybin system. J. Phys A, 38, 3823–3840 (2005).
  * [24] R. Saxton, Dynamic instability of the liquid crystal director. In W. B. Lindquist (ed.), Current Progress in Hyperbolic Systems, 100, 325–330 (1989).
  * [25] M. Wunsch, The generalized Hunter–Saxton system. SIAM J. Math. Anal., 42, 4951–4960 (2010).
  * [26] P. Zhang and Y. Zheng, On the existence and uniqueness of solutions to an asymptotic equation of a variational wave equation. Acta Math. Sin. (Engl. Ser.), 15, 115–130 (1999).
  * [27] P. Zhang and Y. Zheng, Existence and uniqueness of solutions of an asymptotic equation arising from a variational wave equation with general data. Arch. Ration. Mech. Anal., 155, 49–83 (2000).


# Relaxation exponents of OTOCs and overlap with local Hamiltonians

Vinitha Balachandran  Science, Mathematics and Technology Cluster, Singapore
University of Technology and Design, 8 Somapah Road, 487372 Singapore  Dario
Poletti  Science, Mathematics and Technology Cluster, Singapore University of
Technology and Design, 8 Somapah Road, 487372 Singapore  EPD Pillar, Singapore
University of Technology and Design, 8 Somapah Road, 487372 Singapore
MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit UMI 3654,
Singapore  Centre for Quantum Technologies, National University of Singapore
117543, Singapore  The Abdus Salam International Centre for Theoretical
Physics, Strada Costiera 11, 34151 Trieste, Italy

###### Abstract

OTOC has been used to characterize the information scrambling in quantum
systems. Recent studies showed that local conserved quantities play a crucial
role in governing the relaxation dynamics of OTOC in non-integrable systems.
In particular, slow scrambling of OTOC is seen for observables that has an
overlap with local conserved quantities. However, an observable may not
overlap with the Hamiltonian, but with the Hamiltonian elevated to an exponent
larger than one. Here, we show that higher exponents correspond to faster
relaxation, although still algebraic, and with exponents that can increase
indefinitely. Our analytical results are supported by numerical experiments.

##  I Introduction

For generic many body quantum systems, information initially encoded in a few
local degrees of freedom can spread in time over the entire space accessible.
This process is called information scrambling and can be characterized by out
of time ordered correlators (OTOCs) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21]. For quantum systems with a classical
limit, OTOCs can be mapped to Lyapunov exponents [22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33]. Because of this, OTOCs have been applied to understand
the thermalization in many body quantum systems [3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13].

Recent studies have pointed out the relevance of local conserved quantities in
the relaxation dynamics of OTOCs [34, 35, 36, 37, 34, 38, 39, 40]. In
particular, in [39] it was shown that the emergence of algebraic relaxation
can stem from locality of the Hamiltonian, i.e., the ensuing presence of a
Lieb-Robinson bound [41], and the eigestate thermalization hypothesis (ETH)
[42, 43]. Importantly, with the approach developed in [39] it was also
possible to show that algebraic relaxation of the OTOC is typical.

In the scenarios considered until now, the operators in the OTOC would have
non-zero overlap with the Hamiltonian or a local conserved quantity (i.e.
total magnetization). Here we investigate how the relaxation dynamics would be
affected if the operators in the OTOC, e.g. $A$, do not overlap with the
Hamiltonian $H$, but only with one of its powers, i.e. ${\rm tr}(AH)=0$, but
${\rm tr}(AH^{m})\neq 0$ for $m$ being an integer larger than one. We show
that depending on the exponent $m$ at which the overlap becomes non-zero, we
expect an algebraic relaxation of the OTOC in time with an exponent
proportional to $m$. To obtain this result, we also show the relation between
the first non-zero derivative of the diagonals of an operator in the energy
basis, with the exponent $m$ at which ${\rm tr}(AH^{m})\neq 0$.

The paper is organized as follows. In section II we introduce the definition
of OTOCs as well as explain the relaxation dynamics of OTOCs from the
knowledge of the matrix elements of the observables in the eigenenergy basis.
In Sec. III we show analytically our main result, i.e. that any different
exponents can emerge in the relaxation of the OTOC, depending on the order at
which the operators in the OTOC overlap with the Hamiltonian. Our numerical
results are presented in IV. We draw our conclusions in Sec. V.

##  II Emergence of slow scrambling

###  II.1 Definition

Consider the infinite-temperature out-of-time-ordered correlator (OTOC)
between two local observables $A$ and $B$ defined as

$\displaystyle O^{AB}(t)$
$\displaystyle=\frac{1}{2}\langle[A(t),B][A(t),B]^{\dagger}\rangle$ (1)

where $A(t)=U^{\dagger}AU$ is the time evolved operator $A$ due to the unitary
evolution $U=\mathcal{T}e^{-{\rm i}\int_{0}^{t}H(\tau)d\tau}$ from the time-
ordered integration of the (generically) time-dependent Hamiltonian $H(t)$.
Expanding the commutators, we can rewrite Eq.(1) as

$\displaystyle\frac{1}{2}\langle[A(t),B][A(t),B]^{\dagger}\rangle$
$\displaystyle=\langle B^{2}A(t)^{2}\rangle-\langle A(t)BA(t)B\rangle$
$\displaystyle=G^{AB}(t)-F^{AB}(t),$

where $G^{AB}(t)=\langle B^{2}A(t)^{2}\rangle$ is the time-ordered part of
OTOC and $F^{AB}(t)=\langle A(t)BA(t)B\rangle$ is the not-time-ordered part.
We consider only unitary and Hermitian observables for which $G(t)=1$ and
hence we restrict ourselves to $F(t)$ in the remaining part. Taking energy
eigenstates as the basis of the Hilbert space, the time evolution of OTOC can
be written in the eigenenergy basis $|p\rangle$ as

$\displaystyle F^{AB}(t)=\frac{1}{\mathcal{V}}\sum_{p,q,k,l}e^{{\rm
i}(E_{p}-E_{q}+E_{k}-E_{l})t}A_{pq}B_{qk}A_{kl}B_{lp}$ (3)

where $E_{p}$ is the eigenenergy, $A_{pq}=\langle p|A|q\rangle$ and
$B_{qk}=\langle q|B|k\rangle$. We work in units for which $\hbar=1$.

As $t\rightarrow\infty$, dominant terms in the above expression are those for
which $E_{p}-E_{q}+E_{k}-E_{l}=0$. Hence, for generic systems [44, 45], the
infinite-time value of $F^{AB}(t)$ are given by

$\displaystyle F^{AB}(\infty)=\frac{1}{\mathcal{V}}\bigg{(}\sum_{p}$
$\displaystyle A_{pp}^{2}B_{pp}^{2}+\sum_{p,q\neq
p}\big{(}A_{pp}B_{pq}A_{qq}B_{qp}$ $\displaystyle+$ $\displaystyle
A_{pq}B_{qq}A_{qp}B_{pp}\big{)}\bigg{)}.$ (4)

Eq. (II.1) highlights the importance of diagonal elements of $A$ and $B$ in
the eigenenergy basis in the infinite-time value of OTOC. Indeed, a non-zero
diagonal element in $A$ or $B$ is necessary to guarantee a non-zero value of
$F^{AB}(\infty)$.

###  II.2 Conditions for algebraic relaxation of OTOC

Two sufficient conditions for the emergence of algebraic relaxation of OTOC
[39, 40] are

  * •

a Lieb-Robinson bound (or even an algebraic spreading of correlation which
occurs in systems with power-law interactions),

  * •

algebraic scaling of infinite-time value of the OTOC with the system size.

In local and bounded Hamiltonians, the speed of propagation of the
correlations is limited by Lieb-Robinson bound [41, 46]. Hence, an accurate
description of the evolution of OTOC of a thermodynamically large system, can
be obtained simply considering a finite portion of it. Assuming that the
system is maximally scrambled within the region of size $L$, the decay of
$F^{AB}_{L=\infty}(t)$ is bounded by the Lieb-Robinson velocity $v_{\rm LR}$
as

$\displaystyle F^{AB}_{L=\infty}(t)\approx F^{AB}_{L=s\;v_{\rm
LR}\;t}(\infty),$ (5)

where $s$ is a real number larger than $1$. Hence, $L$ increases with time and
is a time-dependent quantity. Therefore, the scaling of $F^{AB}_{L}(\infty)$
is crucial to predict the bound for the relaxation of OTOC. In particular,
when $F^{AB}_{L}(\infty)$ decays algebraically with the system size, e.g.
$F^{AB}_{L}(\infty)\propto L^{-\alpha}$, then the OTOC of the thermodynamic
size system cannot decay faster than algebraically in time, or more precisely
from Eq.(5) one can write that it cannot be faster than

$\displaystyle F^{AB}_{L=\infty}(t)\propto\frac{1}{t^{\alpha}}$ (6)

because $L=s\;v_{\rm LR}\;t$.

The actual decay of the OTOC may even be slower, for example considering cases
in which the system goes through prethermalization [47] or in which the system
is many-body localized [48]. However, the relaxation cannot be faster, hence
the OTOC will have a slow, non-exponential relaxation. A comprehensive
analysis of this is presented in [39].

##  III Generic algebraic relaxation in short-ranged systems

###  III.1 Estimate of the infinite time value of OTOC

In this section, we show how to obtain the approximate value of the infinite-
time, finite-size, OTOC $F^{AB}_{L}(\infty)$

$\displaystyle F^{AB}_{L}(\infty)=$
$\displaystyle\frac{1}{\mathcal{V}}\sum_{p}A_{pp}^{2}B_{pp}^{2}+\frac{1}{\mathcal{V}}\sum_{p,q\neq
p}A_{pp}A_{qq}|B_{pq}|^{2}$ $\displaystyle+\frac{1}{\mathcal{V}}\sum_{p,q\neq
p}B_{pp}B_{qq}|A_{pq}|^{2}$ $\displaystyle\approx$
$\displaystyle\frac{1}{\mathcal{V}}\sum_{p}A_{pp}^{2}B_{pp}^{2}+\frac{1}{\mathcal{V}}\sum_{p}A_{pp}A_{pp}\left[(BB^{\dagger})_{pp}-B_{pp}^{2}\right]$
$\displaystyle+\frac{1}{\mathcal{V}}\sum_{p}B_{pp}B_{pp}\left[(AA^{\dagger})_{pp}-A_{pp}^{2}\right]$
$\displaystyle\approx$
$\displaystyle\frac{1}{\mathcal{V}}\sum_{p}A_{pp}^{2}B_{pp}^{2}+\frac{1}{\mathcal{V}}\sum_{p}\left[{\rm
tr}{(BB^{\dagger})}-B_{pp}^{2}\right]A_{pp}^{2}$
$\displaystyle+\frac{1}{\mathcal{V}}\sum_{p}\left[{\rm
tr}{(AA^{\dagger})}-A_{pp}^{2}\right]B_{pp}^{2}$ $\displaystyle\approx$
$\displaystyle\frac{1}{\mathcal{V}}\sum_{p}A_{pp}^{2}B_{pp}^{2}+\frac{1}{\mathcal{V}}\sum_{p}\left[1-B_{pp}^{2}\right]A_{pp}^{2}$
$\displaystyle+\frac{1}{\mathcal{V}}\sum_{p}\left[1-A_{pp}^{2}\right]B_{pp}^{2}$
$\displaystyle\approx$
$\displaystyle\frac{1}{\mathcal{V}}\sum_{p}\left[A_{pp}^{2}+B_{pp}^{2}-A_{pp}^{2}B_{pp}^{2}\right]$
$\displaystyle\approx$
$\displaystyle\frac{1}{\mathcal{V}}\sum_{p}\left[A_{pp}^{2}+B_{pp}^{2}\right],$
(7)

where we have used steps similar to [45, 39], and a similar discussion can be
found in [40]. Thus, the main contribution of the infinite-time finite-size
OTOC comes from the $A_{pp}^{2}$ and $B_{pp}^{2}$ terms which we will be
discussing in the following.

###  III.2 Structure of the diagonal elements

In short, the diagonal element $A_{pp}$ can be approximated by a function of
eigenenergy $E_{p}$

$\displaystyle|A_{pp}-f_{A}(E_{p}/L)|$ $\displaystyle\leq e^{-(\Omega(L))},$
(8)

where $f_{A}(E_{p}/L)$ can be expanded as

$\displaystyle f_{A}(E_{p}/L)$
$\displaystyle=f_{A}(0)+f^{(1)}_{A}(0)E_{p}/L+\frac{1}{2}f^{(2)}_{A}(0)E_{p}^{2}/L^{2}+...$
$\displaystyle=\sum_{q}\frac{f_{A}^{(q)}}{q!}\left(\frac{E_{p}}{L}\right)^{q}$
(9)

with $f_{A}^{(q)}$ being the $q-$th derivative of $f_{A}$. We also note that,
using Lemma 1 in [45] one can write

$\displaystyle\frac{1}{\mathcal{V}}\sum_{p}E_{p}^{q}$ $\displaystyle=\langle
H^{q}\rangle=O(L^{q/2}).$ (10)

In [45] it was shown that, for traceless operators $f_{A}(0)=0$, and if
$f^{(1)}_{A}(0)\neq 0$ then we can write

$\displaystyle{\rm tr}{(AH)}$
$\displaystyle=\frac{1}{\mathcal{V}}\sum_{p}A_{pp}E_{p}$
$\displaystyle\approx\frac{1}{\mathcal{V}L}\sum_{p}E_{p}^{2}f^{(1)}_{A}(0)$
$\displaystyle\approx\frac{\langle H^{2}\rangle}{L}f^{(1)}_{A}(0)$ (11)

and thus

$\displaystyle f^{(1)}_{A}(0)$ $\displaystyle\approx\frac{{\rm
tr}{(AH)}L}{\langle H^{2}\rangle}.$ (12)

Hence, the first derivative of a local observable $A$ is independent of the
system size. From Eq, (12) we get

$\displaystyle F_{L}^{AB}(\infty)$
$\displaystyle\approx\frac{1}{\mathcal{V}}\sum_{p}(A_{pp}^{2}+B_{pp}^{2})$
$\displaystyle\approx\frac{1}{\mathcal{V}}\sum_{p}\frac{E_{p}^{2}}{L^{2}}\left[\left(f^{(1)}_{A}(0)\right)^{2}+\left(f^{(1)}_{B}(0)\right)^{2}\right]$
$\displaystyle\approx\frac{1}{\mathcal{V}}\sum_{p}\frac{E_{p}^{2}}{L^{2}}\frac{\left[{\rm
tr}{(AH)}^{2}+{\rm tr}{(BH)}^{2}\right]L^{2}}{\langle H^{2}\rangle^{2}}$
$\displaystyle\approx\frac{{\rm tr}{(AH)}^{2}+{\rm tr}{(BH)}^{2}}{\langle
H^{2}\rangle}$ $\displaystyle\propto\frac{1}{L}.$ (13)

The last step stems from the fact that ${\rm tr}(AH)$ and ${\rm tr}(BH)$ are
independent of the system size, while $\langle H^{2}\rangle\propto L$ from Eq.
(10).

If ${\rm tr}(AH)=0$ but, for instance, ${\rm tr}(AH^{p})\neq 0$ only for
$p\geq p_{c}$ then one can generalize the previous result. Considering
$f_{A}^{(q)}(0)$ as the smalles non-zero derivative of $f_{A}$ at zero energy
(with the same parity as $p_{c}$), then we can write

$\displaystyle{\rm tr}(AH^{p_{c}})$
$\displaystyle=\sum_{n}\frac{f_{A}^{(q)}}{q!}\frac{E_{n}^{q+p_{c}}}{L^{q}}$
(14)

which implies that

$\displaystyle f_{A}^{(q)}$ $\displaystyle=\frac{q!{\rm
tr}(AH^{p_{c}})L^{q}}{\langle H^{p_{c}+q}\rangle}.$ (15)

Now, if $q<p_{c}$ then $f_{A}^{(q)}$ would decay as $L^{-(p_{c}-q)/2}$, which
implies that they are $0$, and the non-size dependent $f_{A}^{(q)}(0)$ would
occur exactly at $q=p_{c}$. This implies that the first non-zero derivative of
$f_{A}(0)$ is the $p_{c}-$th one. Thus, when ${\rm tr}(AH^{p})\neq 0$ only for
$p\geq p_{c}$ we can write

$\displaystyle F_{L}^{AB}(\infty)$
$\displaystyle\approx\frac{1}{\mathcal{V}}\sum_{n}\left(\frac{E_{n}}{L}\right)^{2p_{c}}\left[\left(f^{(p_{c})}_{A}(0)\right)^{2}+\left(f^{(p_{c})}_{B}(0)\right)^{2}\right]$
$\displaystyle\approx\frac{1}{\mathcal{V}}\sum_{n}\left(\frac{E_{n}}{L}\right)^{2p_{c}}\frac{\left[{\rm
tr}{(AH^{p_{c}})}^{2}+{\rm tr}{(BH^{p_{c}})}^{2}\right]L^{2p_{c}}}{\langle
H^{2p_{c}}\rangle^{2}}$ $\displaystyle\approx\frac{{\rm
tr}{(AH^{p_{c}})}^{2}+{\rm tr}{(BH^{p_{c}})}^{2}}{\langle H^{2p_{c}}\rangle}$
$\displaystyle\propto\frac{1}{L^{p_{c}}}.$ (16)

Building on Eq.(III.2), and combining it with the Lieb-Robinson bound
$L=s\;v_{\rm LR}\;t$ we can thus guarantee that $F^{AB}$ cannot relax faster
than $t^{-p_{c}}$. Furthermore, for systems in which correlations mostly
spread diffusively, i.e. proportional to $t^{1/2}$, we can can expect $F^{AB}$
to relax as $t^{-p_{c}/2}$. Hence, the structure of the diagonal elements of
the observables, and which is the first non-zero derivative at $0$ energy,
i.e., which is the first exponent of the Hamiltonian that has non-zero overlap
with the operators $A$ and $B$ considered, play an important role in the
relaxation dynamics of the OTOC in the system. This is numerically verified in
detail in the following section.

##  IV Results

###  IV.1 Model

We consider a prototypical non-integrable model, the tilted Ising chain with
Hamiltonian

$\displaystyle H$
$\displaystyle=\sum_{l=1}^{L-1}J_{z}\sigma^{z}_{l}\sigma^{z}_{l+1}+\sum_{l=1}^{L}(h_{x}\sigma^{x}_{l}+h_{z}\sigma^{z}_{l}),$
(17)

where $J_{z}$ is the coupling constant in the z direction, while $h_{x}$ and
$h_{z}$ are the transverse and the longitudinal field strengths. The model is
integrable when either $h_{x}=0$ or $h_{z}=0$. This can be verified by
studying the level spacing statistics which typically follows a Poisson
distribution for integrable systems and a Wigner-Dyson distribution for non-
integrable ones [49, 50]. In particular, one can calculate
$\delta_{n}=E_{n+1}-E_{n}$, the level spacing between two consecutive energy
levels $E_{n}$ and $E_{n+1}$ within a single symmetry sector, define the ratio
$r_{n}=\text{max}(\delta_{n},\delta_{n+1})/\text{min}(\delta_{n},\delta_{n+1})$
and take an average $r=\sum_{n}r_{n}/N$ where N is the number of energy level
differences considered. For a Poisson distribution, $r$ can be computed
analytically and it gives $r=2\ln 2-1\approx 0.386$, while for a Wigner-Dyson
distribution $r$ can be evaluated numerically to be $r\approx 0.529$ [51]. In
the current work, we use parameters $J_{z}=1$, $h_{z}=0.809$ and $h_{x}=0.9$
which result in $r\approx 0.53$ already for a system size of $L=12$ spins.

###  IV.2 Observables and structure of their diagonal elements

To span over a variety of different structures, and to have operators $A$
which have ${\rm tr}(AH^{p})\neq 0$ only for $p\leq p_{c}$ with $p_{c}$ which
can be different from $1$, we analyze both single-site and multi-site
observables in our study. In particular, we consider the following four types
of observables,

$\displaystyle{\rm single-site}\rightarrow$
$\displaystyle\;\;\sigma_{l}^{\alpha}$ (18) $\displaystyle{\rm double-
site}\rightarrow$ $\displaystyle\;\;\sigma_{l}^{\alpha}\sigma_{l+1}^{\alpha}$
(19) $\displaystyle{\rm triple-site}\rightarrow$
$\displaystyle\;\;\sigma_{l-1}^{\alpha}\sigma_{l}^{\alpha}\sigma_{l+1}^{\alpha}$
(20) $\displaystyle{\rm quadruple-site}\rightarrow$
$\displaystyle\;\;\sigma_{l-2}^{\alpha}\sigma_{l-1}^{\alpha}\sigma_{l}^{\alpha}\sigma_{l+1}^{\alpha}$
(21)

where $\alpha=x,y$ or $z$. The diagonal elements of these operators in the
eigenbasis of Hamiltonian Eq. (17) are shown in Fig. 1. The left column is for
$\alpha=x$, center column for $\alpha=y$ and right column for $\alpha=z$. The
rows are for increasing range of operators from top to bottom, with the top
row for single-site operators and the fourth row for four-site operators. In
all the panels, the dashed lines represent the expected algebraic energy
dependence of $f_{A}$ near energy zero from Sec. III.2. We note that these
fits are evaluated directly from calculating $f^{(n)}_{A}(0)$ with Eq.(15)
along with eigenenergies $E_{n}$ for the system Hamiltonian in Eq. (17) with
no fitting parameters.

For single site observables $A=\sigma^{x}_{l}$ and $\sigma^{z}_{l}$, for the
non-integrable Ising chain ${\rm tr}(AH)\neq 0$ and so $f^{(1)}_{A}(0)\neq 0$.
However, with $A=\sigma^{y}_{l}$, ${\rm tr}(AH^{n})=0$ for any $n$. Hence, we
expect a linear variation of the diagonal elements of $\sigma^{x}_{l}$ and
$\sigma^{z}_{l}$ with energy density $E_{n}/L$ and a flat profile for
$\sigma^{y}_{l}$. This can be seen in Fig. 1(a-c). To conform our analytical
predictions, we plot $f^{(1)}_{A}(0)E_{n}/L$ where $f^{(1)}_{A}(0)$ is
calculated explicity form Eq.12.

The two-site observables $A=\sigma^{\alpha}_{L/2}\sigma^{\alpha}_{L/2+1}$, are
shown in the panels (d-f). For $A=\sigma^{x}_{L/2}\sigma^{x}_{L/2+1}$, ${\rm
tr}(AH^{2})\neq 0$ whereas ${\rm tr}(AH)=0$ and, as predicted in Sec. III.2,
we thus observe that $f_{A}$ can be fitted by a parabola
$f^{(2)}_{A}(0)E_{n}^{2}/(2!L^{2})$ indicated by the dashed black lines. Since
${\rm tr}(AH)\neq 0$ for $A=\sigma^{z}_{L/2}\sigma^{z}_{L/2+1}$, we see a
linear scaling of $A_{nn}$ with $E_{n}/L$. For
$A=\sigma^{y}_{L/2}\sigma^{y}_{L/2+1}$ observable, ${\rm tr}(AH^{m})\neq 0$
for $m\geq 3$. Hence, we see a cubic structure of the diagonal elements with a
fitting of the form $f^{(3)}_{A}(0)E_{n}^{3}/(3!L^{3})$.

We also consider triple-site observables
$A=\sigma^{\alpha}_{L/2-1}\sigma^{\alpha}_{L/2}\sigma^{\alpha}_{L/2+1}$. These
are depicted in the panels (g-i). Here, ${\rm tr}(AH^{m})\neq 0$ for $m\geq 3$
for $\sigma^{x}_{l}$ observables and we clearly see a cubic structure for the
diagonal elements that can be fitted with lines of the form
$f^{(3)}_{A}(0)E_{n}^{3}/(3!L^{3})$. Since there are no diagonal elements for
any power of $H$ for the $\sigma^{y}$ observable, a flat profile is seen. With
the $\sigma^{z}$ observable, a parabolic structure is seen since ${\rm
tr}(AH^{2})\neq 0$ whereas ${\rm tr}(AH)=0$. This is also nicely fitted by
$f^{(2)}_{A}(0)E_{n}^{2}/(2!L^{2})$ in the panel (i).

For the four site observable we study
$A=\sigma^{\alpha}_{L/2-2}\sigma^{\alpha}_{L/2-1}\sigma^{\alpha}_{L/2}\sigma^{\alpha}_{L/2+1}$.
For
$A=\sigma^{x}_{L/2-2}\sigma^{x}_{L/2-1}\sigma^{x}_{L/2}\sigma^{x}_{L/2+1}$, a
quartic structure can be seen as ${\rm tr}(AH^{m})\neq 0$ only for $m\geq 4$.
This is fitted by $f^{(4)}_{A}(0)E_{n}^{4}/(4!L^{4})$ (black dashed lines).
With $\sigma^{y}_{l}$ observables, the expected structure is hexic (polynomial
of sixth degree) because ${\rm tr}(AH^{m})\neq 0$ only for $m\geq 6$. Though
it is less clear, we fit it with the expected scaling
$f^{(6)}_{A}(0)E_{n}^{6}/(6!L^{6})$ using the black dashed lines. For
$\sigma^{z}$ observable, we find a parabolic structure in accordance with our
prediction as ${\rm tr}(AH^{2})\neq 0$ whereas ${\rm tr}(AH)=0$. Since we
consider systems of size $L=14$, the results in Fig. 1(j-l) are partially
affected by finite-size effects. Despite this, the numerics are aligned with
our theoretical predictions.

Figure 1: Diagonal elements of the observable in the energyeigen basis for
single site observables $A=\sigma_{L/2}^{\alpha}$ panel(a-c), double site
observables $A=\sigma_{L/2}^{\alpha}\sigma_{L/2+1}^{\alpha}$ (d-f), triple
site observables
$A=\sigma_{L/2-1}^{\alpha}\sigma_{L/2}^{\alpha}\sigma_{L/2+1}^{\alpha}$ (g-i)
and quadruple site observables
$A=\sigma_{L/2-2}^{\alpha}\sigma_{L/2-1}^{\alpha}\sigma_{L/2}^{\alpha}\sigma_{L/2+1}^{\alpha}$
(j-l). Left panels are for $\sigma^{x}_{l}$ ($\alpha=x$) observables, middle
panels are for $\sigma^{y}_{l}$ ($\alpha=y$) observables and right panels are
for $\sigma^{z}_{l}$ ($\alpha=z$) observables. Dashed lines are the lowest
order fits in the Taylor expansion of the observable in Eq. (III.2). Here,
$L=14$, $J_{z}=1$, $h_{x}=0.9$ and $h_{z}=0.809$.

To summarize this section, we observe clearly that the diagonal elements of
operators can have a very different dependence as a function of energy near
zero. In particular we have numerically verified the prediction that
$A_{nn}\sim 1/L^{p}$ where $p$ is the lowest positive integer such that ${\rm
tr}(AH^{p})\neq 0$.

###  IV.3 Scaling of the infinite time value of OTOC

Figure 2: Infinite time values of OTOC corresponding to the single site
observables with $A=\sigma_{L/2-1}^{\alpha}$, $B=\sigma_{L/2}^{\alpha}$
panel(a), double site observables with
$A=\sigma_{L/2-2}^{\alpha}\sigma_{L/2-1}^{\alpha}$,
$B=\sigma_{L/2}^{\alpha}\sigma_{L/2+1}^{\alpha}$ (b), triple site observables
with
$A=\sigma_{L/2-3}^{\alpha}\sigma_{L/2-2}^{\alpha}\sigma_{L/2-1}^{\alpha}$,
$B=\sigma_{L/2}^{\alpha}\sigma_{L/2+1}^{\alpha}\sigma_{L/2+2}^{\alpha}$ (c)
and quadruple site observables with
$A=\sigma_{L/2-4}^{\alpha}\sigma_{L/2-3}^{\alpha}\sigma_{L/2-2}^{\alpha}\sigma_{L/2-1}^{\alpha}$,
$B=\sigma_{L/2}^{\alpha}\sigma_{L/2+1}^{\alpha}\sigma_{L/2+2}^{\alpha}\sigma_{L/2+3}^{\alpha}$
(d). Green lines with diamonds are for observables involving only
$\sigma^{z}_{l}$ $(\alpha=z)$ operators, red lines with circles for
$\sigma^{x}_{l}$ $(\alpha=x)$ and blue lines with stars for $\sigma^{y}_{l}$
$(\alpha=y)$ operators respectively. Black-dashed, brown-dotted and grey-
dashed dotted lines are the fits for $\sigma^{z}_{l}$, $\sigma^{x}_{l}$ and
$\sigma^{y}_{l}$ observables. Here, $J_{z}=1$, $h_{x}=0.9$ and $h_{z}=0.809$.

In Fig. 2 we show numerical confirmation that given the minimum positive
integer $p_{c}$ such that ${\rm tr}(AH^{p_{c}})\neq 0$ or ${\rm
tr}(BH^{p_{c}})\neq 0$, then $F^{AB}_{L=\infty}(t)\propto 1/L^{p_{c}}$. In
each of the panels we show how the infinite time value of the OTOC
$F^{AB}_{L}(t=\infty)$ varies as a function of the system size $L$. In the
different panels we will focus on single-site, panel (a), two-site, panel (b),
three-site, panel (c) and four-site, panel (d), observables. In each panel the
red line with circles corresponds to $\alpha=x$, blue with stars to $\alpha=y$
and green with diamonds to $\alpha=z$. In panel Fig. 2(a), we plot the
infinite time values of OTOC with single site observables
$A,B=\sigma^{\alpha}_{l}$, where $l=L/2$ for observable $B$ and $l=L/2-1$ for
observable $A$. We see that these observables have $p_{c}=1$, and hence they
follow $1/L$ scaling as shown by dashed line. $\sigma^{y}_{l}$ has no overlap
with any local conserved quantities and hence the diagonal elements as well as
the infinite time values of OTOC are zero. Fig. 2(b) is for double site
observables Eq. (19), where $l=L/2$ for observable $B$ and $l=L/2-2$ for
observable $A$. We compare the numerical results with fitted lines, in
particular with $1/L^{2}$ (dotted), $1/L^{3}$ (dashed dotted lines) and $1/L$
(dashed), respectively corresponding to operators with $p_{c}=2,3$ and $1$. We
note that due to the small value of the overlap of
$\sigma^{y}_{j}\sigma^{y}_{j+1}$ with the Hamiltonian, the expected scaling is
followed only at larger system sizes. In panel (c), we plot the triple site
observables Eq. (20) where $j=L/2-3$ for $A$ and $j=L/2$ for $B$. Fitted lines
are for $1/L^{2}$ and $1/L^{3}$ scalings as expected since $\alpha=2$ and $3$
respectively. Since the diagonal elements of
$\sigma^{y}_{j}\sigma^{y}_{j+1}\sigma^{y}_{j+2}$ are zero, the infinite time
value of the OTOC $F^{AB}_{L}(t=\infty)$ is zero. Panel (d) is for quadruple
site observables Eq. (21) where $j=L/2-4$ for $A$ and $j=L/2$ for $B$. The
expected scalings are $p_{c}=4,6,2$. However, due to the fact that the
observables have a large support at initial time, we see that the correct
scaling of $1/L^{4},1/L^{6},1/L^{2}$ are followed only at large system sizes.

###  IV.4 Dynamics of OTOCs

Figure 3: Time evolution of OTOC corresponding to the single site observables
with $A=\sigma_{L/2-1}^{\alpha}$, $B=\sigma_{L/2}^{\alpha}$ panel(a), double
site observables with $A=\sigma_{L/2-2}^{\alpha}\sigma_{L/2-1}^{\alpha}$,
$B=\sigma_{L/2}^{\alpha}\sigma_{L/2+1}^{\alpha}$ (b), triple site observables
$A=\sigma_{L/2-3}^{\alpha}\sigma_{L/2-2}^{\alpha}\sigma_{L/2-1}^{\alpha}$,
$B=\sigma_{L/2}^{\alpha}\sigma_{L/2+1}^{\alpha}\sigma_{L/2+2}^{\alpha}$ (c)
and quadruple site observables
$A=\sigma_{L/2-4}^{\alpha}\sigma_{L/2-3}^{\alpha}\sigma_{L/2-2}^{\alpha}\sigma_{L/2-1}^{\alpha}$,
$B=\sigma_{L/2}^{\alpha}\sigma_{L/2+1}^{\alpha}\sigma_{L/2+2}^{\alpha}\sigma_{L/2+3}^{\alpha}$
(d).Green lines are for observables involving only $\sigma^{z}_{l}$
$(\alpha=z)$ operators, red lines for $\sigma^{x}_{l}$ $(\alpha=x)$ and blue
lines for $\sigma^{y}_{l}$ $(\alpha=y)$ operators respectively. Black-dashed,
brown-dotted and grey-dashed dotted lines are the fits for $\sigma^{z}_{l}$,
$\sigma^{x}_{l}$ and $\sigma^{y}_{l}$ observables discussed in the text. Here,
$L=14$ for lighter shades and $L=12$ for darker shades and $J_{z}=1$,
$h_{x}=0.9$ and $h_{z}=0.809$.

We study the dynamics of OTOC in Fig. 3, where each panel reflects the same
case analyzed in the corresponding panel of Fig. 2. Green lines are for
observables involving only $\sigma^{z}_{l}$ operators, red lines for
$\sigma^{x}_{l}$ and blue lines for $\sigma^{y}_{l}$ operators respectively.
In these plots we need to study the long-time evolution. We thus need to
disregard initial transients. At the same time, though, our results are
affected by finite size, so we would need to concentrate on long, yet,
intermediate times to evaluate the relaxation of the OTOC over time. Light
shades are for $L=14$ and dark shades for $L=12$. Black-dashed, brown-dotted
and grey-dashed dotted lines are the fits for $\sigma^{z}_{l}$,
$\sigma^{x}_{l}$ and $\sigma^{y}_{l}$ observables. Fig. 3(a) is for single-
site observables, as in Eq. (18). We have already seen that since ${\rm
tr}(OH)\neq 0$, for $(O=A,B)$, then the infinite time value of OTOC
$F^{AB}_{L}(t=\infty)$ scales as $1/L$. From our discussion at the end of Sec.
III.2, we thus expect that $F^{AB}(t)\propto 1/t^{1/2}$, and the numerical
result of the dynamics is well fitted by the black dashed line proportional to
$t^{1/2}$.

In Fig. 3(b) we study the two-site observables of Eq. (19) with $l=L/2$ for
observable $B$ and $l=L/2-2$ for observable $A$. As already discussed, the
lowest order terms that has non zero values in the Taylor expansion for these
observables are $1,2$ and $3$ respectively for
$\sigma^{z}_{l},\sigma^{x}_{l},\sigma^{y}_{l}$ observables. In Fig. 2(b) we
showed the scaling of the infinite-time OTOC for these observables as $1/L$,
$1/L^{2}$, $1/L^{3}$. Here we would thus expect a scaling with time of
$1/t^{1/2}$, $1/t$ and $1/t^{3/2}$ which is shown in the plots by dotted,
dashed dotted and dashed lines. We study the evolution of three-site
observables of Eq. (20) in Fig. 3(c). Here $l=L/2-3$ for $A$ and $l=L/2$ for
$B$. Fitted lines are for $1/t$ and $1/t^{3/2}$ scaling as expected since
${\rm tr}(OH)\neq 0$ for $\sigma^{z}_{l}$ and ${\rm tr}(OH^{2})\neq 0$ for
$\sigma^{x}_{l}$ observable. Panel (d) shows the dynamics for four site
observables with $l=L/2-4$ for $A$ and $l=L/2$ for $B$. The expected scaling
is $1/t$, $1/t^{2}$ and $1/t^{3}$ for $\alpha=z,x$ and $y$ respectively, for
which operators the corresponding critical exponent $p_{c}$ that give non-zero
overlap are $2,4$ and $6$.

##  V Conclusions

OTOCs have been studied as a probe for quantum information scrambling. Slow,
algebraic scrambling has been reported in systems with local conserved
quantities [39, 34, 38, 40].

In this paper we showed that higher the exponent at which ones elevates the
Hamiltonian in order to have a non-zero overlap with the operators in the
OTOC, the faster is the relaxation of the OTOC over time. Furthermore, if
there is an exponent such that the overlap is non-zero, then the relaxation,
even if it appears to be fast, is bounded to be, at the fastest, algebraic,
and only if there is no overlap with any power of the Hamiltonian (or other
conserved quantities), then the relaxation can be exponential.

From our resuls it follows that considering single-site operators in the OTOC,
and a local Hamiltonian with only single site and nearest neighbours term,
relaxation can only take a limited set of exponents. It is thus necessary to
consider operators with larger support, such as two-site, three-site and four-
site operators, to observe a larger variety and magnitude of relaxation
exponents. This, however, leads to the difficulty of study the relaxation
numerically due to more pronounced finite-size effects when studying operators
with larger support. Future developments in numerical methods could help to
test our results for larger systems.

In order to derive these results, we also found a relation between the first
non-zero derivative of the function representing the diagonals of an operator
in the energy basis, and the first non-zero exponent of the Hamiltonian which
has non-zero overlap with the operators of the OTOC. Future works could extend
these results to time-dependent systems with other types of conserved
quantities.

Acknowledgments: We acknowledge C. von Keyserlingk who asked a critical and
insightful question which started this project. The computational work for
this article were partially performed on the National Supercomputing Centre,
Singapore [52]. DP and VB are extremely grateful to G. Casati, to whom this
volume is dedicated, for his guidance and example throughout the many years we
had the fortune to learn from, interact and collaborate with.

## References

  * Witten [1998] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998), arXiv:hep-th/9802150 .
  * Maldacena [1999] J. Maldacena, International Journal of Theoretical Physics 38, 1113–1133 (1999).
  * Hayden and Preskill [2007] P. Hayden and J. Preskill, Black holes as mirrors: quantum information in random subsystems, Journal of High Energy Physics 2007, 120–120 (2007).
  * Sekino and Susskind [2008] Y. Sekino and L. Susskind, Fast scramblers, Journal of High Energy Physics 2008, 065–065 (2008).
  * Shenker and Stanford [2014] S. H. Shenker and D. Stanford, Black holes and the butterfly effect, Journal of High Energy Physics 2014, 10.1007/jhep03(2014)067 (2014).
  * Sachdev and Ye [1993] S. Sachdev and J. Ye, Gapless spin-fluid ground state in a random quantum heisenberg magnet, Physical Review Letters 70, 3339 (1993).
  * Kitaev [2015] A. Kitaev, A simple model of quantum holography, Talks at KITP (2015).
  * Lashkari  _et al._ [2013] N. Lashkari, D. Stanford, M. Hastings, T. Osborne, and P. Hayden, Towards the fast scrambling conjecture, Journal of High Energy Physics 2013, 10.1007/jhep04(2013)022 (2013).
  * Roberts and Stanford [2015] D. A. Roberts and D. Stanford, Diagnosing chaos using four-point functions in two-dimensional conformal field theory, Phys. Rev. Lett. 115, 131603 (2015).
  * Cotler  _et al._ [2017] J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stanford, A. Streicher, and M. Tezuka, Black holes and random matrices, Journal of High Energy Physics 2017, 10.1007/jhep05(2017)118 (2017).
  * Roberts  _et al._ [2015] D. A. Roberts, D. Stanford, and L. Susskind, Localized shocks, Journal of High Energy Physics 2015, 10.1007/jhep03(2015)051 (2015).
  * Hosur  _et al._ [2016] P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Chaos in quantum channels, Journal of High Energy Physics 2016, 10.1007/jhep02(2016)004 (2016).
  * Borgonovi  _et al._ [2019] F. Borgonovi, F. M. Izrailev, and L. F. Santos, Timescales in the quench dynamics of many-body quantum systems: Participation ratio versus out-of-time ordered correlator, Phys. Rev. E 99, 052143 (2019).
  * Li  _et al._ [2017] J. Li, R. Fan, H. Wang, B. Ye, B. Zeng, H. Zhai, X. Peng, and J. Du, Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator, Phys. Rev. X 7, 031011 (2017).
  * Gärttner  _et al._ [2017] M. Gärttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall, J. J. Bollinger, and A. M. Rey, Measuring out-of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet, Nature Physics 13, 781–786 (2017).
  * Landsman  _et al._ [2019] K. A. Landsman, C. Figgatt, T. Schuster, N. M. Linke, B. Yoshida, N. Y. Yao, and C. Monroe, Verified quantum information scrambling, Nature 567, 61–65 (2019).
  * Niknam  _et al._ [2020] M. Niknam, L. F. Santos, and D. G. Cory, Sensitivity of quantum information to environment perturbations measured with a nonlocal out-of-time-order correlation function, Phys. Rev. Research 2, 013200 (2020).
  * Joshi  _et al._ [2020] M. K. Joshi, A. Elben, B. Vermersch, T. Brydges, C. Maier, P. Zoller, R. Blatt, and C. F. Roos, Quantum information scrambling in a trapped-ion quantum simulator with tunable range interactions, Phys. Rev. Lett. 124, 240505 (2020).
  * Blok  _et al._ [2021] M. S. Blok, V. V. Ramasesh, T. Schuster, K. O’Brien, J. M. Kreikebaum, D. Dahlen, A. Morvan, B. Yoshida, N. Y. Yao, and I. Siddiqi, Quantum information scrambling on a superconducting qutrit processor, Phys. Rev. X 11, 021010 (2021).
  * Mi  _et al._ [2021] X. Mi, P. Roushan, C. Quintana, S. Mandra, J. Marshall, C. Neill, F. Arute, K. Arya, J. Atalaya, R. Babbush, J. C. Bardin, R. Barends, A. Bengtsson, S. Boixo, A. Bourassa, M. Broughton, B. B. Buckley, D. A. Buell, B. Burkett, N. Bushnell, Z. Chen, B. Chiaro, R. Collins, W. Courtney, S. Demura, A. R. Derk, A. Dunsworth, D. Eppens, C. Erickson, E. Farhi, A. G. Fowler, B. Foxen, C. Gidney, M. Giustina, J. A. Gross, M. P. Harrigan, S. D. Harrington, J. Hilton, A. Ho, S. Hong, T. Huang, W. J. Huggins, L. B. Ioffe, S. V. Isakov, E. Jeffrey, Z. Jiang, C. Jones, D. Kafri, J. Kelly, S. Kim, A. Kitaev, P. V. Klimov, A. N. Korotkov, F. Kostritsa, D. Landhuis, P. Laptev, E. Lucero, O. Martin, J. R. McClean, T. McCourt, M. McEwen, A. Megrant, K. C. Miao, M. Mohseni, W. Mruczkiewicz, J. Mutus, O. Naaman, M. Neeley, M. Newman, M. Y. Niu, T. E. O’Brien, A. Opremcak, E. Ostby, B. Pato, A. Petukhov, N. Redd, N. C. Rubin, D. Sank, K. J. Satzinger, V. Shvarts, D. Strain, M. Szalay, M. D. Trevithick, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, I. Aleiner, K. Kechedzhi, V. Smelyanskiy, and Y. Chen, Information scrambling in computationally complex quantum circuits (2021), arXiv:2101.08870 [quant-ph] .
  * Braumüller  _et al._ [2021] J. Braumüller, A. H. Karamlou, Y. Yanay, B. Kannan, D. Kim, M. Kjaergaard, A. Melville, B. M. Niedzielski, Y. Sung, A. Vepsäläinen, R. Winik, J. L. Yoder, T. P. Orlando, S. Gustavsson, C. Tahan, and W. D. Oliver, Probing quantum information propagation with out-of-time-ordered correlators (2021), arXiv:2102.11751 [quant-ph] .
  * Rozenbaum  _et al._ [2017] E. B. Rozenbaum, S. Ganeshan, and V. Galitski, Lyapunov exponent and out-of-time-ordered correlator’s growth rate in a chaotic system, Phys. Rev. Lett. 118, 086801 (2017).
  * Hashimoto  _et al._ [2017] K. Hashimoto, K. Murata, and R. Yoshii, Out-of-time-order correlators in quantum mechanics, Journal of High Energy Physics 2017, 10.1007/jhep10(2017)138 (2017).
  * Cotler  _et al._ [2018] J. S. Cotler, D. Ding, and G. R. Penington, Out-of-time-order operators and the butterfly effect, Annals of Physics 396, 318–333 (2018).
  * García-Mata  _et al._ [2018] I. García-Mata, M. Saraceno, R. A. Jalabert, A. J. Roncaglia, and D. A. Wisniacki, Chaos signatures in the short and long time behavior of the out-of-time ordered correlator, Phys. Rev. Lett. 121, 210601 (2018).
  * Chávez-Carlos  _et al._ [2019] J. Chávez-Carlos, B. López-del Carpio, M. A. Bastarrachea-Magnani, P. Stránský, S. Lerma-Hernández, L. F. Santos, and J. G. Hirsch, Quantum and classical lyapunov exponents in atom-field interaction systems, Physical Review Letters 122, 024101 (2019).
  * Fortes  _et al._ [2019] E. M. Fortes, I. García-Mata, R. A. Jalabert, and D. A. Wisniacki, Gauging classical and quantum integrability through out-of-time-ordered correlators, Phys. Rev. E 100, 042201 (2019).
  * Rammensee  _et al._ [2018] J. Rammensee, J. D. Urbina, and K. Richter, Many-body quantum interference and the saturation of out-of-time-order correlators, Phys. Rev. Lett. 121, 124101 (2018).
  * Prakash and Lakshminarayan [2020] R. Prakash and A. Lakshminarayan, Scrambling in strongly chaotic weakly coupled bipartite systems: Universality beyond the ehrenfest timescale, Physical Review B 101, 121108(R) (2020).
  * Bergamasco  _et al._ [2019] P. D. Bergamasco, G. G. Carlo, and A. M. F. Rivas, Out-of-time ordered correlators, complexity, and entropy in bipartite systems, Physical Review Research 1, 033044 (2019).
  * Rozenbaum  _et al._ [2020] E. B. Rozenbaum, L. A. Bunimovich, and V. Galitski, Early-time exponential instabilities in nonchaotic quantum systems, Phys. Rev. Lett. 125, 014101 (2020).
  * Wang  _et al._ [2020] J. Wang, G. Benenti, G. Casati, and W.-g. Wang, Complexity of quantum motion and quantum-classical correspondence: A phase-space approach, Phys. Rev. Research 2, 043178 (2020).
  * Wang  _et al._ [2021] J. Wang, G. Benenti, G. Casati, and W.-g. Wang, Quantum chaos and the correspondence principle, Phys. Rev. E 103, L030201 (2021).
  * Rakovszky  _et al._ [2018] T. Rakovszky, F. Pollmann, and C. W. von Keyserlingk, Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation, Phys. Rev. X 8, 031058 (2018).
  * Nahum  _et al._ [2017] A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev. X 7, 031016 (2017).
  * Nahum  _et al._ [2018] A. Nahum, S. Vijay, and J. Haah, Operator spreading in random unitary circuits, Phys. Rev. X 8, 021014 (2018).
  * von Keyserlingk  _et al._ [2018] C. W. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Operator hydrodynamics, otocs, and entanglement growth in systems without conservation laws, Phys. Rev. X 8, 021013 (2018).
  * Khemani  _et al._ [2018] V. Khemani, A. Vishwanath, and D. A. Huse, Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws, Phys. Rev. X 8, 031057 (2018).
  * Balachandran  _et al._ [2021] V. Balachandran, G. Benenti, G. Casati, and D. Poletti, From the eigenstate thermalization hypothesis to algebraic relaxation of OTOCs in systems with conserved quantities, Physical Review B 104, 10.1103/physrevb.104.104306 (2021).
  * Balachandran  _et al._ [2022] V. Balachandran, L. F. Santos, M. Rigol, and D. Poletti, Effect of symmetries in out-of-time ordered correlators in interacting integrable and nonintegrable many-body quantum systems (2022), arXiv:2211.07073 [cond-mat.stat-mech] .
  * Lieb and Robinson [1972] E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, Communications in Mathematical Physics 28, 251 (1972).
  * Srednicki [1996] M. Srednicki, Thermal fluctuations in quantized chaotic systems, Journal of Physics A: Mathematical and General 29, L75–L79 (1996).
  * Deutsch [1991] J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991).
  * Srednicki [1999] M. Srednicki, The approach to thermal equilibrium in quantized chaotic systems, Journal of Physics A: Mathematical and General 32, 1163 (1999).
  * Huang  _et al._ [2019] Y. Huang, F. G. S. L. Brandão, and Y.-L. Zhang, Finite-size scaling of out-of-time-ordered correlators at late times, Phys. Rev. Lett. 123, 010601 (2019).
  * Cheneau  _et al._ [2012] M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauß, T. Fukuhara, C. Gross, I. Bloch, C. Kollath, and S. Kuhr, Light-cone-like spreading of correlations in a quantum many-body system, Nature 481, 484 (2012).
  * Luitz  _et al._ [2020] D. J. Luitz, R. Moessner, S. L. Sondhi, and V. Khemani, Prethermalization without temperature, Phys. Rev. X 10, 021046 (2020).
  * Lee  _et al._ [2019] J. Lee, D. Kim, and D.-H. Kim, Typical growth behavior of the out-of-time-ordered commutator in many-body localized systems, Phys. Rev. B 99, 184202 (2019).
  * Bohigas  _et al._ [1984] O. Bohigas, M. J. Giannoni, and C. Schmit, Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws, Phys. Rev. Lett. 52, 1 (1984).
  * Casati  _et al._ [1980] G. Casati, F. Valz-Gris, and I. Guarnieri, Connection between quantization of nonintegrable systems and statistical theory of spectra, Lett. Nuovo Cimento 28, 279 (1980).
  * Oganesyan and Huse [2007] V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B 75, 155111 (2007).
  * [52] https://www.nscc.sg/, access from January 1st to September 30th 2022.