  * March 2022

# Integrable Fractional Modified Korteweg-de Vries, Sine-Gordon, and Sinh-
Gordon Equations

Mark J. Ablowitz  Department of Applied Mathematics, University of Colorado,
Boulder, Colorado, U.S.A<EMAIL_ADDRESS>Joel B. Been  Department
of Applied Mathematics and Statistics and Department of Physics, Colorado
School of Mines, Golden, Colorado, U.S.A<EMAIL_ADDRESS>Lincoln D. Carr
Quantum Engineering Program, Department of Applied Mathematics and Statistics,
and Department of Physics, Colorado School of Mines, Golden, Colorado, U.S.A.
<EMAIL_ADDRESS>
###### Abstract

The inverse scattering transform allows explicit construction of solutions to
many physically significant nonlinear wave equations. Notably, this method can
be extended to fractional nonlinear evolution equations characterized by
anomalous dispersion using completeness of suitable eigenfunctions of the
associated linear scattering problem. In anomalous diffusion, the mean squared
displacement is proportional to $t^{\alpha}$, $\alpha>0$, while in anomalous
dispersion, the speed of localized waves is proportional to $A^{\alpha}$,
where $A$ is the amplitude of the wave. Fractional extensions of the modified
Korteweg-deVries (mKdV), sine-Gordon (sineG) and sinh-Gordon (sinhG) and
associated hierarchies are obtained. Using symmetries present in the linear
scattering problem, these equations can be connected with a scalar family of
nonlinear evolution equations of which fractional mKdV (fmKdV), fractional
sineG (fsineG), and fractional sinhG (fsinhG) are special cases. Completeness
of solutions to the scalar problem is obtained and, from this, the nonlinear
evolution equation is characterized in terms of a spectral expansion. In
particular, fmKdV, fsineG, and fsinhG are explicitly written. One-soliton
solutions are derived for fmKdV and fsineG using the inverse scattering
transform and these solitons are shown to exhibit anomalous dispersion.

##  1 Introduction

Fractional calculus has been effectively applied to describe physical systems
with anomalous behavior associated with multi-scale media. The underlying
fractional mathematical formulation, originally designed to interpolate
between integer derivative orders, has been used to describe new phenomena,
such as novel forms of transport in biology [29, 10, 33, 27], amorphous
materials [30, 25, 15], porous media [8, 9, 21], and climate science [19]
amongst others. Fractional equations often predict physically measurable
quantities that follow power laws. For example, in anomalous diffusion, the
mean squared displacement is related to time by a power law $t^{\alpha}$,
$\alpha>0$ [31, 23, 34, 32]. Similarly, for integrable soliton equations,
fractional generalizations predict _anomalous dispersion_ where the speed of
localized solitonic waves are related to their amplitude by a power law [1].

Integrable evolution equations are key elements in the study of nonlinear
dynamics because they have deep mathematical structure and provide exactly
solvable models whose results can be compared with experimental and numerical
data. Some important and well-known examples of integrable evolution equations
are the Korteweg-de Vries (KdV), modified Korteweg-de Vries (mKdV), nonlinear
Schrodinger (NLS), sine-Gordon (sineG), and sinh-Gordon (sinhG) equations.
These equations are solvable by the inverse scattering transformation (IST), a
nonlinear generalization of Fourier transforms where the nonlinear equation is
associated with a linear scattering problem. They also admit an infinite set
of conservation laws and have soliton solutions which are robust localized
traveling waves [2, 3].

In [1], we obtained and analyzed the integrable fractional Korteweg-de Vries
(fKdV) and integrable fractional nonlinear Schrödinger (fNLS) equations. These
were two examples of a hierarchy of fractional equations that can be
constructed. In the case of NLS, the hierarchy is written in terms of $2\times
2$ matrix operators. In this article, we demonstrate that this process can be
applied to define and solve key, physically relevant nonlinear evolution
equations — namely the fmKdV, fsineG, and fsinhG equations in terms of scalar
operators. Although the fmKdV, fsineG, and fsinhG equations can be written in
terms of matrix operators the scalar system is considerably simpler, more
compact and, provides a direct analog of the scalar fKdV operator.

The fractional operators of these integrable systems are are nonlinear
generalizations of the well-established Riesz fractional derivative. Although
there are many fractional derivatives, the Riesz formulation is particularly
intuitive and accessible for physicists who do not specialize in this area of
mathematics. The Riesz fractional derivative is defined by its Fourier
multiplier $|k|^{2\epsilon}$, $0<\epsilon<1$, and can be understood as the
fractional power of $-\partial_{x}^{2}$. Fractional equations defined using
the Riesz fractional derivative (alternately termed the Riesz transform [28]
or fractional Laplacian [20]) are effective tools when describing behavior in
complex systems because the Riesz fractional derivative is closely related to
non-Gaussian statistics [22]. It has found physical applications in describing
movement of water in porous media [24], transport of temperature in fluid
dynamics [13], and power law attenuation in materials [16] amongst many others
[26, 11, 12].

The KdV equation describes quadratic nonlinear waves with weak dispersion; it
was discovered in water waves over one hundred years ago [18]. The KdV
equation admits solitary wave solutions which are localized waves of permanent
form that propagate unidirectionally and whose speed and amplitude are
linearly related. Seventy years later, using numerical methods, KdV solitary
waves were found to interact elastically; they were termed solitons [35]. Soon
afterwards the KdV equation with decaying initial data was linearized and
soliton solutions were obtained analytically using inverse scattering methods
[14]. A few years later the NLS equation was found to be solvable via inverse
scattering and to have soliton solutions. In [6], the linearization procedure
was generalized with the NLS, mKdV, and sineG (in light cone coordinates)
equations as special cases. The procedure was termed the Inverse Scattering
Transform (IST). These equations arise in numerous physical contexts [5, 4].
Remarkably, all of these equations have fractional extensions which pave the
way for applications to anomalous dispersion and multi-scale behavior.

In this article, we define and solve the fmKdV, fsineG, and fsinhG equations
on the line with suitable initial data using three ingredients: a general
nonlinear equation solvable by the IST, a completeness relation for squared
eigenfunctions, and an anomalous dispersion relation. We develop a scalar
reduction of the Ablowitz-Kaup-Newell-Segur (AKNS) system in which we find the
fmKdV, fsineG, and fsinhG equations as special cases using power law
dispersion relations. Then, we characterize a completeness relation for
squared eigenfunctions of this scalar system. This completeness relation
provides a spectral representation for the fractional operators in the fmKdV,
fsineG, and fsinhG equations, giving the equations an explicit representation
in physical space. From basic IST theory we derive a one-soliton solutions to
the whole class of nonlinear equations described by the scalar reduction; in
particular, we give those for the fmKdV and fsineG equations. We also verify
these results using the completeness relation for squared eigenfunctions. The
velocity of these solitons are related to their amplitude by a power law.
Therefore, the fmKdV and fsineG equations predict anomalous dispersion.

##  2 AKNS Scattering System and Scalar Reduction

The inverse scattering transformation relies on associating the nonlinear
problem we want to solve to a linear scattering problem by taking the
potential of the linear problem to be the solution of the nonlinear problem.
For many nonlinear evolution equations, e.g., the mKdV, sineG, and NLS
equations, the associated linear scattering problem is the AKNS system (also
often called the AKNS eigenvalue problem). The nonlinear evolution equations
are linearized by the scattering problem. We previously demonstrated that the
AKNS system can linearize the _fractional_ Nonlinear Schrödinger equation [1]
via a $2\times 2$ matrix operator.

Here, we will show that given a symmetry reduction, the vector valued
nonlinear evolution equations associated to the AKNS scattering problem
becomes a scalar nonlinear evolution equation. This family of equations is
then shown to contain the mKdV, sineG, and sinhG equations:

$\displaystyle q_{t}$ $\displaystyle\mp 6q^{2}q_{x}+q_{xxx}=0,$ (1)
$\displaystyle u_{xt}$ $\displaystyle=\sin{u},$ (2)

where $r=\pm q$ for KdV and $r=-q$, with $u_{x}/2=-q$ for sineG with $q$ real.
We also note that with $r=q=u_{x}/2$ and $q$ real we find the sinhG equation:

$u_{xt}=\sinh u$ (3)

Below, we show that this family of scalar evolution equations also contains
fmKdV, fsineG, and fsinhG as well as their hierarchies. First, we will outline
scattering theory of the AKNS system and show how this leads to the scalar
scattering problem.

###  2.1 AKNS Scattering Problem

The Ablowitz-Kaup-Newell-Segur (AKNS) system is the $2\times 2$ scattering
problem for the vector-valued function
$\mathbf{v}(x)=\left(v_{1}(x),v_{2}(x)\right)^{T}$ ($T$ represents transpose)

$\displaystyle v^{(1)}_{x}$ $\displaystyle=-ikv^{(1)}+q(x,t)v^{(2)},$ (4)
$\displaystyle v^{(2)}_{x}$ $\displaystyle=+ikv^{(2)}+r(x,t)v^{(1)},$ (5)

where $q$ and $r$ act as potentials and $k$ is an eigenvalue. We can associate
to this scattering problem a vector-valued family of integrable nonlinear
equations [6]

$\displaystyle\sigma_{3}\partial_{t}\mathbf{u}+2A_{0}(\mathbf{L}^{A})\mathbf{u}=0,\quad\sigma_{3}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},$ (6)

where $\mathbf{u}=\left(r,q\right)^{T}$ decays sufficiently rapidly at
infinity and the $2\times 2$ matrix operator

$\displaystyle\mathbf{L}^{A}\equiv\frac{1}{2i}\begin{pmatrix}\partial_{x}-2rI_{-}q&2rI_{-}r\\\
-2qI_{-}q&-\partial_{x}+2qI_{-}r\end{pmatrix},$ (7)

with $I_{-}=\int_{-\infty}^{x}dy$. $\mathbf{L}^{A}$ is the adjoint of

$\displaystyle\mathbf{L}\equiv\frac{1}{2i}\begin{pmatrix}-\partial_{x}-2qI_{+}r&-2qI_{+}q\\\
2rI_{+}r&\partial_{x}+2rI_{+}q\end{pmatrix},$ (8)

with $I_{+}=\int_{x}^{\infty}dy$. The function $A_{0}$ has been traditionally
considered to be meromorphic. The family of equations represented by (6) is
commonly related to cases when
$A_{0}(\mathbf{L}^{A})=\left(\mathbf{L}^{A}\right)^{n}$, $n=1,2,3...$.
However, using the completeness relation for squared eigenfunctions which is
discussed in the next section, it was shown that this can be extended to much
more general $A_{0}(k)$ [1]. The operator $A_{0}(\mathbf{L}^{A})$ can also be
related to the dispersion relation of the linearization of (6). Specifically,
if we put $q=e^{i(kx-\omega((k)t)}$ into the linearization of (6), we have

$\displaystyle A_{0}\left(\frac{k}{2}\right)=-\frac{i}{2}\omega(-k).$ (9)

We can obtain the Nonlinear Schrödinger equation from equation (6) by putting
$r=\mp q^{*}$ with its linear dispersion relation $\omega(k)=-k^{2}$.
Similarly, sineG, sinhG, and mKdV follow from $r=-q$, $\omega(k)=k^{-1}$;
$r=+q$, $\omega(k)=k^{-1}$; and $r=\pm q$, $\omega(k)=-k^{3}$, respectively
with $q$ real. In [1], it was shown that fNLS could be obtained from $r=\pm
q$, $\omega(k)=-k^{2}|k|^{\epsilon}$. The associated hierarchy of integrable
equations follows by taking $\omega(k)=-k^{n}|k|^{\epsilon},n=3,4,...$.

We will take the linearization of fmKdV, fsineG, and fsinhG to be

$\displaystyle q_{t}+(-\partial_{x}^{2})^{\epsilon}q_{xxx}$ $\displaystyle=0,$
(10) $\displaystyle u_{tx}=(-\partial_{x}^{2})^{\epsilon}u,~{}~{}$
$\displaystyle
q_{t}=\int_{-\infty}^{x}(-\partial_{\xi}^{2})^{\epsilon}q(\xi,t)\,d\xi,$ (11)

where $(-\partial_{x}^{2})^{\epsilon}$ is the Riesz fractional derivative.
Notice that both fsineG and fsinhG have the same linear equation (11). The
linearization of fKdV has dispersion relation
$\omega(k)=-k^{3}|k|^{2\epsilon}$ and that of fsineG and fsinhG is
$\omega(k)=|k|^{2\epsilon}/k$. Therefore, fKdV can be obtained from (6) with
$r=\pm q$ and $A_{0}(k)=-4ik^{3}|2k|^{2\epsilon}$ and similarly fsineG (sinhG)
are (6) with $r=-q$ ($r=+q$) and $A_{0}(k)=i|2k|^{2\epsilon}/(4k)$.

###  2.2 Scattering Data for the AKNS System

With sufficient decay and smoothness of $\mathbf{u}$, we define eigenfunctions
functions for the AKNS system as solutions to equations (4) and (5) satisfying
the boundary conditions

$\displaystyle\bm{\phi}(x;k,t)$ $\displaystyle\sim\begin{pmatrix}1\\\
0\end{pmatrix}e^{-ikx},~{}~{}\overline{\bm{\phi}}(x;k,t)$
$\displaystyle\sim\begin{pmatrix}0\\\ 1\end{pmatrix}e^{+ikx},~{}~{}$
$\displaystyle x\to-\infty,$ (12) $\displaystyle\bm{\psi}(x;k,t)$
$\displaystyle\sim\begin{pmatrix}0\\\
1\end{pmatrix}e^{+ikx},~{}~{}\overline{\bm{\psi}}(x;k,t)$
$\displaystyle\sim\begin{pmatrix}1\\\ 0\end{pmatrix}e^{-ikx},~{}~{}$
$\displaystyle x\to+\infty.$ (13)

As the eigenfunctions $\bm{\psi}$, $\overline{\bm{\psi}}$ are linearly
independent, we can write $\bm{\phi}$ and $\overline{\bm{\phi}}$ as

$\displaystyle\bm{\phi}(x;k,t)$
$\displaystyle=b(k,t)\bm{\psi}(x;k,t)+a(k,t)\overline{\bm{\psi}}(x;k,t),$ (14)
$\displaystyle\overline{\bm{\phi}}(x;k,t)$
$\displaystyle=\overline{a}(k,t)\bm{\psi}(x;k,t)+\overline{b}(k,t)\overline{\bm{\psi}}(x;k,t).$
(15)

Then, we can write the scattering data explicitly in terms of the
eigenfunctions as

$\displaystyle a(k,t)$
$\displaystyle=W(\bm{\phi},\bm{\psi}),\quad\overline{a}(k,t)$
$\displaystyle=W(\overline{\bm{\psi}},\overline{\bm{\phi}}),$ (16)
$\displaystyle b(k,t)$
$\displaystyle=W(\overline{\bm{\psi}},\bm{\phi}),\quad\overline{b}(k,t)$
$\displaystyle=W(\overline{\bm{\phi}},\bm{\psi}),$ (17)

with the Wronskian given by $W(u,v)=u^{(1)}v^{(2)}-u^{(2)}v^{(1)}$. The
transmission and reflection coefficients, $\tau(k,t)$, $\overline{\tau}(k,t)$
and $\rho(k,t)$, $\overline{\rho}(k,t)$, are defined by

$\displaystyle\tau(k,t)$ $\displaystyle=\frac{1}{a(k,t)},\quad$
$\displaystyle\rho(k,t)=\frac{b(k,t)}{a(k,t)},$ (18)
$\displaystyle\overline{\tau}(k,t)$
$\displaystyle=\frac{1}{\overline{a}(k,t)},\quad$
$\displaystyle\overline{\rho}(k,t)=\frac{\overline{b}(k,t)}{\overline{a}(k,t)}.$
(19)

We also define the mixed reflection coefficient by

$\displaystyle\tilde{\rho}(k,t)$
$\displaystyle=\frac{\overline{b}(k,t)}{a(k,t)}.$ (20)

The zeros of $a$ and $\overline{a}$ at $k_{j}=\xi_{j}+i\eta_{j}$,
$\eta_{j}>0$, $j=1,2,...,J$ and
$\overline{k}_{j}=\overline{\xi}_{j}+i\overline{\eta}_{j}$,
$\overline{\eta}_{j}<0$, $j=1,2,...,\overline{J}$, respectively, are
eigenvalues of the AKNS system corresponding to bound states. With decaying
data, these eigenvalues exist only when $r=-q$. We assume the eigenvalues are
‘proper’, i.e., they are simple zeros of $a$/$\overline{a}$, they are not on
the real $k$ axis, and $J=\overline{J}$; cf. [3]. The bound state
eigenfunctions are related by

$\displaystyle\bm{\phi}_{j}(x,t)=b_{j}(t)\bm{\psi}_{j}(x,t),\quad\overline{\bm{\phi}}_{j}(x,t)=\overline{b}_{j}(t)\overline{\bm{\psi}}_{j}(x,t),$
(21)

where $b_{j}(t)=b(k_{j},t)$. We also define the norming constants by

$\displaystyle C_{j}(t)$
$\displaystyle=b_{j}(t)/a_{j}^{\prime}(t),\quad\overline{C}_{j}(t)=\overline{b}_{j}(t)/\overline{a}_{j}^{\prime}(t),$
(22) $\displaystyle\tilde{C}_{j}(t)$
$\displaystyle=\overline{b}_{j}(t)/a_{j}^{\prime}(t),$ (23)

where $a_{j}^{\prime}(t)=\partial_{k}a(k,t)|_{k=k_{j}}$, etc. When $r=\mp
q^{*}$ in (4)-(5), we have the symmetry reductions

$\displaystyle\overline{\bm{\psi}}(x,k,t)$
$\displaystyle=\sigma\bm{\psi}^{*}(x,k^{*},t),\quad\overline{\bm{\phi}}(x,k,t)=\sigma^{-1}\bm{\phi}^{*}(x,k^{*},t),$
(24)

for the eigenfunctions and $\overline{a}(k,t)=a^{*}(k^{*},t)$ and
$\overline{b}(k,t)=\mp b^{*}(k^{*},t)$ for the scattering data where

$\displaystyle\sigma_{\pm}=\begin{pmatrix}0&1\\\ \pm
1&0\end{pmatrix},\quad\sigma_{\pm}^{-1}=\begin{pmatrix}0&\pm 1\\\ \
1&0\end{pmatrix}.$ (25)

When $r=\pm q$, $q$ real, we have the symmetry reductions

$\displaystyle\overline{\bm{\psi}}(x;k,t)$
$\displaystyle=\sigma_{\pm}\bm{\psi}(x;-k,t),\quad$
$\displaystyle\overline{\bm{\phi}}(x;k,t)=\sigma_{\pm}^{-1}\bm{\phi}(x;-k,t),$
(26)

for the eigenfunctions and

$\displaystyle\overline{a}(k,t)$ $\displaystyle=a(-k,t),\quad$
$\displaystyle\overline{b}(k,t)=\pm b(-k,t),$ (27)

for the scattering data. From the scattering eigenfunctions $\bm{\psi}$ and
$\bm{\phi}$, we can construct the eigenfunctions of the operator $\mathbf{L}$,
$\mathbf{\Psi}(x,k,t)$ and $\overline{\mathbf{\Psi}}(x,k,t)$, and its adjoint
$\mathbf{L}^{A}$, $\mathbf{\Psi}^{A}(x,k,t)$ and
$\overline{\mathbf{\Psi}}^{A}(x,k,t)$ by

$\displaystyle\mathbf{\Psi}(x,k,t)$
$\displaystyle=\left((\psi^{(1)}(x,k,t))^{2},(\psi^{(2)}(x,k,t))^{2}\right)^{T},$
(28) $\displaystyle~{}\overline{\mathbf{\Psi}}(x,k,t)$
$\displaystyle=\left((\overline{\psi}^{(1)}(x,k,t))^{2},(\overline{\psi}^{(2)}(x,k,t))^{2}\right)^{T},$
(29) $\displaystyle\mathbf{\Psi}^{A}(x,k,t)$
$\displaystyle=\left((\phi^{(2)}(x,k,t))^{2},-(\phi^{(1)}(x,k,t))^{2}\right)^{T},$
(30) $\displaystyle\overline{\mathbf{\Psi}}^{A}(x,k,t)$
$\displaystyle=\left((\overline{\phi}^{(2)}(x,k,t))^{2},-(\overline{\phi}^{(1)}(x,k,t))^{2}\right)^{T}.$
(31)

Notice that these are all written in terms of squared eigenfunctions of the
AKNS system. Explicitly, we have

$\displaystyle\mathbf{L}\mathbf{\Psi}$ $\displaystyle=k\mathbf{\Psi},~{}~{}$
$\displaystyle\mathbf{L}\overline{\mathbf{\Psi}}=k\overline{\mathbf{\Psi}},$
(32) $\displaystyle\mathbf{L}^{A}\mathbf{\Psi}^{A}$
$\displaystyle=k\mathbf{\Psi}^{A},~{}~{}$
$\displaystyle\mathbf{L}^{A}\overline{\mathbf{\Psi}}^{A}=k\overline{\mathbf{\Psi}}^{A}.$
(33)

###  2.3 Scalar Scattering System

The scattering equations for the scalar system, obtained from the symmetry
reduction $r=\pm q$, are

$\displaystyle v^{(1)}_{x}$ $\displaystyle=-ikv^{(1)}+q(x,t)v^{(2)},$ (34)
$\displaystyle v^{(2)}_{x}$ $\displaystyle=+ikv^{(2)}\pm q(x,t)v^{(1)}.$ (35)

To construct a family of nonlinear evolution equations for this system, which
is a subset of the class in equation (6), we use the eigenvalue relations in
equations (32) and (33) in addition to an orthogonality relation from the AKNS
system. Taking $r=\pm q$ and writing out
$\mathbf{L}\mathbf{\Psi}=k\mathbf{\Psi}$ in components, we have

$\displaystyle 2ik(\psi^{(1)})^{2}$
$\displaystyle=-\frac{\partial(\psi^{(1)})^{2}}{\partial
x}-2qI_{+}\left[q(\pm(\psi^{(1)})^{2}+(\psi^{(2)})^{2})\right],$ (36)
$\displaystyle 2ik(\psi^{(2)})^{2}$
$\displaystyle=\frac{\partial(\psi^{(2)})^{2}}{\partial
x}+2qI_{+}\left[q((\psi^{(1)})^{2}\pm(\psi^{(2)})^{2})\right].$ (37)

We define the functions

$\displaystyle\mu_{-}(x,k)$
$\displaystyle=(\psi^{(1)}(x,k))^{2}+(\psi^{(2)}(x,k))^{2},$ (38)
$\displaystyle\mu_{+}(x,k)$
$\displaystyle=(\psi^{(1)}(x,k))^{2}-(\psi^{(2)}(x,k))^{2}.$ (39)

Taking $r=+q$, adding (36) to (37) and using (38) and (39) we have

$\displaystyle 2ik\mu_{-}=-\frac{\partial\mu_{+}}{\partial x}.$ (40)

Subtracting equation (37) from (36) yields

$\displaystyle 2ik\mu_{+}=-\frac{\partial\mu_{-}}{\partial
x}-4qI_{+}\left[q\mu_{-}\right].$ (41)

Putting equation (40) into (41) we get the following scalar eigenvalue
equation

$\displaystyle L_{+}\mu_{+}$ $\displaystyle=k^{2}\mu_{+},~{}~{}$
$\displaystyle L_{+}=-\frac{1}{4}\frac{\partial^{2}}{\partial
x^{2}}+q^{2}+qI_{+}q_{y}.$ (42)

We can similarly show that

$\displaystyle L_{+}^{A}\nu_{+}$ $\displaystyle=k^{2}\nu_{+},~{}~{}$
$\displaystyle L_{+}^{A}=-\frac{1}{4}\frac{\partial^{2}}{\partial
x^{2}}+q^{2}+q_{x}I_{-},$ (43)

where

$\displaystyle\nu_{+}(x,k)$
$\displaystyle=(\phi^{(1)}(x,k))^{2}+(\phi^{(2)}(x,k))^{2},$ (44)
$\displaystyle\nu_{-}(x,k)$
$\displaystyle=(\phi^{(1)}(x,k))^{2}-(\phi^{(2)}(x,k))^{2}.$ (45)

For $r=-q$, we have

$\displaystyle L_{-}\mu_{-}$ $\displaystyle=k^{2}\mu_{-},~{}~{}$
$\displaystyle L_{-}=-\frac{1}{4}\frac{\partial^{2}}{\partial
x^{2}}-q^{2}-qI_{+}q_{y},$ (46) $\displaystyle L_{-}^{A}\nu_{-}$
$\displaystyle=k^{2}\nu_{-},~{}~{}$ $\displaystyle
L_{-}^{A}=-\frac{1}{4}\frac{\partial^{2}}{\partial x^{2}}-q^{2}-q_{x}I_{-}q.$
(47)

Equations (42), (43), (46), and (47) define the operators and squared
eigenfunctions of the scalar scattering system. For the AKNS scattering
problem, we know that the following orthogonality relation holds [6]

$\displaystyle\int_{-\infty}^{\infty}\left\\{\left(r_{t}+2\Omega(k)r\right)(\psi^{(1)})^{2}+\left(-q_{t}+2\Omega(k)q\right)(\psi^{(2)})^{2}\right\\}dx=0,$
(48)

where $\Omega(k)$ is a suitable function of $k$, taken to be meromorphic in
[7]. With $r=\pm q$, $\Omega(k)=ik\Theta(k^{2})$, and $\mu_{1}$ and $\mu_{2}$
in (38) and (39), we may write

$\displaystyle\int_{-\infty}^{\infty}\left\\{q_{t}\mu_{\pm}+2qik\Theta(k^{2})\mu_{\mp}\right\\}dx=0.$
(49)

Noting that we have

$\displaystyle 2ik\mu_{-}=-\frac{\partial\mu_{+}}{\partial x}\text{ for
}r=+q,$ (50) $\displaystyle 2ik\mu_{+}=-\frac{\partial\mu_{-}}{\partial
x}\text{ for }r=-q,$ (51)

and using the extension of equations (42) and (46)

$\displaystyle\Theta(L_{\pm})\mu_{\pm}$
$\displaystyle=\Theta(k^{2})\mu_{\pm},$ (52)

we can write

$\displaystyle\int_{-\infty}^{\infty}\left\\{q_{t}\mu_{\pm}+q_{x}\Theta(L_{\pm})\mu_{\pm}\right\\}dx=0,$
(53)

using integration by parts. We can then shift $\Theta(L_{\pm})$ from operating
on $\mu_{\pm}$ to $q_{x}$ using the adjoint of $L$,

$\displaystyle L_{\pm}^{A}=-\frac{1}{4}\partial_{x}^{2}\pm q^{2}\pm
q_{x}I_{-}q,$ (54)

to give

$\displaystyle\int_{-\infty}^{\infty}\left\\{q_{t}+\Theta(L_{\pm}^{A})q_{x}\right\\}\mu_{\pm}dx=0,$
(55)

which implies

$\displaystyle q_{t}+\Theta(L_{\pm}^{A})q_{x}=0.$ (56)

This defines the family of nonlinear evolution equations associated to the
scalar scattering system in equations (34) and (35). Notice that if we take
$\Theta(L_{\pm}^{A})=-4L_{\pm}^{A}$, equation (56) gives mKdV

$\displaystyle q_{t}+q_{xxx}\mp 6q^{2}q_{x}=0.$ (57)

We can relate the operator $\Theta$ directly to the dispersion relation of the
linearization of equation 56. As $L_{\pm}^{A}\to-\frac{1}{4}\partial_{x}^{2}$
implies $\Theta(L_{\pm}^{A})\to\Theta(-\frac{1}{4}\partial_{x}^{2})$, this
linearization is

$\displaystyle q_{t}+\Theta\left(-\partial_{x}^{2}/4\right)q_{x}=0.$ (58)

Putting $q=e^{i(kx-\omega(k)t)}$ gives

$\displaystyle\Theta(k^{2})=\frac{\omega(2k)}{2k}.$ (59)

Therefore, using our definitions of linear fmKdV and linear fsineG and fsinhG,
with dispersion relations $\omega(k)=-k^{3}|k|^{2\epsilon}$ and
$\omega(k)=|k|^{2\epsilon}/k$ where $0<\epsilon<1$, respectively, we have
$\Theta(L_{\pm}^{A})=-4L_{\pm}^{A}|4L_{\pm}^{A}|^{\epsilon}$,
$\Theta(L_{-}^{A})=\frac{|4L_{-}^{A}|^{\epsilon}}{4L_{-}}$, and
$\Theta(L_{+}^{A})=\frac{|4L_{+}^{A}|^{\epsilon}}{4L_{+}}$, respectively
(recall that fmKdV has $r=\pm q$ while fsineG and fsinhG have $r=-q$ and
$r=+q$, respectively). Therefore, we can write fmKdV, fsineG, and fsinhG as

$\displaystyle q_{t}-4L_{\pm}^{A}|2L_{\pm}^{A}|^{\epsilon}q_{x}$
$\displaystyle=0,$ (60) $\displaystyle
q_{t}+\frac{|4L_{-}^{A}|^{\epsilon}}{4L_{-}^{A}}q_{x}=0,~{}~{}u_{tx}$
$\displaystyle+\frac{|4L_{-}^{A}|^{\epsilon}}{4L_{-}^{A}}u_{xx},$ (61)
$\displaystyle
q_{t}+\frac{|4L_{+}^{A}|^{\epsilon}}{4L_{+}^{A}}q_{x}=0,~{}~{}u_{tx}$
$\displaystyle+\frac{|4L_{+}^{A}|^{\epsilon}}{4L_{+}^{A}}u_{xx}.$ (62)

Notice that as both $L_{\pm}\to-\partial_{x}^{2}/4$, in the linear limit,
fsineG and fsinhG both converge to (11). Currently, the meaning of
$|L_{\pm}^{A}|^{\epsilon}$ is not clear; it will be defined in the next
section using a spectral expansion in terms of the squared eigenfunctions
$\mu_{\pm}$ and $\nu_{\pm}$.

###  2.4 Completeness of Squared Scalar Eigenfunctions

In [17] it was shown that the eigenfunctions $\mathbf{\Psi}$ and
$\mathbf{\Psi}^{A}$, equations (28) and (30), of the AKNS system are complete.
Specifically, for a sufficiently smooth and decaying vector-valued function
$\mathbf{v}(x)=\left(v^{(1)}(x),v^{(2)}(x)\right)^{T}$, we have

$\displaystyle\mathbf{v}(x)$
$\displaystyle=\sum_{n=1}^{2}\int_{\Gamma_{\infty}^{(n)}}dkf_{n}(k)\int_{-\infty}^{\infty}dy\,\mathbf{G}_{n}(x,y,k)\mathbf{v}(y),$
(63) $\displaystyle\mathbf{G}_{1}(x,y,k)$
$\displaystyle=\mathbf{\Psi}(x,k)\mathbf{\Psi}^{A}(y,k)^{T},\quad
f_{1}(k)=-\tau^{2}(k)/\pi,$ $\displaystyle\mathbf{G}_{2}(x,y,k)$
$\displaystyle=\overline{\mathbf{\Psi}}(x,k)\overline{\mathbf{\Psi}}^{A}(y,k)^{T},\quad
f_{2}(k)=\overline{\tau}^{2}(k)/\pi,$

where $\Gamma_{R}^{(1)}$ ($\Gamma_{R}^{(2)}$) is the semicircular contour in
the upper (lower) half plane evaluated from $-R$ to $+R$ and $\tau(k)$ and
$\overline{\tau}(k)$ are transmission coefficients defined in equations (14)
and (15). Notice that $\mathbf{G}_{n}$, $n=1,2$ are $2\times 2$ matrices.
However, here we need to use the adjoint completeness relation, which may be
found directly from (63) using the inner product
$(\mathbf{u},\mathbf{v}):=\int_{-\infty}^{\infty}\mathbf{u}(x)^{T}\mathbf{v}(x)dx$
where $\mathbf{u}$, $\mathbf{v}$ are $2\times 1$ column vectors. To do this,
we expand $\mathbf{v}$ using the above completeness relation, and then
exchange the order of integration to find an expansion for $\mathbf{u}$. This
procedure gives us

$\displaystyle\mathbf{v}(x)$
$\displaystyle=\sum_{n=1}^{2}\int_{\Gamma_{\infty}^{(n)}}dkf_{n}(k)\int_{-\infty}^{\infty}dy\,\mathbf{G}_{n}^{A}(x,y,k)\mathbf{v}(y),$
(64) $\displaystyle\mathbf{G}_{1}^{A}(x,y,k)$
$\displaystyle=\mathbf{\Psi}^{A}(x,k)\mathbf{\Psi}(y,k)^{T},\quad
f_{1}(k)=-\tau^{2}(k)/\pi,$ $\displaystyle\mathbf{G}_{2}^{A}(x,y,k)$
$\displaystyle=\overline{\mathbf{\Psi}}^{A}(x,k)\overline{\mathbf{\Psi}}(y,k)^{T},\quad
f_{2}(k)=\overline{\tau}^{2}(k)/\pi.$

However, when $r=\pm q$, the symmetry reductions in equations (26) and (27)
give

$\displaystyle\overline{\mathbf{\Psi}}(k)=\sigma_{+}\mathbf{\Psi}(-k),~{}~{}\overline{\mathbf{\Psi}}^{A}(k)=-\sigma_{+}\mathbf{\Psi}^{A}(-k).$
(65)

Therefore, the adjoint completness relation in (64) reduces to

$\displaystyle\mathbf{v}(x)=$
$\displaystyle-\int_{\Gamma_{\infty}^{(1)}}dk\frac{\tau^{2}(k)}{\pi}\int_{-\infty}^{\infty}dy\mathbf{\Psi}^{A}(x,k)\mathbf{\Psi}(y,k)^{T}\mathbf{v}(y)$
(66)
$\displaystyle-\int_{\Gamma_{\infty}^{(2)}}dk\frac{\tau^{2}(-k)}{\pi}\int_{-\infty}^{\infty}dy\sigma_{+}\mathbf{\Psi}^{A}(x,-k)\mathbf{\Psi}(y,-k)^{T}\sigma_{+}\mathbf{v}(y).$
(67)

The second integral may be rewritten with the substitution $\xi=-k$ as

$\displaystyle-\int_{\Gamma_{\infty}^{(1)}}d\xi\frac{\tau^{2}(\xi)}{\pi}\int_{-\infty}^{\infty}dy\sigma_{+}\mathbf{\Psi}^{A}(x,\xi)\mathbf{\Psi}(y,\xi)^{T}\sigma_{+}\mathbf{v}(y).$
(68)

Therefore, we have

$\displaystyle\mathbf{v}(x)=\int_{\Gamma_{\infty}^{(1)}}d\xi\frac{\tau^{2}(\xi)}{\pi}\int_{-\infty}^{\infty}dy\left[\mathbf{\Psi}^{A}(x,k)\mathbf{\Psi}(y,k)^{T}-\sigma_{+}\mathbf{\Psi}^{A}(x,k)\mathbf{\Psi}(y,k)^{T}\sigma_{+}\right]\mathbf{v}(y).$
(69)

If we put $\mathbf{v}(x)=\mathbf{h}(x)=\left(h(x),\mp h(x)\right)^{T}$, we can
reduce the completeness relation to

$\displaystyle
h(x)=\mp\int_{\Gamma_{\infty}^{(1)}}dk\int_{-\infty}^{\infty}dy\,g_{\pm}(x,y,k)h(y),$
(70)

for the scalar function $h(x)$ where

$\displaystyle
g_{\pm}(x,y,k)=\frac{\tau^{2}(k)}{\pi}\nu_{\pm}(x,k)\mu_{\pm}(y,k),$ (71)

with $\nu_{\pm}(x,k)=(\phi^{(1)}(x,k))^{2}\pm(\phi^{(2)}(x,k))^{2}$ and
$\mu_{\pm}(x,k)=(\psi^{(1)}(x,k))^{2}\mp(\psi^{(2)}(x,k))^{2}$. Then, the
action of the operator $\Theta(L_{\pm}^{A})$ on the function $h(x)$ may be
written as

$\displaystyle\Theta(L_{\pm}^{A})h(x)=\mp\int_{\Gamma_{\infty}^{(1)}}dk\Theta(k^{2})\int_{-\infty}^{\infty}dy\,g_{\pm}(x,y,k)h(y),$
(72)

and the famility of nonlinear evolution equations in equation (56) becomes

$\displaystyle
q_{t}\mp\int_{\Gamma_{\infty}^{(1)}}dk\Theta(k^{2})\int_{-\infty}^{\infty}dy\,g_{\pm}(x,y,k)\partial_{y}q(y)=0.$
(73)

In particular, fmKdV can be represented as

$\displaystyle
q_{t}\mp\int_{\Gamma_{\infty}^{(1)}}dk|2k|^{2\epsilon}\int_{-\infty}^{\infty}dy\,g_{\pm}(x,y,k)\left[q_{yyy}\mp
6q^{2}\right]=0,$ (74)

and fsineG and fsinhG are given by

$\displaystyle q_{t}$
$\displaystyle\mp\int_{\Gamma_{\infty}^{(1)}}dk|2k|^{2(1-\epsilon)}\int_{-\infty}^{\infty}dy\,g_{-}(x,y,k)\partial_{y}q(y)=0,$
(75) $\displaystyle q_{t}$
$\displaystyle\mp\int_{\Gamma_{\infty}^{(1)}}dk|2k|^{2(1-\epsilon)}\int_{-\infty}^{\infty}dy\,g_{+}(x,y,k)\partial_{y}q(y)=0.$
(76)

Because the $k$ integral in equation (70) is over the semicircle in the upper
half plane, it can be expressed instead in terms of an integral along the real
line and a sum over the residues using contour integration. This is a useful
representation because it explicitly separates the continuous and discrete
spectra, where the latter corresponds to bound states at $k=k_{j}$,
$j=1,2,...,J$. Using the closed contour composed of $\Gamma^{(1)}$ and an
integral along the real line from $\infty$ to $-\infty$, we can write

$\displaystyle\Theta(L_{\pm}^{A})h(x)=$
$\displaystyle\mp\int_{-\infty}^{\infty}dk\Theta(k^{2})\int_{-\infty}^{\infty}dy\,g_{\pm}(x,y,k)h(y),$
(77) $\displaystyle\pm 2\pi
i\sum_{j=1}^{J}\text{Res}\left(\Theta(k^{2})\int_{-\infty}^{\infty}dy\,g_{\pm}(x,y,k)h(y),k_{j}\right).$

Because $\nu_{\pm}$ and $\mu_{\pm}$ are all analytic in the upper half plane,
the only residues come from the poles of $\tau^{2}$, or zeros of $a^{2}$.
These occur at $k_{j}=\xi_{j}+i\eta_{j}$, $j=1,2,...,J$ and are assumed to be
simple, meaning $\tau^{2}$ has a double pole at $k_{j}$. Therefore, we can
compute the residue at $k_{j}$ as

Res
$\displaystyle\left(\int_{-\infty}^{\infty}dy\,g_{\pm}(x,y,k)h(y),k_{j}\right)=\lim_{k\to
k_{j}}\frac{\partial}{\partial
k}\left[(k-k_{j})^{2}\Theta(k^{2})\int_{-\infty}^{\infty}dy\,g_{\pm}(x,y,k)h(y)\right],$
$\displaystyle=\frac{\Theta(k_{j}^{2})}{\pi(a_{j}^{\prime})^{2}}\int_{-\infty}^{\infty}dy\left\\{\partial_{k}\nu_{\pm}(x,k)\mu_{\pm}(y,k)+\nu_{\pm}(x,k)\partial_{k}\mu_{\pm}(x,k)\right\\}_{k=k_{j}}h(y),$
(78)
$\displaystyle+\left(\frac{2k_{j}\Theta^{\prime}(k_{j}^{2})}{\pi(a_{j}^{\prime})^{2}}-\frac{a_{j}^{\prime\prime}\Theta(k_{j}^{2})}{\pi(a_{j}^{\prime})^{3}}\right)\int_{-\infty}^{\infty}dy\,\nu_{\pm}(x,k_{j})\mu_{\pm}(y,k_{j})h(y).$

Defining

$\displaystyle g_{\pm,j}^{(1)}(x,y)$
$\displaystyle=\frac{2i}{(a_{j}^{\prime})^{2}}\left\\{\partial_{k}\nu_{\pm}(x,k)\mu_{\pm}(y,k)+\nu_{\pm}(x,k)\partial_{k}\mu_{\pm}(y,k)\right\\}_{k=k_{j}},$
(79) $\displaystyle g_{\pm,j}^{(2)}(x,y)$
$\displaystyle=\frac{2i}{(a_{j}^{\prime})^{2}}\nu_{\pm}(x,k_{j})\mu_{\pm}(y,k_{j}),$
(80) $\displaystyle g_{\pm,j}^{(3)}(x,y)$
$\displaystyle=-\frac{2ia_{j}^{\prime\prime}}{(a_{j}^{\prime})^{3}}\nu_{\pm}(x,k_{j})\mu_{\pm}(y,k_{j}),$
(81)

we have

$\displaystyle h(x)=$
$\displaystyle\mp\int_{-\infty}^{\infty}dk\Theta(k^{2})\int_{-\infty}^{\infty}dy\,g_{\pm}(x,y,k)h(y),$
(82)
$\displaystyle\pm\sum_{j=1}^{J}\int_{-\infty}^{\infty}dy\left\\{\Theta(k_{j}^{2})g_{\pm,j}^{(1)}(x,y)+2k_{j}\Theta^{\prime}(k_{j}^{2})g^{(2)}_{\pm,j}(x,y)+\Theta(k_{j}^{2})g_{\pm,j}^{(3)}(x,y)\right\\}h(y).$

Notice that if we take $\Theta(k^{2})=1$, then we have the identity in (70)
written with continuous and discrete spectra separated.

##  3 The IST for Fractional Modified KdV, SineG, and SinhG

Solving nonlinear evolution equations with the IST is analogous to solving
linear evolution equations with Fourier transforms. To solve linear problems,
the Fourier transform is taken to map the problem into Fourier space where the
time evolution is described by a simple set of differential equations. These
equations are then solved to give the solution at any time $t$ in Fourier
space. Finally, the solution is mapped back to physical space using the
inverse Fourier transform, which amounts to evaluating an integral. Mapping
the initial condition into scattering space via direct scattering is analogous
to taking the Fourier transform, time evolution in scattering space is nearly
identical to that in Fourier space, and inverse scattering maps the solution
to the nonlinear problem back into physical space just as the inverse Fourier
transform does. The major difference between Fourier transforms and the IST is
that performing integrals for the Fourier transform and inverse Fourier
transform is replaced by solving linear integral equations for direct
scattering and inverse scattering. In the following we, outline direct
scattering, time evolution, and inverse scattering for the scalar scattering
system.

###  3.1 Direct Scattering

To solve the nonlinear evolution equation, equation (56), by the inverse
scattering transform, we first map the initial condition into scattering
space; this is analogous to taking the Fourier transform of a linear partial
differential equation. This process involves analyzing linear integral
equations for the eigenfunctions, determining their analytic properties, and
then obtaining the scattering data using Wronskian relations.

Eigenfunctions of the scalar scattering problem are precisely $\bm{\phi}$ and
$\bm{\psi}$ of the AKNS system with the symmetry reduction in equations (26)
and (27). They are solutions to equations (34) and (35) subject to the
boundary conditions in equations (12) and (13) with the scattering data
defined by equation (14). It is convenient to express the scattering functions
in terms of _Jost solutions_ by taking

$\displaystyle\mathbf{M}(x,k,t)$ $\displaystyle=e^{ikx}\bm{\phi}(x,k,t),\quad$
$\displaystyle\mathbf{N}(x,k,t)=e^{-ikx}\bm{\psi}(x,k,t).$ (83)
$\displaystyle\overline{\mathbf{M}}(x,k,t)$
$\displaystyle=\sigma^{-1}_{\pm}\mathbf{M}(x,-k,t),\quad$
$\displaystyle\overline{\mathbf{N}}(x,k,t)=\sigma_{\pm}\mathbf{N}(x,-k,t),$
(84)

where the symmetry reductions for $r=\pm q$ are in terms of

$\displaystyle\sigma_{\pm}=\begin{pmatrix}0&1\\\ \pm
1&0\end{pmatrix},\quad\sigma^{-1}_{\pm}=\begin{pmatrix}0&\pm 1\\\ \
1&0\end{pmatrix}.$ (85)

Then, the boundary conditions become constant

$\displaystyle\mathbf{M}(x,k,t)$ $\displaystyle\sim\begin{pmatrix}1\\\
0\end{pmatrix},~{}~{}x\to-\infty,~{}~{}~{}\mathbf{N}(x,k,t)\sim\begin{pmatrix}0\\\
1\end{pmatrix},~{}~{}x\to\infty,$ (86)

and the scattering equation, where either $\mathbf{M}$ or $\mathbf{N}$ is
represented generically as $\bm{\chi}=\bm{\chi}(x,k,t)$, becomes

$\displaystyle\partial_{x}\bm{\chi}=ik\mathbf{B}\bm{\chi}+\mathbf{Q}\bm{\chi},$
(87)

where

$\displaystyle\mathbf{B}=\begin{pmatrix}0&0\\\
0&2\end{pmatrix},\quad\mathbf{Q}=\begin{pmatrix}0&q\\\ \pm
q&0\end{pmatrix}.\quad$ (88)

This differential equation can be converted to an integral equation for
$\mathbf{M}$ and $\mathbf{N}$, cf. [4],

$\displaystyle\mathbf{M}(x,k,t)$ $\displaystyle=\begin{pmatrix}1\\\
0\end{pmatrix}+\int_{-\infty}^{\infty}\mathbf{G}(x-\xi,k,t)\mathbf{Q}(\xi,t)\mathbf{M}(\xi,k,t)\,d\xi,$
(89) $\displaystyle\mathbf{N}(x,k,t)$ $\displaystyle=\begin{pmatrix}0\\\
1\end{pmatrix}+\int_{-\infty}^{\infty}\overline{\mathbf{G}}(x-\xi,k,t)\mathbf{Q}(\xi,t)\mathbf{N}(\xi,k,t)\,d\xi,$
(90)

where

$\displaystyle\mathbf{G}(x,k,t)$ $\displaystyle=\theta(x)\begin{pmatrix}1&0\\\
0&e^{2ikx}\end{pmatrix},$ (91) $\displaystyle\overline{\mathbf{G}}(x,k,t)$
$\displaystyle=-\theta(-x)\begin{pmatrix}e^{-2ikx}&0\\\ 0&1\end{pmatrix}.$
(92)

So long as $q,r\in L^{1}(\mathbb{R})$, these Volterra integral equations have
absolutely and uniformly convergent Neumann series in the upper half $k$-plane
[3]. Therefore, the functions $\mathbf{M}$ and $\mathbf{N}$ are analytic
functions of $k$ for $\text{Im}\,k>0$ and continuous for $\text{Im}\,k\geq 0$.
This also implies that $\overline{\mathbf{M}}$ and $\overline{\mathbf{N}}$ are
analytic for $\text{Im}\,k<0$ and continuous for $\text{Im}\,k\leq 0$ from
their relations in equation (84). Using these integral equations and the
intial condition at $t=0$, the Jost solutions $\mathbf{M}$ and $\mathbf{N}$
can be constructed at $t=0$ and, subsequently, the scattering functions from
the relations in equation (83). Then, the initial scattering data may be
derived from the Wronskian relations in equations (16) and (17).

We will also need the asymptotic properties of $\mathbf{N}$ and $\mathbf{M}$
to reconstruct the solution in inverse scattering; so, expanding equations
(89) and (90) in large $k$, after integrating by parts, we have

$\displaystyle\mathbf{M}(x,k,t)$
$\displaystyle=\begin{pmatrix}1-\frac{1}{2ik}\int_{-\infty}^{x}q(\xi,t)r(\xi,t)\,d\xi\\\
-\frac{1}{2ik}r(x,t)\end{pmatrix}+\mathcal{O}(k^{-2})$ (93)
$\displaystyle\mathbf{N}(x,k,t)$
$\displaystyle=\begin{pmatrix}\frac{1}{2ik}q(x,t)\\\
1-\frac{1}{2ik}\int_{x}^{\infty}q(\xi,t)r(\xi,t)\,d\xi\end{pmatrix}+\mathcal{O}(k^{-2})$
(94)

###  3.2 Time Evolution

After the initial condition is projected into scattering space by
reconstructing the scattering functions and scattering data, the data is
evolved in time by solving a simple set of ordinary differential equations.
The scattering functions evolve in time according to

$\displaystyle\mathbf{v}_{t}=\begin{pmatrix}A&B\\\
C&-A\end{pmatrix}\mathbf{v}$ (95)

where $A$, $B$, $C$ are functions of $x$, $k$, $t$ which cannot be represented
generally. However, their asymptotic properties can be used to characterize
the time evolution of the scattering data [4] as

$\displaystyle a(k,t)$ $\displaystyle=a(k,0),~{}~{}$ $\displaystyle
b(k,t)=b(k,0)e^{-2ik\Theta(k^{2})t},$ (96) $\displaystyle\rho(k,t)$
$\displaystyle=\rho(k,0)e^{-2ik\Theta(k^{2})t},~{}~{}$ $\displaystyle
C_{j}(t)=C_{j}(0)e^{-2ik_{j}\Theta(k_{j}^{2})t},$ (97)

for $j=1,2,...,J$. We recall that $\Theta(k^{2})$ is related to the linear
dispersion relation as in equation (59). To characterize the spectral
expansion in equation (73), we need to be able to compute the time evolution
of the scattering functions $\bm{\psi}$ and $\bm{\phi}$. Although equation
(95) does not give this in a simple way, the scattering functions can be
evolved in time using inverse scattering, which is discussed next.

###  3.3 Inverse Scattering

Inverse scattering allows the construction of the solution to the nonlinear
evolution equation $q(x,t)$ and the scattering functions $\bm{\psi}(x,k,t)$
and $\bm{\phi}(x,k,t)$ from the scattering data, $a(k,t)$ and $b(k,t)$
obtained from equation (96). For the Jost solutions (83), we can write the
scattering data in equation (14), using (18), as

$\displaystyle\bm{\mu}(x,k,t)=\overline{\mathbf{N}}(x,k,t)+\rho(k,t)e^{2ikx}\mathbf{N}(x,k,t),\quad\bm{\mu}(x,k,t)=\mathbf{M}(x,k,t)/a(k,t),$
(98)

where $\mathbf{M}$ is analytic in the upper half plane, $\bm{\mu}$ is
meromorphic with simple poles at the zeros of $a$, and $\overline{\mathbf{N}}$
is analytic in the lower half plane ($\rho$ is not analytic in general).
Therefore, equation (98) defines the “jump” condition of a Riemann-Hilbert
problem which we will transform onto an integral equation for $\mathbf{N}$.
This equation will allow the construction of the scattering function
$\bm{\psi}$ and the solution $q(x,t)$. We will also outline how the same
method can be used to derive an equation for $\mathbf{M}$.

We assume that $a$ has simple zeros, and hence $\bm{\mu}$ has simple poles in
the upper half plane at $k_{j}$ for $j=1,2,...,J$ with no zeros along the real
line. Then, as $\bm{\mu}$ has only simple poles, we can represent it as

$\displaystyle\bm{\mu}(x,k,t)=\mathbf{h}(x,k,t)+\sum_{j=1}^{J}\frac{\mathbf{A}_{j}(x,t)}{k-k_{j}},$
(99)

where $\mathbf{h}$ is analytic in $k$ for $\text{Im}\,k>0$. By integrating in
a small neighborhood around each $k_{j}$, and using equation (98), we find
that $\mathbf{A}_{j}$ is given by

$\displaystyle\mathbf{A}_{j}(x,t)=C_{j}(t)e^{2ik_{j}x}\mathbf{N}_{j}(x,t),\text{
for }j=1,2,...,J,$ (100)

where $C_{j}(t)=b_{j}(t)/a_{j}^{\prime}(t)$ and
$\mathbf{N}_{j}(x,t)=\mathbf{N}(x,k_{j},t)$. We then define the projection
operators

$\displaystyle P^{\pm}[f](k)=\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{f(\xi)}{\xi-(k\pm i0)}d\xi.$ (101)

If $f_{+}$ ($f_{-}$) is analytic in the upper (lower) half plane and
$f_{\pm}(k)\to 0$ as $|k|\to\infty$ for $\text{Im}\,k>0$ ($\text{Im}\,k<0$),
then

$\displaystyle P^{\pm}[f_{\pm}]=\pm f_{\pm},\quad P^{\pm}[f_{\mp}]=0.$ (102)

Taking equation (98) and subtracting $(1,0)^{T}$ and the simple poles of
$\bm{\mu}$, we have

$\displaystyle\mathbf{h}(x,k,t)-\begin{pmatrix}1\\\
0\end{pmatrix}=\overline{\mathbf{N}}(x,k,t)-\begin{pmatrix}1\\\
0\end{pmatrix}-\sum_{j=1}^{J}\frac{\mathbf{A}_{j}(x,t)}{k-k_{j}}+\rho(k,t)e^{2ikx}\mathbf{N}(x,k,t).$
(103)

The left side is analytic in the upper half plane and approaches zero as
$|k|\to\infty$; $\overline{\mathbf{N}}-(1,0)^{T}$ is also analytic in the
lower half plane and vanishes asymptotically. Therefore, applying $P^{-}$ to
(103) gives

$\displaystyle\overline{\mathbf{N}}(x,k,t)=\begin{pmatrix}1\\\
0\end{pmatrix}+\sum_{j=1}^{J}\frac{\mathbf{A}_{j}(x,t)}{k-k_{j}}+\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{\rho(\xi,t)e^{2i\xi
x}}{\xi-(k-i0)}\mathbf{N}(x,\xi,t)\,d\xi.$ (104)

Noting that $\overline{\mathbf{N}}(x,k,t)=\sigma\mathbf{N}(x,-k,t)$ and using
the expression for $\mathbf{A}_{j}$ in equation (100), we find the integral
equation for $\mathbf{N}$

$\displaystyle\mathbf{N}(x,k,t)=\begin{pmatrix}0\\\
1\end{pmatrix}-\sum_{j=1}^{J}\frac{C_{j}(t)e^{2ik_{j}x}}{k+k_{j}}\sigma^{-1}\mathbf{N}_{j}(x,t)+\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{\rho(\xi,t)e^{2i\xi
x}}{\xi+k+i0}\sigma^{-1}\mathbf{N}(x,\xi,t)\,d\xi.$ (105)

Evaluating this at $k_{\ell}$ for $\ell=1,2,...,J$, we obtain an equation for
$\mathbf{N}_{\ell}(x,t)$.

$\displaystyle\mathbf{N}_{\ell}(x,t)=\begin{pmatrix}0\\\
1\end{pmatrix}-\sum_{j=1}^{J}\frac{C_{j}(t)e^{2ik_{j}x}}{k_{\ell}+k_{j}}\sigma^{-1}\mathbf{N}_{j}(x,t)+\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{\rho(\xi,t)e^{2i\xi
x}}{\xi+k_{\ell}}\sigma^{-1}\mathbf{N}(x,\xi,t)\,d\xi.$ (106)

We can similarly show that $\mathbf{M}$ and $\mathbf{M}_{j}$ solve

$\displaystyle\mathbf{M}(x,k,t)=\begin{pmatrix}1\\\
0\end{pmatrix}+\sum_{j=1}^{J}\frac{\tilde{C}_{j}(t)e^{-2ik_{j}x}}{k+k_{j}}\sigma\mathbf{M}_{j}(x,t)-\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{\tilde{\rho}(\xi,t)e^{-2i\xi
x}}{\xi+k+i0}\sigma\mathbf{M}(x,\xi,t)\,d\xi,$ (107)
$\displaystyle\mathbf{M}_{\ell}(x,t)=\begin{pmatrix}1\\\
0\end{pmatrix}+\sum_{j=1}^{J}\frac{\tilde{C}_{j}(t)e^{-2ik_{j}x}}{k_{\ell}+k_{j}}\sigma\mathbf{M}_{j}(x,t)-\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{\tilde{\rho}(\xi,t)e^{-2i\xi
x}}{\xi+k_{\ell}}\sigma\mathbf{M}(x,\xi,t)\,d\xi.$ (108)

These equations can then be expanded for large $k$ and, by comparing these
expansions to those for the direct scattering problem in equations (93) and
(94), we can recover the solution at any time $q(x,t)$ from
$\mathbf{N}(x,k,t)$ as

$\displaystyle q(x,t)=\mp
2i\sum_{j=1}^{J}e^{2ik_{j}x}C_{j}(t)N_{j}^{(2)}(x,t)\pm\frac{1}{\pi}\int_{-\infty}^{\infty}\rho(\xi,t)e^{2i\xi
x}N^{(2)}(x,\xi,t)\,d\xi.$ (109)

Notice that just as the spectral expansion of the $\Theta(L_{\pm}^{A})$ split
into discrete and continuous spectra in equation (82), the solution $q$ is
composed of a sum over discrete contributions and an integral over continuous
contributions. These equations can be converted into the following GLM type
integral equations [6]

$\displaystyle\mathbf{K}(x,y;t)\pm\begin{pmatrix}1\\\
0\end{pmatrix}F(x+y;t)+\int_{x}^{\infty}\sigma^{-1}\mathbf{K}(x,s;t)F(s+y;t)\,ds=0,$
(110) $\displaystyle\mathbf{L}(x,y;t)\pm\begin{pmatrix}0\\\
1\end{pmatrix}G(x+y;t)+\int_{-\infty}^{x}\sigma^{-1}\mathbf{L}(x,s;t)G(s+y;t)\,ds=0,$
(111)

where

$\displaystyle F(x;t)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\rho(\xi,t)e^{+i\xi
x}d\xi-i\sum_{j=1}^{J}C_{j}(t)e^{ik_{j}x},$ (112) $\displaystyle G(x;t)$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\tilde{\rho}(\xi,t)e^{-i\xi
x}d\xi-i\sum_{j=1}^{J}\tilde{C}_{j}(t)e^{ik_{j}x},$ (113)

and the Jost eigenfunctions are related to the triangular kernel by

$\displaystyle\mathbf{N}(x;k,t)$ $\displaystyle=\begin{pmatrix}0\\\
1\end{pmatrix}+\int_{x}^{\infty}\mathbf{K}(x,s;t)e^{-ik(x-s)}ds,~{}\text{Im}\,k>0,$
(114) $\displaystyle\mathbf{M}(x;k,t)$ $\displaystyle=\begin{pmatrix}1\\\
0\end{pmatrix}-\int_{-\infty}^{x}\mathbf{L}(x,s;t)e^{+ik(x-s)}\,ds,~{}\text{Im}\,k>0.$
(115)

The solution of the nonlinear scalar equation can then be obtained from

$\displaystyle q(x,t)=-2K^{(1)}(x,x;t),$ (116)

where $K^{(1)}$ denotes the $1$st component of the vector $\mathbf{K}$.

##  4 The One Soliton Solution

Pure soliton solutions of the scalar general evolution equation (56) are
reflectionless, i.e., $\rho(k,t)=0$ on the real line. They are also bound
states corresponding to the discrete eigenvalues at the zeros of $a$. We note
that soliton solutions for $r=q$ do not exist when we assume $q$ and $r$
vanish sufficiently rapidly as $|x|\to\infty$. Because of this, we will only
consider $r=-q$ for the remainder of our study. For a given initial condition,
the number of discrete eigenvalues $k_{j}=\xi_{j}+i\eta_{j}$ $j=1,2,...,J$
gives the number of solitons. The general $J$-soliton solution can be reduced
to solving a linear algebraic system. For simplicity we consider the one-
soliton solution, $J=1$, although the argument we lay out can be used also for
larger $J$. We find the one-soliton solution by constructing
$\mathbf{N}(x,k,t)$ from equation (105) and (106) and then recovering $q(x,t)$
using (109). We will then explicitly verify that this solution is in fact a
solution of the general evolution equation characterized in physical space by
the completeness relation, equation (73), using complex variable methods. This
will require that we construct $\mathbf{N}(x,k,t)$ and $\mathbf{M}(x,k,t)$
(thereby $\mu_{\pm}(x,k,t)$, $\nu_{\pm}(x,k,t)$ and $\tau(k,t)$).

###  4.1 Deriving the One Soliton from Inverse Scattering

Putting $\rho=0$ and $J=1$ into equation (106) and taking $k_{1}=i\eta$ and
$C_{1}(0)=-2i\eta e^{2\eta x_{0}}$ such that $C_{1}(t)=-2i\eta e^{2\eta
x_{0}+2\eta\Theta(-\eta^{2})t}$ gives an algebraic equation for
$\mathbf{N}_{1}(x,t)$

$\displaystyle\mathbf{N}_{1}(x,t)=\begin{pmatrix}0\\\
1\end{pmatrix}+e^{-z_{t}(x)}\sigma^{-1}\mathbf{N}_{1}(x,t),$ (117)

where $z_{t}(x)=2\eta(x-x_{0})-2\eta\Theta(-\eta^{2})t$. The eigenvalue
$k_{1}$ is imaginary because for $r=-q$, all discrete eigenvalues come in
pairs $\\{k_{j},-k_{j}^{*}\\}_{j=1}^{J}$. Solving this gives

$\displaystyle
N_{1}^{(1)}(x,t)=-\frac{1}{2}\text{sech}\\{z_{t}(x)\\},~{}~{}N_{1}^{(2)}(x,t)=\frac{1}{2}\left(1+\text{tanh}\\{z_{t}(x)\\}\right),$
(118)

and then putting these components into equation (109) yields

$\displaystyle q(x,t)=2\eta\,\text{sech}\\{z_{t}(x)\\}.$ (119)

###  4.2 Verifying the One Soliton

To verify that the one soliton given in equation is truly a solution to the
general evolution equation in physical space (56), we evaluate the spectral
expansion of $\Theta(L_{\pm}^{A})$ in equation (72) at time $t$; this means we
need to know $\mu_{\pm}$, $\nu_{\pm}$ and $\tau$ at time $t$. These can be
recovered from the scattering functions $\bm{\psi}$ and $\bm{\phi}$ which are
related to $\mathbf{N}$ and $\mathbf{M}$ by equation (83). We first recover
the Jost functions. The $\mathbf{N}$ function can be constructed from
$\mathbf{N}_{1}$ using equation (105) which gives

$\displaystyle\mathbf{N}(x,k,t)=\begin{pmatrix}0\\\
1\end{pmatrix}+\frac{2i\eta
e^{-z_{t}(x)}}{k+i\eta}\sigma^{-1}\mathbf{N}_{1}(x,t)=\left(-i\eta\frac{\text{sech}\\{z_{t}(x)\\}}{k+i\eta},\frac{k+i\eta\text{tanh}\\{z_{t}(x)\\}}{k+i\eta}\right)^{T}.$
(120)

We can find the $\mathbf{M}$ Jost solutions in a similar manner to
$\mathbf{N}$ using equations (107) and (108) noting that
$\tilde{C}_{1}(0)=2i\eta e^{-2\eta x_{0}}$ and so $\tilde{C}_{1}(t)=2i\eta
e^{-2\eta x_{0}+2\eta\Theta(-\eta^{2})t}$; this is $C_{1}(t)$ with $\eta$
replaced by $-\eta$. We find

$\displaystyle
M_{1}^{(1)}(x,t)=\frac{1}{2}\left(1-\text{tanh}\\{z_{t}(x)\\}\right),~{}~{}M_{1}^{(2)}(x,t)=-\frac{1}{2}\text{sech}\\{z_{t}(x)\\},$
(121)
$\displaystyle\mathbf{M}(x,k,t)=\left(\frac{k-i\eta\text{tanh}\\{z_{t}(x)\\}}{k+i\eta},-i\eta\frac{\text{sech}\\{z_{t}(x)\\}}{k+i\eta}\right)^{T}.$
(122)

From these, we can construct $\bm{\psi}$ and $\bm{\phi}$ using equation (83).
Using the wronskian relation in equation (16) and the fact that $\tau=1/a$, we
have

$\displaystyle\tau(k)=\frac{k+i\eta}{k-i\eta}.$ (123)

We then build $\mu_{-}$ and $\nu_{-}$, the eigenfunctions for the scalar
system. These are

$\displaystyle\mu_{-}(x,k,t)$
$\displaystyle=e^{2ikx}\frac{\left(\eta^{2}-k^{2}+2ik\eta\text{tanh}\\{z_{t}(x)\\}\right)}{\left(k+i\eta\right)^{2}},$
(124) $\displaystyle\nu_{-}(x,k,t)$
$\displaystyle=e^{-2ikx}\frac{\left(k^{2}-\eta^{2}-2ik\eta\text{tanh}\\{z_{t}(x)\\}+2\eta^{2}\text{sech}^{2}\\{z_{t}(x)\\}\right)}{\left(k+i\eta\right)^{2}}.$
(125)

Therefore, we can construct the kernel
$g_{-}(x,y,k,t)=\tau^{2}(k)\nu_{-}(x,k,t)\mu_{-}(y,k,t)/\pi$ inside of the
spectral definition of $\Theta(L_{-}^{A})$ in equation (72). We can now show
that the soliton solution for $q$ given in equation (119) is truly a solution
to the general evolution equation (56) by explicitly demonstrating that
equation (73) holds. We will evaluate the operator

$\displaystyle\Theta(L_{-}^{A})\partial_{x}q(x,t)=\int_{\Gamma_{\infty}^{(1)}}dk\Theta(k^{2})\int_{-\infty}^{\infty}dy\,g_{-}(x,y,k,t)\partial_{y}q(y,t),$
(126)

and show that it is equivalent to $-q_{t}$ where $q$ is defined by equation
(119). It is most prudent to split this into its continuous and discrete parts
as in equation (82). As we will see, the portion of the operator related to
the continuous spectra will vanish while that associated to the single
eigenvalue $k_{1}$ will satisfy the above equation. This comes from the
integral

$\displaystyle
I(k,t)=\int_{-\infty}^{\infty}\mu_{-}(y,k,t)\partial_{y}q(y,t)\,dy=0,$ (127)

for all $k$. Looking at equation (82), we can see that this statement implies
that the continuous portion, the integral of $g_{-}(x,y,k)$ over the real
line, vanishes. It also implies that the integrals associated to the discrete
kernels $g^{(2)}_{-,1}(x,y)$ and $g^{(3)}_{-,1}(x,y)$ and the first half of
the $g^{(1)}_{-,1}(x,y)$ integral are zero. The single term that does not
vanish is

$\displaystyle\Theta(L_{-}^{A})\partial_{x}q(x,t)=-\frac{2i}{(a_{1}^{\prime})^{2}}\Theta(-\eta^{2})\nu_{-}(x,i\eta,t)\int_{-\infty}^{\infty}\partial_{k}\mu_{-}(y,k,t)|_{k=i\eta}\partial_{y}q(y)\,dy.$
(128)

Hence, we expect $-q_{t}(x,t)=\Theta(L_{-}^{A})\partial_{x}q(x,t)$. By making
the change of variables $\xi=z_{t}(y)$, $d\xi=z_{t}^{\prime}(y)dy=2\eta dy$
where $z_{t}(x)=2\eta(x-x_{0})-2\eta\Theta(-\eta^{2})t$, the above integral
becomes

$\displaystyle\int_{-\infty}^{\infty}\partial_{k}\mu_{-}(y,k,t)|_{k=i\eta}\partial_{y}q(y)\,dy=-ie^{z_{t}(x)-2\eta
x}\int_{-\infty}^{\infty}\left(2\eta
x-z_{t}(x)+\xi\right)\text{sech}^{2}\xi\text{tanh}\xi\,d\xi.$ (129)

Of the three terms in the integral, the first two vanish because the function
is odd. The final term, i.e., $\xi\text{sech}^{2}\xi\text{tanh}\xi$, can be
evaluated using integration by parts and the fundamental theorem of calculus
to give

$\displaystyle\int_{-\infty}^{\infty}\xi\text{sech}^{2}\xi\text{tanh}\xi\,d\xi=\frac{1}{2}\int_{-\infty}^{\infty}\text{sech}^{2}\xi\,d\xi=1$
(130)

Therefore, evaluating $\nu_{-}(x,k,t)$ in equation (125) at $i\eta$ and using
$(a^{\prime}_{1})^{2}=-4\eta^{2}$ we find

$\displaystyle\Theta(L_{-}^{A})\partial_{x}q(x,t)=-4\eta^{2}\Theta(-\eta^{2})\text{sech}\\{z_{t}(x)\\}\text{tanh}\\{z_{t}(x)\\}.$
(131)

Comparing this to $q_{t}$

$\displaystyle
q_{t}(x,t)=4\eta^{2}\Theta(-\eta^{2})\text{sech}\\{z_{t}(x)\\}\text{tanh}\\{z_{t}(x)\\},$
(132)

we notice that equation (126) is true provided equation (127) holds, which we
will now confirm. Again, we make the change of variables $\xi=z_{t}(y)$ so
that the integral $I(k,t)$ becomes

$\displaystyle
I(k,t)=A(k,t)\int_{-\infty}^{\infty}\tilde{\mu}_{-}(\xi,k)q^{\prime}(\xi)\,d\xi$
(133)

where $A(k,t)=e^{2ikx_{0}+2ik\Theta(-\eta^{2})t}$ and

$\displaystyle\tilde{\mu}_{-}(\xi,k)$
$\displaystyle=e^{ik\xi/\eta}\frac{\left(k^{2}-\eta^{2}+2ik\eta\text{tanh}\xi\right)}{\left(k+i\eta\right)^{2}},$
(134) $\displaystyle q^{\prime}(\xi)$
$\displaystyle=-2\eta\,\text{sech}\xi\text{tanh}\xi.$ (135)

If we introduce

$\displaystyle I_{1}(k)$
$\displaystyle=\int_{-\infty}^{\infty}e^{ik\xi/\eta}\text{sech}\xi\text{tanh}\xi\,d\xi,$
(136) $\displaystyle I_{2}(k)$
$\displaystyle=\int_{-\infty}^{\infty}e^{ik\xi/\eta}\text{sech}\xi\text{tanh}^{2}\xi\,d\xi,$
(137)

$I(k,t)$ can be written as

$\displaystyle
I(k,t)=-A(k,t)\frac{2\eta}{(k+i\eta)^{2}}\left[(k^{2}-\eta^{2})I_{1}(k)+2ik\eta
I_{2}(k)\right].$ (138)

Using integration by parts, we can put $I_{1}$ in terms of the Fourier
transform of the hyperbolic secant.

$\displaystyle
I_{1}(k)=-\int_{-\infty}^{\infty}e^{ik\xi/\eta}\partial_{\xi}\text{sech}\xi\,d\xi=\frac{ik}{\eta}\int_{-\infty}^{\infty}e^{ik\xi/\eta}\text{sech}\xi\,d\xi$
(139)

This Fourier transform can be evaluated by forming a rectangular contour in
the complex plane with corners at $z=-R$, $z=R$, $z=R+i\pi$ and $z=-R+i\pi$.
Integrating around this contour, the pole at $z=i\pi/2$ is the only residue.
Then — in the limit $R\to\infty$ — the left and right sides of the contour
integral vanish, and the top contour can be written in terms of the bottom
contour (which is the integral along the real line we want when $R\to\infty$)
because
$e^{ik(\xi+i\pi)/\eta}\text{sech}(\xi+i\pi)=-e^{-k\pi/\eta}e^{ik\xi/\eta}\text{sech}\xi$.
With these notes, $I_{1}$ is found to be

$\displaystyle I_{1}(k)=i\frac{k\pi}{\eta}\text{sech}\frac{k\pi}{2\eta}.$
(140)

Then, using a tricky manipulation, $I_{2}$ can be written in terms of $I_{1}$
as

$\displaystyle
I_{2}(k)=\left[\frac{ik}{2\eta}+\frac{\eta}{2ik}\right]I_{1}(k).$ (141)

Taking equations (140) and (141), we find that

$\displaystyle I(k,t)=0.$ (142)

Therefore,

$\displaystyle\Theta(L_{-}^{A})\partial_{x}q(x,t)=-q_{t}(x,t),$ (143)

and the one soliton in (119) is a solution to the general evolution equation
in (56).

###  4.3 The One Soliton for fmKdV and fSG

The general nonlinear evolution equation becomes the fmKdV equation when we
put $\Theta(L_{-}^{A})=-4L_{-}^{A}|4L_{-}^{A}|^{\epsilon}$ and it becomes the
fsineG equations with
$\Theta(L_{-}^{A})=\frac{|4L_{-}^{A}|^{\epsilon}}{4L_{-}^{A}}$ where
$0<\epsilon<1$. Therefore, for these two equations, the one-soliton solution
given in (119) becomes

$\displaystyle q_{m}(x,t)$
$\displaystyle=2\eta\,\text{sech}\left\\{2\eta(x-x_{0})-(2\eta)^{3+2\epsilon}t\right\\},$
(144) $\displaystyle q_{SG}(x,t)$
$\displaystyle=2\eta\,\text{sech}\left\\{2\eta(x-x_{0})+(2\eta)^{-1+2\epsilon}t\right\\}.$
(145)

We also find the ”kink” solution $u$ from $q_{SG}=u_{x}/2$ to be

$\displaystyle
u(x,t)=\arctan{\text{sinh}\left\\{2\eta(x-x_{0})+(2\eta)^{2\epsilon-1}t\right\\}}.$
(146)

Both solutions are traveling waves which propagate without dissipating. The
speed of the two solitons are given by

$\displaystyle v_{m}(\eta)$ $\displaystyle=(2\eta)^{2+2\epsilon}$ (147)
$\displaystyle v_{SG}(\eta)$ $\displaystyle=(2\eta)^{-2+2\epsilon}$ (148)

Notice that the fractional equations predict power law relationships between
the speed of the wave and the amplitude of the wave, $\eta$. Therefore, the
fmKdV and fsineG equations predict anomalous dispersion, showing that this is
a common characteristic of fractional nonlinear systems.

Figure 1:  Localized waves predicted by the fmKdV and fsineG equations,
equations (147-148), show super-dispersive transport as their velocity
increases for $0<\epsilon<1$. Notice that the fmKdV velocity is scaled by
$1/12$. Also, at $\epsilon=1$, the resulting nonlinear equations are described
by integer operators. Just as in anomalous diffusion where the mean squared
displacement is proportional to $t^{\alpha}$, the velocity in anomalous
dispersion is proportional to $A^{\epsilon}$, where $A$ is the amplitude of
the wave. Here $\eta=3/2$.

##  5 Conclusion

We developed fractional extensions of the modified KdV, sine-Gordon, and sinh-
Gordon equations on the line with decaying data. This process requires three
key steps: a general evolution equation solvable by the inverse scattering
transformation, completeness of squared eigenfunctions, and an anomalous
dispersion relation. We demonstrated these three elements by developing a
scalar general evolution equation using a symmetry reduction of the AKNS
scattering system. Then, we found the fmKdV, fsineG, and fsinhG equations as a
special case of this general evolution equation using the anomalous dispersion
relations of the linear fmKdV, fsineG, and fsinhG equations, respectively.
From scattering theory for the AKNS system, we found squared eigenfunctions
and their associated operators for the scalar scattering problem. We then re-
expressed completeness of the AKNS system in terms of these scalar squared
eigenfunctions to give a spectral representation of the operator
$\Theta(L_{\pm})$ in the general evolution equation. We developed the direct
scattering, time evolution, and inverse scattering for the scalar scattering
system and used these to derive the one-soliton solution for fmKdV and fsineG.
We used the completeness relation to verify that these one-soliton solutions
were truly solutions of fmKdV and fsineG. Finally, we showed that the one-
soliton solutions of fmKdV and fsineG have power law relationships between the
soliton’s amplitude and velocity. This super-dispersive transport is an
experimentally testable prediction of this theory.

## Acknowledgements

We thank U. Al-Khawaja, J. Lewis, and A. Gladkina for useful discussions. This
project was partially supported by NSF under grants DMS-2005343 and
DMR-2002980.

## References

  * [1] M. Ablowitz, J. Been and L. Carr  “Fractional Integrable Nonlinear Soliton Equations”  In _Phys. Rev. Lett., in press_ American Physical Society, 2022
  * [2] MA Ablowitz, PA Clarkson and Peter A Clarkson  “Solitons, nonlinear evolution equations and inverse scattering”  Cambridge university press, 1991
  * [3] MA Ablowitz, B Prinari and AD Trubatch  “Discrete and continuous nonlinear Schrödinger systems”  Cambridge University Press, 2004
  * [4] Mark J Ablowitz  “Nonlinear dispersive waves: asymptotic analysis and solitons”  Cambridge University Press, 2011
  * [5] Mark J Ablowitz and Harvey Segur  “Solitons and the inverse scattering transform”  SIAM, 1981
  * [6] Mark J Ablowitz, David J Kaup, Alan C Newell and Harvey Segur  “The inverse scattering transform-Fourier analysis for nonlinear problems”  In _Studies in Applied Mathematics_ 53.4 Wiley Online Library, 1974, pp. 249–315
  * [7] Mark J Ablowitz, David J Kaup, Alan C Newell and Harvey Segur  “The inverse scattering transform-Fourier analysis for nonlinear problems”  In _Studies in Applied Mathematics_ 53.4 Wiley Online Library, 1974, pp. 249–315
  * [8] David A Benson, Stephen W Wheatcraft and Mark M Meerschaert  “Application of a fractional advection-dispersion equation”  In _Water resources research_ 36.6 Wiley Online Library, 2000, pp. 1403–1412
  * [9] David A Benson, Rina Schumer, Mark M Meerschaert and Stephen W Wheatcraft  “Fractional dispersion, Lévy motion, and the MADE tracer tests”  In _Transport in porous media_ 42.1 Springer, 2001, pp. 211–240
  * [10] I. Bronstein et al.  “Transient Anomalous Diffusion of Telomeres in the Nucleus of Mammalian Cells”  In _Phys. Rev. Lett._ 103 American Physical Society, 2009, pp. 018102  DOI: 10.1103/PhysRevLett.103.018102
  * [11] Claudia Bucur and Enrico Valdinoci  “Nonlocal Diffusion and Applications”  In _Lecture Notes of the Unione Matematica Italiana_ Springer International Publishing, 2016  DOI: 10.1007/978-3-319-28739-3
  * [12] José Antonio Carrillo et al.  “Nonlocal and nonlinear diffusions and interactions: new methods and directions”  Springer, 2017
  * [13] Peter Constantin and Jiahong Wu  “Behavior of solutions of 2D quasi-geostrophic equations”  In _SIAM journal on mathematical analysis_ 30.5 SIAM, 1999, pp. 937–948
  * [14] Clifford S Gardner, John M Greene, Martin D Kruskal and Robert M Miura  “Method for solving the Korteweg-deVries equation”  In _Physical review letters_ 19.19 APS, 1967, pp. 1095
  * [15] Qing Gu et al.  “Non-Gaussian Transport Measurements and the Einstein Relation in Amorphous Silicon”  In _Phys. Rev. Lett._ 76 American Physical Society, 1996, pp. 3196–3199  DOI: 10.1103/PhysRevLett.76.3196
  * [16] Sverre Holm  “Waves with power-law attenuation”  Springer, 2019
  * [17] D.J Kaup  “Closure of the squared Zakharov-Shabat eigenstates”  In _Journal of Mathematical Analysis and Applications_ 54.3, 1976, pp. 849–864  DOI: https://doi.org/10.1016/0022-247X(76)90201-8
  * [18] Diederik Johannes Korteweg and Gustav De Vries  “XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves”  In _The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science_ 39.240 Taylor & Francis, 1895, pp. 422–443
  * [19] Eva Koscielny-Bunde et al.  “Indication of a universal persistence law governing atmospheric variability”  In _Physical Review Letters_ 81.3 APS, 1998, pp. 729
  * [20] Anna Lischke et al.  “What is the fractional Laplacian? A comparative review with new results”  In _Journal of Computational Physics_ 404, 2020, pp. 109009  DOI: https://doi.org/10.1016/j.jcp.2019.109009
  * [21] Mark M Meerschaert, Yong Zhang and Boris Baeumer  “Tempered anomalous diffusion in heterogeneous systems”  In _Geophysical Research Letters_ 35.17 Wiley Online Library, 2008
  * [22] Mark M. Meerschaert and Alla Sikorskii  “Stochastic Models for Fractional Calculus”  De Gruyter, 2011  DOI: doi:10.1515/9783110258165
  * [23] Ralf Metzler and Joseph Klafter  “The random walk’s guide to anomalous diffusion: a fractional dynamics approach”  In _Physics Reports_ 339.1, 2000, pp. 1–77  DOI: https://doi.org/10.1016/S0370-1573(00)00070-3
  * [24] Arturo Pablo, Fernando Quirós, Ana Rodrı́guez and Juan Luis Vázquez  “A fractional porous medium equation”  In _Advances in Mathematics_ 226.2 Elsevier, 2011, pp. 1378–1409
  * [25] G. Pfister and H. Scher  “Time-dependent electrical transport in amorphous solids: ${\mathrm{As}}_{2}$ ${\mathrm{Se}}_{3}$”  In _Phys. Rev. B_ 15 American Physical Society, 1977, pp. 2062–2083  DOI: 10.1103/PhysRevB.15.2062
  * [26] Constantine Pozrikidis  “The Fractional Laplacian”  CRC Press, 2018
  * [27] Benjamin M Regner et al.  “Anomalous diffusion of single particles in cytoplasm”  In _Biophysical journal_ 104.8 Elsevier, 2013, pp. 1652–1660
  * [28] Marcel Riesz  “L’intégrale de Riemann-Liouville et le probléme de Cauchy”  In _Acta mathematica_ 81, 1949, pp. 1–222
  * [29] Michael J. Saxton  “A Biological Interpretation of Transient Anomalous Subdiffusion. I. Qualitative Model”  In _Biophysical Journal_ 92.4, 2007, pp. 1178–1191  DOI: https://doi.org/10.1529/biophysj.106.092619
  * [30] Harvey Scher and Elliott W. Montroll  “Anomalous transit-time dispersion in amorphous solids”  In _Phys. Rev. B_ 12 American Physical Society, 1975, pp. 2455–2477  DOI: 10.1103/PhysRevB.12.2455
  * [31] M.. Shlesinger, B.. West and J. Klafter  “Lévy dynamics of enhanced diffusion: Application to turbulence”  In _Phys. Rev. Lett._ 58 American Physical Society, 1987, pp. 1100–1103  DOI: 10.1103/PhysRevLett.58.1100
  * [32] Wei Wang et al.  “Fractional Brownian motion with random diffusivity: emerging residual nonergodicity below the correlation time”  In _Journal of Physics A: Mathematical and Theoretical_ 53.47 IOP Publishing, 2020, pp. 474001  DOI: 10.1088/1751-8121/aba467
  * [33] Aubrey V Weigel, Blair Simon, Michael M Tamkun and Diego Krapf  “Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking”  In _Proceedings of the National Academy of Sciences_ 108.16 National Acad Sciences, 2011, pp. 6438–6443
  * [34] Bruce J. West, Paolo Grigolini, Ralf Metzler and Theo F. Nonnenmacher  “Fractional diffusion and Lévy stable processes”  In _Phys. Rev. E_ 55 American Physical Society, 1997, pp. 99–106  DOI: 10.1103/PhysRevE.55.99
  * [35] Norman J Zabusky and Martin D Kruskal  “Interaction of” solitons” in a collisionless plasma and the recurrence of initial states”  In _Physical review letters_ 15.6 APS, 1965, pp. 240