# The nonlinear Schrödinger equation:
A mathematical model with its wide-ranging applications

Natanael Karjanto
Department of Mathematics, University College, Natural Science Campus
Sungkyunkwan University, Suwon 16419, Republic of Korea 🖂<EMAIL_ADDRESS>
###### Abstract

The nonlinear Schrödinger equation (NLSE) models the slowly varying envelope
dynamics of a weakly nonlinear quasi-monochromatic wave packet in dispersive
media. In the context of Bose-Einstein condensate (BEC), it is often referred
to as the Gross-Pitaevskii equation (GPE). The NLSE is one example of
integrable systems of a nonlinear partial differential equation (PDE) in
$(1+1)$D and it possesses an infinite set of conservation laws. This nonlinear
evolution equation arises in various physical settings and admits a wide range
of applications, including but not limited to, surface gravity waves,
superconductivity, nonlinear optics, and BEC. This chapter discusses not only
the modeling aspect of the NLSE but also provides an overview of the
applications in these four exciting research areas. The former features
derivations of the NLSE heuristically and by employing the method of multiple-
scale from other mathematical models as governing equations. Depending on how
the variables are interpreted physically, the resulting NLSE can model a
different dynamics of the wave packet. Furthermore, depending on the adopted
assumptions and the chosen governing equations, each approach may provide
different values for the corresponding dispersive and nonlinear coefficients.

###### Contents

  1. 1 Introduction
  2. 2 Hydrodynamics
    1. 2.1 Heuristic derivation for the NLSE
    2. 2.2 Derivation of the temporal NLSE
    3. 2.3 Derivation of the spatial NLSE
    4. 2.4 Applications in surface gravity waves
  3. 3 Superconductivity
    1. 3.1 NLSE derivation from the nonlinear Klein-Gordon equation
    2. 3.2 Applications of sine-Gordon model
  4. 4 Nonlinear optics
    1. 4.1 NLSE derivation from Maxwell’s and Helmholtz’ equations
    2. 4.2 Applications in nonlinear optics
  5. 5 Bose-Einstein condensation
    1. 5.1 NLSE derivation from Bose-Einstein condensed state
    2. 5.2 Applications in BEC
  6. 6 Conclusion
  7. References

##  1 Introduction

A lot of physical and natural phenomena can be modeled mathematically using
partial differential equations (PDEs). In the realm of linear theory,
solutions of PDEs obey the principle of linear superposition, and in some
cases, they possess explicit analytical expressions. However, the laws of the
nature are not always linear, and nonlinear PDEs often play an essential role
in modeling these phenomena. Having applications in practically all areas of
the natural sciences, the theory of PDEs, both linear and nonlinear, is one of
the largest and most active areas of modern mathematics.

Historically, the subject of PDEs sprang as a study on the geometry of
surfaces and tackled various problems in mechanics. It continues with the
historically developed calculus of variations to bridge the theory of surfaces
with the understanding of physical problems. Particularly, the study of wave
propagation problems has, in turn, stimulated further developments in the
general theory of nonlinear PDEs. Some examples of famous nonlinear PDEs
amongst others are the nonlinear kinematic wave equation, the nonlinear Klein-
Gordon equation, the Burgers equation, the Fischer equation, the Boussinesq
equation, the Korteweg-de Vries equation (KdVE), the nonlinear Schrödinger
equation (NLSE), the Benjamin-Ono equation, the Benjamin-Bona-Mahoney
equation, the Kadomtsev-Petviashvili equation, the Davey-Stewartson equation,
and the Camassa-Holm equation [Debnath 2012].

This chapter specifically deals with the NLSE. In particular, we will provide
an overview on modeling and application aspects of the model in $(1+1)$D, the
simplest domain of space and time variables. The nonlinear term in the NLSE
that we are discussing is in the form of the cubic power and thus the equation
is also sometimes called the cubic Schrödinger equation (CSE). In the context
of Bose-Einstein condensation (BEC), the NLSE is known as the Gross-Pitaevskii
equation (GPE). The NLSE models an evolution equation for slowly varying
envelope dynamics of a weakly nonlinear quasi-monochromatic wave packet in
dispersive media. It possesses an infinite set of conservation laws and the
model is a completely integrable system by the inverse scattering transform
[Ablowitz et al 1974, Ablowitz and Segur 1981, Ablowitz and Clarkson 1991].
Indeed, having fundamental knowledge on the NLSE is essential in understanding
the general theory of nonlinear dispersive waves.

In the absence of the nonlinear term, the NLSE reduces to the well-known
Schrödinger equation, a linear PDE that governs a wave function characterizing
the state of a quantum-mechanical system. The equation is named after the
Austrian physicist Erwin Schrödinger who developed a number of fundamental
results in the field of quantum theory [Schrödinger 1926]. The Schrödinger
equation can be derived using the mathematical formulation of quantum
mechanics in terms of operators in Hilbert space introduced by [Dirac 1930]
and [von Neumann 1932], known as the Dirac-von Neumann axioms. The readers who
are interested in more detailed discussion on the Schrödinger equation may
consult any textbook in quantum mechanics, such as [Griffiths and Schroeter
2018, Phillips 2003] or [Shankar 1994]. The discussion in this chapter will
focus more on the NLSE instead of the (linear) Schrödinger equation.

The NLSE arises in various physical settings in fluid mechanics and
hydrodynamics describing the evolution of surface gravity water waves. It also
has extensive applications in nonlinear optics, plasma physics and
magnetohydrodynamics, solid state physics characterizing the propagation of a
heat pulse in a solid, superconductivity describing solitary waves propagation
in piezoelectric semiconductors, condensed matter such as BEC, and even
mathematical finance. In this chapter, we will focus only on four of these
applications: water waves, superconductivity, nonlinear optics and BEC.

This chapter is organized as follows. After this introduction, Section 2
discusses NLSE modeling and application in hydrodynamics. We derive both
linear and nonlinear Schrödinger equations heuristically. By implementing the
method of multiple-scale [Kevorkian and Cole 2012, Kevorkian and Cole 2013,
Nayfeh 2008], it follows with the derivation of the temporal and the spatial
NLSEs. An overview of applications in surface water waves will be presented.
Section 3 covers the NLSE derivation from the nonlinear Klein-Gordon equation
using the same technique as in Section 2. We also consider applications of the
sine-Gordon models in superconductivity. Section 4 reviews modeling and
applications of the NLSE in the field of nonlinear optics. We adopt Maxwell’s
and Helmholtz’ equations as a starting point for the derivation of the NLSE.
Section 5 describes the derivation of the GPE by means of the mean-field
theory and its applications in the emerging field of BEC. Finally, Section 6
provides concluding remarks to our discussion.

##  2 Hydrodynamics

###  2.1 Heuristic derivation for the NLSE

The following heuristic derivation for the NLSE is based on [Debnath 1994,
343–345, Dingemans 1997, 888–890]. For a linear dispersive wave equation, we
can express its general solution in terms of a Fourier transform
representation. Suppose we have the following linear equation governing the
evolution of the surface elevation $\eta(x,t)$:

$\partial_{t}\eta+i\Omega(-i\partial_{x})\eta=0,$ (1)

then its general solution $\eta(x,t)$ expressed by the Fourier representation
is given by

$\eta(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\zeta)\,e^{i(kx-\omega
t)}\,\mathrm{d}\zeta.$ (2)

Here, we can replace the variable $\zeta$ either with wavenumber $k$ or with
frequency $\omega$, and both are related by the linear dispersion relationship
$\omega=\Omega(k)$ or $k=K(\omega)$, where $K=\Omega^{-1}$. The spectrum
function $F(\zeta)$ will be determined from a given initial or boundary
condition. For an initial value problem (IVP), $F(k)$ is the Fourier transform
of the initial condition $\eta(x,0)$. Correspondingly, the Fourier transform
of the initial signal $\eta(0,t)$ is given by $F(\omega)$ in the case of a
boundary value problem (BVP). They are given as follows:

$\displaystyle F(k)$
$\displaystyle=\int_{-\infty}^{\infty}\eta(x,0)\,e^{-ikx}\,\mathrm{d}x$ (3)
$\displaystyle F(\omega)$
$\displaystyle=\int_{-\infty}^{\infty}\eta(0,t)\,e^{i\omega t}\,\mathrm{d}t.$
(4)

We adopt an assumption of a slowly modulated wave as it propagates in a
dispersive medium, and hence $F(k)$ and $F(\omega)$ are narrow-banded spectra
around $k_{0}$ and $\omega_{0}$, respectively. The linear dispersion
relationship can be expressed in its Taylor-expansion series about the basic
state wavenumber $k_{0}$ and frequency $\omega_{0}$, written as follows:

$\displaystyle\Omega(k)$
$\displaystyle=\omega_{0}+\Omega^{\prime}(k_{0})(k-k_{0})+\frac{1}{2!}\Omega^{\prime\prime}(k_{0})(k-k_{0})^{2}+\dots$
(5) $\displaystyle K(\omega)$
$\displaystyle=k_{0}+K^{\prime}(\omega_{0})(\omega-\omega_{0})+\frac{1}{2!}K^{\prime\prime}(\omega_{0})(\omega-\omega_{0})^{2}+\dots.$
(6)

Therefore, we can rewrite (2) as
$\eta(x,t)=A(\xi,\tau)e^{i(k_{0}x-\omega_{0}t)}$, where $A(\xi,\tau)$ is the
corresponding complex-valued amplitude of the wave packet $\eta(x,t)$, written
in two different versions:

$\displaystyle A(\xi_{1},\tau_{1})$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(k_{0}+\kappa)\,e^{i(\xi_{1}-\Omega_{\textmd{res}}(k_{0})\tau_{1}/\kappa^{2})}\,\mathrm{d}\kappa,$
(7) $\displaystyle A(\xi_{2},\tau_{2})$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega_{0}+\nu)\,e^{-i(\tau_{2}-K_{\textmd{res}}(\omega_{0})\xi_{2}/\nu^{2})}\,\mathrm{d}\nu.$
(8)

Here, $\kappa=k-k_{0}=\mathcal{O}(\epsilon)$,
$\xi_{1}=\kappa(x-\Omega^{\prime}(k_{0})t)$, $\tau_{1}=\kappa^{2}t$,
$\nu=\omega-\omega_{0}=\mathcal{O}(\epsilon)$, $\xi_{2}=\nu^{2}x$,
$\tau=\nu(t-K^{\prime}(\omega_{0})x)$, and $0<\epsilon\ll 1$ is a small
positive parameter. The residual terms appearing in the exponential term read

$\displaystyle\Omega_{\textmd{res}}(k_{0})$
$\displaystyle=\Omega(k)-[\omega_{0}+\Omega^{\prime}(k_{0})\kappa]=\kappa^{2}\left(\frac{1}{2!}\Omega^{\prime\prime}(k_{0})+\frac{1}{3!}\Omega^{\prime\prime\prime}(k_{0})\kappa+\dots\right)$
$\displaystyle K_{\textmd{res}}(\omega_{0})$
$\displaystyle=K(\omega)-[k_{0}+K^{\prime}(\omega_{0})\nu]=\nu^{2}\left(\frac{1}{2!}K^{\prime\prime}(\omega_{0})+\frac{1}{3!}K^{\prime\prime\prime}(\omega_{0})\nu+\dots\right).$

From the complex-amplitude representations (7) and (8), it follows that
$\kappa$ (respectively $\nu$) are associated with the differential operator
$i\partial_{\xi}$ (respectively $-i\partial_{\tau}$) and thus,
$\kappa^{2}=-\partial_{\xi}^{2}$ (respectively
$\nu^{2}=-\partial_{\tau}^{2}$). Thus, the complex-valued amplitude $A$
satisfies

$\displaystyle\partial_{\tau}A+i\Omega_{\textmd{res}}(i\partial_{\xi})A$
$\displaystyle=0$ (9)
$\displaystyle\partial_{\xi}A+iK_{\textmd{res}}(-i\partial_{\tau})A$
$\displaystyle=0.$ (10)

For a narrow-banded spectrum, the equations (9) and (10) reduce to approximate
equations called the temporal and the spatial “linear Schrödinger” equations
by approximating $\Omega_{\textmd{res}}$ and $K_{\textmd{res}}$ in their
lowest-order terms, respectively

$\displaystyle i\partial_{\tau}A+\beta_{1}\partial_{\xi}^{2}A$
$\displaystyle=0,$ (11) $\displaystyle
i\partial_{\xi}A+\beta_{2}\partial_{\tau}^{2}A$ $\displaystyle=0.$ (12)

Here, the dispersion coefficients
$\beta_{1}=\frac{1}{2}\Omega^{\prime\prime}(k_{0})$ and
$\beta_{2}=-\frac{1}{2}K^{\prime\prime}(\omega_{0})=\frac{1}{2}\frac{\Omega^{\prime\prime}(k_{0})}{[\Omega^{\prime}(k_{0})]^{3}}$.

Before proceeding to derive the NLSE heuristically, we make a quick note on
the local existence and uniqueness for an IVP of the temporal linear
Schrödinger equation (LSE) (11). Physically, this evolution equation exhibits
a dispersion phenomenon. It means that the group velocity $\Omega^{\prime}(k)$
depends on the wavenumber $k$ and waves with different frequencies travel at
different speed. Consequently, a traveling and localized wave packet will
dissolve.

######  Definition 1 (Fundamental solution).

The function

$\Psi(\xi,\tau):=\frac{1}{\sqrt{4\pi
i\beta_{1}\tau}}e^{\frac{i|\xi|^{2}}{4\beta_{1}\tau}},\qquad\xi\in\mathbb{R},\quad\tau\neq
0.$ (13)

is called the fundamental solution of the LSE (11).

######  Theorem 2.

The corresponding initial value problem LSE (11) with an initial condition
$A(\xi,0)$ admits an exact solution using the fundamental solution (13), given
explicitly as follows:

$A(\xi,\tau)=\frac{1}{\sqrt{4\pi
i\beta_{1}\tau}}\int_{-\infty}^{\infty}e^{-\frac{(\xi-\zeta)^{2}}{4i\beta_{1}\tau}}A(\zeta,0)\,\mathrm{d}\zeta,\qquad\xi\in\mathbb{R},\quad\tau>0.$
(14)

The proof of this theorem can be found in [Evans 2010, Fibich 2015] using the
Fourier transform method and we have adopted the following convention for the
Fourier transform definition in this chapter.

######  Definition 3 (Fourier transform).

Let $f$ be a defined function on $\xi\in\mathbb{R}$, then the Fourier
transform of $f$ and its inverse Fourier transform are given as follows:

$\displaystyle\hat{f}(\kappa)={\cal F}\left\\{f(\xi)\right\\}$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(\xi)e^{-i\kappa\xi}\,\mathrm{d}\xi$
(15) $\displaystyle f(\xi)={\cal F}^{-1}\left\\{\hat{f}(\kappa)\right\\}$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\hat{f}(\kappa)e^{i\kappa\xi}\,\mathrm{d}\kappa.$
(16)

###### Proof.

Let $\hat{A}(\kappa,\tau)={\cal F}\\{A(\xi,\tau)\\}$ be the Fourier transform
of the complex-valued amplitude $A(\xi,\tau)$. Taking the Fourier transform of
the LSE (11) and its initial condition yields the following, respectively

$i\partial_{\tau}\hat{A}-\beta_{1}\kappa^{2}\hat{A}=0,\qquad\qquad\text{and}\qquad\qquad\hat{A}(\kappa,0).$
(17)

The solution to this ODE is

$\hat{A}(\kappa,\tau)=\hat{A}(\kappa,0)e^{-i\beta_{1}\kappa^{2}\tau}.$ (18)

Taking the inverse of the Fourier transform (18), we obtain the solution in
convolution form

$A(\xi,\tau)=\frac{1}{\sqrt{2\pi}}A(\xi,0)\ast{\cal
F}^{-1}\left\\{e^{-i\beta_{1}\kappa^{2}\tau}\right\\}.$ (19)

Using the fact that the Fourier transform of a Gaussian-shape function is also
Gaussian-shape profile

${\cal
F}\left\\{e^{-\frac{p}{2}\xi^{2}}\right\\}=\frac{1}{\sqrt{p}}e^{-\frac{\kappa^{2}}{2p}},\qquad\text{and}\qquad{\cal
F}^{-1}\left\\{e^{-i\beta_{1}\kappa^{2}\tau}\right\\}=\frac{1}{\sqrt{2i\beta_{1}\tau}}e^{-\frac{\xi^{2}}{4i\beta_{1}\tau}},$
(20)

with $p=1/(2i\beta_{1}\tau)$, we then obtain the desired expression

$A(\xi,\tau)=\frac{1}{\sqrt{4\pi
i\beta_{1}\tau}}\int_{-\infty}^{\infty}e^{-\frac{(\xi-\zeta)^{2}}{4i\beta_{1}\tau}}A(\zeta,0)\,\mathrm{d}\zeta,\qquad\xi\in\mathbb{R},\quad\tau>0.$
(21)

∎

This solution does make sense for all positive time $t>0$ provided that the
initial condition $A(\xi,0)\in L^{1}(\mathbb{R})$. From this expression, we
obtain the following basic $L^{\infty}$-estimate [Strauss 1989, Cazenave
1996a, Teschl 2014]

$\displaystyle\left\|A(\xi,\tau)\right\|_{L^{\infty}(\mathbb{R})}=\sup_{\xi\in\mathbb{R}}\left|A(\xi,\tau)\right|$
$\displaystyle\leq\frac{1}{\sqrt{4\pi\beta_{1}\tau}}\int_{-\infty}^{\infty}\left|A(\xi,0)\right|\,\mathrm{d}\xi$
$\displaystyle=\frac{1}{\sqrt{4\pi\beta_{1}\tau}}\left\|A(\xi,0)\right\|_{L^{1}(\mathbb{R})}.$
(22)

This indicates that any solution with spatially localized initial conditions
will decay uniformly toward zero with a rate of $\sqrt{\tau}$ [Schneider
2011]. Furthermore, using Plancherel’s theorem, if the initial condition
$A(\xi,0)\in L^{1}(\mathbb{R})\cap L^{2}(\mathbb{R})$, then the $L^{2}$-norm
is preserved

$\left\|A(\xi,\tau)\right\|_{L^{2}(\mathbb{R})}=\left\|A(\xi,0)\right\|_{L^{2}(\mathbb{R})},\qquad\qquad\forall\quad
t>0.$ (23)

In the following, we will derive the NLSE from the nonlinear dispersion
relationship that includes a slowly-varying real-valued amplitude
$a(x,t)\sim\mathcal{O}(\epsilon)$:

$\omega=\Omega(k,a^{2})\qquad\qquad\text{or}\qquad\qquad k=K(\omega,a^{2}).$
(24)

We perform a Taylor series expansion about the basic state wavenumber
$k=k_{0}$ (or the basic state frequency $\omega=\omega_{0}$) and the zero
amplitude $|a|^{2}=0$

$\displaystyle\omega$
$\displaystyle=\Omega(k_{0},0)+\frac{\partial\Omega}{\partial
k}(k_{0},0)(k-k_{0})+\frac{\partial^{2}\Omega}{\partial|a|^{2}}(k_{0},0)|a|^{2}+\frac{1}{2}\frac{\partial^{2}\Omega}{\partial
k^{2}}(k_{0},0)(k-k_{0})^{2}$
$\displaystyle\quad+\frac{\partial^{2}\Omega}{\partial
k\partial|a|^{2}}(k_{0},0)(k-k_{0})|a|^{2}+\frac{1}{2}\frac{\partial^{2}\Omega}{\partial|a|^{4}}(k_{0},0)|a|^{4}+\dots$
(25)

Rewriting and considering only the essential terms, we obtain

$\nu-\Omega^{\prime}(k_{0})\kappa-\frac{1}{2}\Omega^{\prime\prime}(k_{0})\kappa^{2}-\epsilon^{2}\frac{\partial^{2}\Omega}{\partial|A|^{2}}(k_{0},0)|A|^{2}-\dots=0.$
(26)

Associating $\nu$ and $\kappa$ with differential operators $i\partial_{t}$ and
$-i\partial_{x}$, and acting on the complex-valued amplitude $A$, it yields

$i(\partial_{t}+\Omega^{\prime}(k_{0})\partial_{x})A+\frac{1}{2}\Omega^{\prime\prime}(k_{0})\partial_{x}^{2}A-\epsilon^{2}\frac{\partial^{2}\Omega}{\partial|a|^{2}}(k_{0},0)|A|^{2}A=0.$
(27)

Introducing the slowly-moving coordinate
$\xi=\epsilon(x-\Omega^{\prime}(k_{0})t)$ and slower time variable
$\tau=\epsilon^{2}t$, we obtain the temporal NLSE

$i\partial_{\tau}A+\beta_{1}\partial_{\xi}^{2}A+\gamma_{1}|A|^{2}A=0$ (28)

where the dispersion and the nonlinear coefficients are respectively given by

$\beta_{1}=\frac{1}{2}\Omega^{\prime\prime}(k_{0})\qquad\qquad\text{and}\qquad\qquad\gamma_{1}=-\frac{\partial^{2}\Omega}{\partial|A|^{2}}(k_{0},0).$
(29)

For the temporal NLSE (28), the following result on the local existence and
uniqueness solution for an initial value problem in Sobolev spaces is known in
the literature. See, for instance, [Schneider 2011, Krämer 2013] for a
detailed proof of the following theorem.

######  Theorem 4.

Let $m\geq 1$, let $H^{m}(\mathbb{R})$ be the Sobolev space, the space of $m$
times weakly differentiable functions $u:\mathbb{R}\rightarrow\mathbb{R}$ with
derivatives in $L^{2}$-space for $j\in\\{0,1,\dots,m\\}$. Let the space
$H^{m}$ be equipped with the norm

$\|u\|_{H^{m}}=\max_{j\in\\{0,1,\dots,m\\}}\|\partial_{\xi}^{j}u\|_{L^{2}}.$
(30)

Let $A_{0}\in H^{m}(\mathbb{R})$ be a complex-valued function. Then there
exists a time $\tau_{0}=\tau_{0}\left(\|A_{0}\|_{H^{m}}\right)>0$ and a unique
solution $A\in\mathbb{C}\left(\left[0,\tau_{0}\right],H^{m}\right)$ of the
temporal NLSE (28) with the initial condition $A_{0}$.

The readers who are interested in the well-posedness of the Cauchy problem and
the long-time behavior of the corresponding global solutions to the NLSE may
consult [Strauss 1989, Cazenave 1996a, Cazenave 1996b, Ginibre 1997, Bourgain
1999].

Similarly, writing out the Taylor series expansion in two variables for
$k=K(\omega,a^{2})$ and retaining only the essential terms, we have

$\displaystyle k$ $\displaystyle=K(\omega_{0},0)+\frac{\partial
K}{\partial\omega}(\omega_{0},0)(\omega-\omega_{0})+\frac{1}{2}\frac{\partial^{2}K}{\partial\omega^{2}}(\omega_{0},0)(\omega-\omega_{0})^{2}+\frac{\partial^{2}\Omega}{\partial|a|^{2}}(\omega_{0},0)|a|^{2}+\dots$

Rearranging the terms to the left-hand side yields

$\kappa-K^{\prime}(\omega_{0})\nu-\frac{1}{2}K^{\prime\prime}(\omega_{0})\nu^{2}-\frac{\partial^{2}\Omega}{\partial|a|^{2}}(\omega_{0},0)|a|^{2}-\dots=0.$
(31)

Corresponding the parameters $\kappa$ and $\nu$ with the differential
operators $-i\partial_{x}$ and $i\partial_{t}$, respectively, and acting on
the complex-valued amplitude $A$, we obtain

$i\left(\partial_{x}+K^{\prime}(\omega_{0})\partial_{t}\right)A-\frac{1}{2}K^{\prime\prime}(\omega_{0})\partial_{t}^{2}A+\epsilon\frac{\partial^{2}\Omega}{\partial|A|^{2}}(\omega_{0},0)|A|^{2}A=0.$
(32)

Introducing the slowly-moving variables $\xi=\epsilon^{2}x$ and
$\tau=\epsilon(t-K^{\prime}(\omega_{0})x)$, we obtain the spatial NLSE

$i\partial_{\xi}A+\beta_{2}\partial_{\tau}^{2}A+\gamma_{2}|A|^{2}A=0$ (33)

where the dispersion and the nonlinear coefficients are respectively given by

$\beta_{2}=-\frac{1}{2}K^{\prime\prime}(\omega_{0})\qquad\qquad\text{and}\qquad\qquad\gamma_{2}=\frac{\partial^{2}\Omega}{\partial|A|^{2}}(\omega_{0},0).$
(34)

###  2.2 Derivation of the temporal NLSE

The following derivation, as well as the derivation for the spatial NLSE in
the next subsection, follow the argument presented by [van Groesen 1998,
Cahyono 2002, Karjanto 2006]. Consider a KdV type of equation with an exact
dispersion relationship property, given as follows

$\partial_{t}\eta+i\Omega(-i\partial_{x})\eta+c\,\eta\partial_{x}\eta=0.$ (35)

Here, $c\in\mathbb{R}$ is a nonlinear coefficient for the KdV equation (35).
We will observe that it also contributes to the nonlinear coefficient for the
NLSE. The function $\Omega$ acts as both a differential operator and a
dispersion relationship.

We seek a solution for the surface wave elevation $\eta(x,t)$ in the form of a
wave packet, or a wave group. This wave packet consists of a superposition of
the first-order harmonic wave, the second-order double harmonic wave, and the
second-order non-harmonic long wave, explicitly given as follows:

$\eta(x,t)=\epsilon
A(\xi,\tau)e^{i\theta}+\epsilon^{2}[B(\xi,\tau)e^{2i\theta}+C(\xi,\tau)]+\textmd{c.c.},$
(36)

where $0<\epsilon\ll 1$ is a small positive parameter as used commonly in
perturbation theory. The term in the exponent is
$\theta(x,t)=k_{0}x-\omega_{0}t$, where $k_{0}$ and $\omega_{0}$ are related
by the linear dispersion relation. The functions $A(\xi,\tau)$, $B(\xi,\tau)$,
and $C(\xi,\tau)$ are complex-valued wave packet envelopes and they are
allowed to vary slowly in the slower-moving frame of reference, where the
spatial and temporal variables are given by
$\xi=\epsilon(x-\Omega^{\prime}(k_{0})t)$ and $\tau=\epsilon^{2}t$,
respectively. As usual, c.c. denotes the complex conjugation of the preceding
terms.

Substituting the Ansatz (36) into the KdV equation (35) yields the residue
$R(x,t)$ that can be expressed in the following form:

$R(x,t)=\sum_{n,m}\epsilon^{n}R_{nm}e^{i\,m\theta}+\textmd{c.c.},$ (37)

where $n\geq 1$, $m\geq 0$, and where the coefficients $R_{nm}$ contain
expressions in $A(\xi,\tau)$, $B(\xi,\tau)$, and $C(\xi,\tau)$ and their
partial derivatives. All coefficients of $R(x,t)$ must vanish in order to
satisfy the KdV equation (35). We obtain the vanishing of the first order
residue coefficients $R_{1m}=0$, $m\geq 0$ when $k_{0}$ and $\omega_{0}$ are
related by the linear dispersion relationship. The second order residue
coefficients read

$\displaystyle R_{20}$ $\displaystyle=0;\qquad R_{21}=0;\qquad R_{2m}=0,\quad
m\geq 3;$ $\displaystyle R_{22}$
$\displaystyle=i([\Omega(2k_{0})-2\omega_{0}]B+c\,k_{0}A^{2}).$

Vanishing of $R_{22}$ leads to an expression of $B(\xi,\tau)$ as a function of
$A(\xi,\tau)$:

$B(\xi,\tau)=\frac{c\,k_{0}A^{2}(\xi,\tau)}{2\omega_{0}-\Omega(2k_{0})}.$ (38)

The third order residue coefficients read

$\displaystyle R_{30}$
$\displaystyle=[\Omega^{\prime}(0)-\Omega^{\prime}(k_{0})]\partial_{\xi}C+\frac{1}{2}c\,\partial_{\xi}|A|^{2};$
$\displaystyle R_{31}$
$\displaystyle=\partial_{\tau}A-\frac{1}{2}i\Omega^{\prime\prime}(k_{0})\partial_{\xi}^{2}A+i\,ck_{0}(A^{\ast}B+AC+AC^{\ast}).$

Requiring $R_{30}$ to vanish yields an expression for $C(\xi,\tau)$ as a
function of $A(\xi,\tau)$ and a $\tau$-dependent constant of integration
$\alpha_{T}$, given as follows:

$C(\xi,\tau)=\frac{1}{2}\frac{c\,|A(\xi,\tau)|^{2}}{\Omega^{\prime}(k_{0})-\Omega^{\prime}(0)}+\alpha_{T}(\tau).$
(39)

To prevent a resonance, $R_{31}$ has to vanish as well, and this leads to an
evolution equation for $A(\xi,\tau)$:

$\partial_{\tau}A+i\beta_{1}\partial_{\xi}^{2}A+i\gamma_{1}|A|^{2}A+2ik_{0}c\,\textmd{Re}[\alpha_{T}(\tau)]A=0.$
(40)

A similar assumption of unidirectional wave propagation, applying the “gauge
transformation” by multiplying the evolution equation by
$e^{2ik_{0}c\int\textmd{Re}[\alpha_{T}(\tau)]d\tau}$ [Mei 1983], we obtain the
temporal NLSE for $A(\xi,\tau)$:

$\partial_{\tau}A+i\beta_{1}\partial_{\xi}^{2}A+i\gamma_{1}|A|^{2}A=0,$ (41)

where the dispersion and the nonlinear coefficients are respectively given by

$\displaystyle\beta_{1}$
$\displaystyle=-\frac{1}{2}\Omega^{\prime\prime}(k_{0})$ (42)
$\displaystyle\gamma_{1}$
$\displaystyle=k_{0}c^{2}\left(\frac{1}{\Omega^{\prime}(k_{0})-\Omega^{\prime}(0)}+\frac{k_{0}}{2\omega_{0}-\Omega(2k_{0})}\right).$
(43)

This temporal NLSE as well as the spatial NLSE derived in the following
subsection are valid for intermediate-water wave models.

In the following, we will derive the corresponding “energy equation” and
“nonlinear dispersion relationship” for the temporal NLSE. Write the complex-
valued amplitude $A(\xi,\tau)$ in the physical, faster-moving variables as
$A_{1}(x,t)=\epsilon A(\xi,\tau)$ using the relationship above
$\xi=\epsilon(x-\Omega^{\prime}(k_{0})t)$ and $\tau=\epsilon^{2}t$. The
temporal NLSE in the physical variables is given by

$\partial_{t}A_{1}+\Omega^{\prime}(k_{0})\partial_{x}A_{1}+i\beta_{1}\partial_{x}^{2}A_{1}+i\gamma_{1}|A_{1}|^{2}A_{1}=0.$
(44)

Apply the Madelung transformation by writing the complex-valued amplitude
$A_{1}(x,t)$ in its polar form $A_{1}(x,t)=a(x,t)e^{i\phi(x,t)}$, where the
amplitude $a(x,t)$ and the phase $\phi(x,t)$ are both real-valued functions
[Ma- delung 1927]. Upon substitution to (44), removing the factor
$e^{i\phi(x,t)}$ and separating the real and the imaginary parts, it yields
the following coupled phase-amplitude equations

$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle{\partial_{t}a+\Omega^{\prime}(k_{0})\partial_{x}a-\beta_{1}\left(a\partial_{x}^{2}\phi+2\partial_{x}a\partial_{x}\phi\right)}&=0\\\
\displaystyle{\partial_{t}\phi+\Omega^{\prime}(k_{0})\partial_{x}\phi+\beta_{1}\left(\frac{\partial_{x}^{2}a}{a}-(\partial_{x}\phi)^{2}\right)+\gamma_{1}a^{2}}&=0.\end{array}\right.$
(47)

Expressing the wavenumber $k(x,t)$ and the frequency $\omega(x,t)$ in the
following form

$k(x,t)=k_{0}+\kappa=k_{0}+\partial_{x}\phi\qquad\qquad\omega(x,t)=\omega_{0}+\nu=\omega_{0}-\partial_{t}\phi.$
(48)

The local wavenumber $\kappa=\partial_{x}\phi$ and the local frequency
$\nu=-\partial_{t}\phi$ act as modulational quantities. Using these
quantities, the phase-amplitude equations can be written in a more compact
form. The amplitude and the phase equations are the first and the second
expressions in (47), respectively. Expressing $\partial_{x}\phi$ in terms of
the local wavenumber $k(x,t)$ and multiplying the amplitude equation with
$a(x,t)$, we obtain

$\frac{1}{2}\partial_{t}(a^{2})+\frac{1}{2}\partial_{x}\left\\{\left[\Omega^{\prime}(k_{0})+\Omega^{\prime\prime}(k_{0})(k-k_{0})\right]a^{2}\right\\}=0.$
(49)

By noting the terms inside the square brackets on the second term above as a
linear approximation for $\Omega^{\prime}(k)$, we arrive at the conservation
of energy equation for the temporal NLSE. It reads

$\partial_{t}(a^{2})+\partial_{x}[\Omega^{\prime}(k)a^{2}]=0.$ (50)

By expressing the local wavenumber $\partial_{x}\phi=k-k_{0}$ and the local
frequency $-\partial_{t}\phi=\omega-\omega_{0}$, we obtain the following

$\omega-\left[\omega_{0}+\Omega^{\prime}(k_{0})(k-k_{0})+\frac{1}{2}\Omega^{\prime\prime}(k_{0})(k-k_{0})^{2}\right]=\beta_{1}\frac{\partial_{x}^{2}a}{a}+\gamma_{1}\,a^{2}.$
(51)

By taking the terms inside the square brackets as a quadratic approximation
for $\Omega(k)$, the phase equation leads to the nonlinear dispersion
relationship

$\omega-\Omega(k)=\beta_{1}\frac{\partial_{x}^{2}a}{a}+\gamma_{1}a^{2}.$ (52)

###  2.3 Derivation of the spatial NLSE

A similar technique using the method of multiple-scale can be applied to the
KdVE with an exact dispersion relationship (35) to obtain the spatial NLSE.
The difference is in the choice of the slow-moving spatial and temporal
variables, respectively chosen as $\xi=\epsilon^{2}x$ and
$\tau=\epsilon(t-x/\Omega^{\prime}(k_{0}))$. Following an equivalent procedure
as in the previous subsection, we obtain identical first-order and second-
order residue coefficients

$\displaystyle R_{1m}$ $\displaystyle=0,\quad m\geq 0$ $\displaystyle R_{20}$
$\displaystyle=0;\qquad R_{21}=0;\qquad R_{2m}=0,\quad m\geq 3;$
$\displaystyle R_{22}$
$\displaystyle=i([\Omega(2k_{0})-2\omega_{0}]B+c\,k_{0}A^{2}).$

The third-order residue coefficients are given as follows:

$\displaystyle R_{30}$
$\displaystyle=\left(1-\frac{\Omega^{\prime}(0)}{\Omega^{\prime}(k_{0})}\right)\partial_{\tau}C-\frac{1}{2}\frac{c}{\Omega^{\prime}(k_{0})}\partial_{\tau}|A|^{2}$
(53) $\displaystyle R_{31}$
$\displaystyle=\Omega^{\prime}(k_{0})\partial_{\xi}A-\frac{1}{2}i\frac{\Omega^{\prime\prime}(k_{0})}{[\Omega^{\prime}(k_{0})]^{2}}\partial_{\tau}^{2}A+i\,ck_{0}(A^{\ast}B+AC+AC^{\ast})$
(54) $\displaystyle R_{32}$
$\displaystyle=-\frac{1}{\Omega^{\prime}(k_{0})}\left(\Omega^{\prime}(2k_{0})\partial_{\tau}B+\frac{1}{2}c\partial_{\tau}A^{2}\right)$
(55) $\displaystyle R_{33}$ $\displaystyle=3ik_{0}c\,AB.$ (56)

Requiring $R_{30}$ to vanish leads to an expression for $C(\xi,\tau)$ as a
function of $A(\xi,\tau)$ and a $\xi$-dependent constant of integration
$\alpha_{S}(\xi)$ for all $\theta(x,t)$:

$C(\xi,\tau)=\frac{1}{2}\frac{c\,|A(\xi,\tau)|^{2}}{\Omega^{\prime}(k_{0})-\Omega^{\prime}(0)}+\alpha_{S}(\xi).$
(57)

In order to prevent resonance, $R_{31}$ has to vanish which leads to a dynamic
evolution equation for $A(\xi,\tau)$:

$\partial_{\xi}A+i\beta_{2}\partial_{\tau}^{2}A+i\gamma_{2}|A|^{2}A+\frac{2ik_{0}c}{\Omega^{\prime}(k_{0})}\textmd{Re}\left[\alpha_{S}(\xi)\right]A=0.$
(58)

We can remove the term containing Re$[\alpha_{S}(\xi)]$ by applying the gauge
transformation [Mei 1983]. Multiply the evolution equation by
$e^{\frac{2ik_{0}c}{\Omega^{\prime}(k_{0})}\int\textmd{Re}[\alpha_{S}(\xi)]d\xi}$,
then the new complex amplitude
$\tilde{A}(\xi,\tau)=e^{\frac{2ik_{0}c}{\Omega^{\prime}(k_{0})}\int\textmd{Re}[\alpha_{S}(\xi)]d\xi}A(\xi,\tau)$
satisfies the spatial NLSE which, after dropping the tilde, can be written in
the following form:

$\partial_{\xi}A+i\beta_{2}\partial_{\tau}^{2}A+i\gamma_{2}|A|^{2}A=0.$ (59)

The dispersion coefficient $\beta$ and the nonlinear coefficient $\gamma$ are
given as follows

$\displaystyle\beta_{2}$
$\displaystyle=\beta(k_{0})=-\frac{1}{2}\frac{\Omega^{\prime\prime}(k_{0})}{[\Omega^{\prime}(k_{0})]^{3}}$
(60) $\displaystyle\gamma_{2}$
$\displaystyle=\gamma(k_{0},c^{2})=\frac{k_{0}c^{2}}{\Omega^{\prime}(k_{0})}\left(\frac{1}{\Omega^{\prime}(k_{0})-\Omega^{\prime}(0)}+\frac{k_{0}}{2\omega_{0}-\Omega(2k_{0})}\right).$
(61)

We apply a similar approach as in the previous section to derive the
corresponding energy equation and nonlinear dispersion relationship for the
spatial NLSE. Write $A_{2}(x,t)=\epsilon A(\xi,\tau)$, where
$\xi=\epsilon^{2}x$ and $\tau=\epsilon(t-x/\Omega^{\prime}(k_{0}))$. The
spatial NLSE in the physical variables is expressed as follows:

$\partial_{x}A_{2}+\frac{1}{\Omega^{\prime}(k_{0})}\partial_{t}A_{2}+i\beta_{2}\partial_{t}^{2}A_{2}+i\gamma_{2}|A_{2}|^{2}A_{2}=0.$
(62)

Apply the Madelung transformation by writing $A_{2}(x,t)$ in its polar form
$A_{2}(x,t)=a(x,t)e^{i\phi(x,t)}$, with $a(x,t)$ and $\phi(x,t)$ are real-
valued quantities [Ma- delung 1927]. After substituting to the spatial NLSE
(62), removing the factor $e^{i\phi(x,t)}$, and collecting the real and the
imaginary parts, we obtain the following coupled phase-amplitude equations in
the original physical variables:

$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle{\partial_{x}a+\frac{\partial_{t}a}{\Omega^{\prime}(k_{0})}-\beta_{2}\left(a\partial_{t}^{2}\phi+2\partial_{t}a\partial_{t}\phi\right)}&=0\\\
\displaystyle{\partial_{x}\phi+\frac{\partial_{t}\phi}{\Omega^{\prime}(k_{0})}+\beta_{2}\left(\frac{\partial_{t}^{2}a}{a}-(\partial_{t}\phi)^{2}\right)+\gamma_{2}a^{2}}&=0.\end{array}\right.$
(65)

Using the previous definition for wavenumber and frequency expressed in terms
of modulated local wavenumber and local frequency, respectively, we can write
these phase-amplitude equations in a more compact form. We also adopt the
linear dispersion relationship $\omega=\Omega(k)$ or $k=K(\omega)$, where
$K=\Omega^{-1}$. From $k=K[\Omega(k)]$, we can derive the relationship between
its derivatives, up to the second order given explicitly as follows:

$K^{\prime}(\omega_{0})=\frac{1}{\Omega^{\prime}(k_{0})}\qquad\textmd{and}\qquad
K^{\prime\prime}(\omega_{0})=-\frac{\Omega^{\prime\prime}(k_{0})}{[\Omega^{\prime}(k_{0})]^{3}}.$
(66)

Expressing the local frequency $-\partial_{t}\phi=\omega-\omega_{0}$ and
multiplying the amplitude equation with $a(x,t)$, we obtain

$\frac{1}{2}\partial_{x}(a^{2})+\frac{1}{2}\partial_{t}\left\\{\left[K^{\prime}(\omega_{0})+K^{\prime\prime}(\omega_{0})(\omega-\omega_{0})\right]a^{2}\right\\}=0.$
(67)

Noting the terms in the square brackets as a linear approximation for
$K^{\prime}(\omega)$, we can write the amplitude equation as the energy
equation

$\partial_{x}(a^{2})+\partial_{t}[K^{\prime}(\omega)a^{2}]=0.$ (68)

By expressing the local wavenumber $\partial_{x}\phi=k-k_{0}$ and the local
frequency $-\partial_{t}\phi=\omega-\omega_{0}$, we can write the phase
equation as follows:

$\left[k_{0}+K^{\prime}(\omega_{0})(\omega-\omega_{0})+\frac{1}{2}K^{\prime\prime}(\omega_{0})(\omega-\omega_{0})^{2}\right]-k=\beta_{2}\frac{\partial_{t}^{2}a}{a}+\gamma_{2}\,a^{2}.$
(69)

By considering the terms inside the square brackets as a quadratic
approximation for $K(\omega)$, the phase equation leads to the nonlinear
dispersion relationship

$K(\omega)-k=\beta_{2}\frac{\partial_{t}^{2}a}{a}+\gamma_{2}\,a^{2}.$ (70)

The nonlinear dispersion relationship (52) or (70) describes the relationship
between wavenumber $k$ and frequency $\omega$ in the dispersion plane
$(k,\omega)$. Since generally the right-hand side of (52) or (70) does not
vanish, then any combination of $(k,\omega)$ does not always satisfy the
linear dispersion relationship.

The ratio $\frac{\partial_{x}^{2}a}{a}$ in (52) or
$\frac{\partial_{t}^{2}a}{a}$ in (70) is coined as the “Chu-Mei quotient” by
[Karjanto and van Groesen 2007a] when they discussed “phase singularity” and
“wavefront dislocation” in surface gravity waves. The unboundedness of this
quotient at the vanishing real-valued amplitude $a(x,t)$ is responsible for
the occurrence of the phenomena. Several authors refer to this quotient as the
“Fornberg-Whitham term” [Infeld and Rowlands 2000, 100–102], referring to the
paper written by [Fornberg and Whitham 1978]. Even though the quotient has
already appeared earlier in the literature in the context of modulated waves
in nonlinear media [Karpman 1967, Karpman and Krushkal 1969], it was [Chu and
Mei 1970, Chu and Mei 1971] who introduced it for the first time when deriving
the modulation equations of Whitham’s theory for slowly varying Stokes’ waves.

###  2.4 Applications in surface gravity waves

The temporal NLSE derived in Subsection 2.2 models absolute dynamics while the
spatial NLSE derived in Subsection 2.3 models convective dynamics [Sulem and
Sulem 1999]. Together with an initial condition, the temporal NLSE constitutes
an initial value problem. Given a wave packet profile at a particular time,
the NLSE governs an evolution in time of the wave packet so that we could find
out the shape of the wave packet at some time in the future. Together with a
boundary condition, the spatial NLSE composes a boundary value problem, or
signaling problem. This model is suitable for a wave signal generation in a
wave tank of a hydrodynamic laboratory. By inputting an initial wave signal to
a wavemaker, letting it propagate along the wave tank, we can measure the
“experimental” wave signal at several frontal positions and compared it with
the “theoretical” wave signal predicted by the spatial NLSE.

In particular, the spatial NLSE and its family of solitons on a non-vanishing
background describing a nonlinear extension of modulational instability have
been utilized as a model for deterministic freak wave generation in
intermediate water depth at the high-speed wave basin of the Dutch Maritime
Research Institute, the Netherlands (MARIN) [van Groesen, Andonowati and
Karjanto 2006, Karjanto 2006, Andonowati, Karjanto and van Groesen 2007]. The
experimental results confirm an occurrence of phase singularity of the wave
packet envelope at the location where the wave signals reach maximum
amplitude, where the phenomenon and its related counterpart of wavefront
dislocation have been predicted theoretically [Karjanto and van Groesen
2007a]. Even though the model does not quantitatively predict the signal and
the spectrum evolution in accurate detail, it exhibits an extraordinary
qualitative agreement. Our experimental results confirm similar behavior with
other testings where the corresponding wave spectra demonstrate frequency
downshift as the wave signals propagate along the wave tank [Karjanto and van
Groesen 2010, Lake et al 1977]. The authors also coined the term “Wessel
curves” indicating the evolution of the real-part and the imaginary-part of
the complex-valued amplitude $A(\xi,\tau)$ exclusive of the oscillatory part
contributed from the continuous-wave or the plane-wave solution.

Modulational instability, also known as sideband instability, is a well-known
phenomenon in both fluid dynamics and nonlinear optics. In the context of
hydrodynamics, it is known as the Benjamin-Feir instability where [Benjamin
and Feir 1967, Benjamin 1967] predicted the onset of the instability of Stokes
wave trains in deep water. Under the theory of linear perturbation analysis,
the plane-wave solution of the NLSE is modulationally unstable and its
nonlinear extension is given by the “Akhmediev-Eleonskii-Kulagin breather”,
also known as the “solitons on a non-vanishing background” [Akhmediev,
Eleonskii and Kulagin 1987, Akhmediev and Ankiewicz 1997, Ablowitz and Herbst
1990, Karjanto 2009]. In the spectral domain, the effect of nonlinearity which
reinforces periodic wave trains, leads to the generation of spectral sidebands
and an eventual breakup of the waveform into a train of pulses. The breather’s
analytical expression for $\beta_{1}=1$ and $\gamma_{1}=2$ in the temporal
NLSE (41) can be written as follows:

$A_{\text{AEK}}(\xi,\tau)=e^{2i\tau}\left(\frac{\nu^{3}\cosh\left[\sigma(\tau-\tau_{0})\right]+i\nu\sigma\sinh\left[\sigma(\tau-\tau_{0})\right]}{2\nu\cosh\left[\sigma(\tau-\tau_{0})\right]-\sigma\cos\left[\nu(\xi-\xi_{0})\right]}-1\right).$
(71)

The family of solitons on a non-vanishing background is defined for the
parameter values $0<\nu<2$ and $\sigma=\nu\sqrt{4-\nu^{2}}$. The soliton is a
holomorphic function for $\sqrt{3}<\nu<2$ and it reaches local maxima and
local minima at $(\xi,\tau)=(\xi_{0}+2n\pi/\nu,\tau_{0})$ and
$(\xi,\tau)=(\xi_{0}+[2n+1]\pi/\nu,\tau_{0})$, respectively, for
$n\in\mathbb{Z}$.

Other NLSE solutions with a non-vanishing background type of soliton have been
proposed for a hydrodynamic freak wave formation, called “breather solutions”
[Dysthe and Trulsen 1999]. One family of solitons on a non-vanishing
background is called by a new name: “Kusnetsov-Ma breather” [Kibler et al.
2012], derived independently by [Kuznetsov 1977] and [Ma 1979]. For the
corresponding dispersive and nonlinear coefficients in the temporal NLSE (41),
$\beta_{1}=1$ and $\gamma_{1}=2$, it is explicitly given as follows:

$A_{\text{KM}}(\xi,\tau)=e^{2i\tau}\left(\frac{\mu^{3}\cos\left[\rho(\tau-\tau_{0})\right]+i\mu\rho\sin\left[\rho(\tau-\tau_{0})\right]}{2\mu\cos\left[\rho(\tau-\tau_{0})\right]-\rho\cosh\left[\mu(\xi-\xi_{0})\right]}+1\right)$
(72)

where $\rho=\mu\sqrt{4+\mu^{2}}$, $\mu\in\mathbb{R}$. Another exact solution
is known as the “Peregrine breather” or the “rational soliton” solution
[Peregrine 1983]. It is given as follows:

$A_{\text{PR}}(\xi,\tau)=e^{2i\tau}\left(\frac{4(1+4i(\tau-\tau_{0})}{1+16(\tau-\tau_{0})^{2}+4(\xi-\xi_{0})^{2}}-1\right).$
(73)

This solution can be obtained as a limiting case for both the Akhmediev-
Eleonskii-Kulagin breather and the Kusnetsov-Ma breather when the parameters
$\nu$ and $\mu$ approach zero [Karjanto and van Groesen 2007b]. The Peregrine
breather has also been successfully generated experimentally as a rogue wave
in a water wave tank [Chabchoub et al 2011].

The NLSE has also been considered as a model to investigate the oceanic rogue
wave formation caused by a nonlinear energy transfer in the open ocean, both
deterministically and stochastically [Henderson, Peregrine and Dold 1999,
Onorato et al 2001, Osborne 2001]. Extensive reports on progress in the
physical mechanisms of oceanic rogue wave phenomenon are available [Kharif and
Pelinovsky 2003, Dysthe, Krogstad and Müller 2008, Kharif, Pelinovsky and
Slunyaev 2009, Pelinovsky and Kharif 2016]. A similar phenomenon has also been
proposed, predicted, observed, and studied in other fields than hydrodynamics
where the NLSE has been used as a mathematical model, including but not
limited to, in optical rogue waves [Solli et al 2007], atmospheric rogue waves
[Stenflo and Marklund 2010], matter rogue waves in Bose-Einstein condensates
[Bludov et al 2009], and financial rogue waves [Ivancevic 2010, Yan 2010].

##  3 Superconductivity

###  3.1 NLSE derivation from the nonlinear Klein-Gordon equation

Consider a line of pendula positioned very close together and they hang
vertically under the influence of gravity. There exists a horizontal torsion
wire for which each pendulum can twist. Let $u(x,t)$ be the twist angle of the
pendulum at position $x$ and time $t$, then its motion can be modeled by a
sine-Gordon equation

$\partial^{2}_{t}u-a\partial^{2}xu+b\sin u=0,\qquad\qquad a,\;b\geq 0.$ (74)

In this model, the term $-b\sin u$ is an external force due to gravitational
acceleration and the term $a\partial_{x}^{2}u$ models the force caused by the
effect of the twist. Assume that one end of the pendulum chain is wiggled with
a small amplitude motion with frequency $\omega$, then the term $\sin u$ can
be approximated by its Maclaurin series about $u=0$. Keeping only the first
two terms, we obtain a nonlinear Klein-Gordon equation with a cubic
nonlinearity.

The following derivation of the NLSE from a nonlinear Klein-Gordon equation
follows the argument in [Sharma and Buti 1976, Newell 1985, Sulem and Sulem
1999, Krämer 2013, Schneider 2011]. Consider a cubic nonlinearity Klein-Gordon
equation describing a model for a wave packet $u(x,t)$ that moves at a
constant group velocity $c$, presented as the following initial value problem
(IVP):

$\displaystyle\partial_{t}^{2}u-a\partial_{x}^{2}u+bu-\lambda u^{3}$
$\displaystyle=0,\qquad\qquad a,\;b\geq 0,\quad\lambda\in\mathbb{R}$ (75)
$\displaystyle u(x,0)=u_{0},\qquad\qquad\partial_{t}u(x,0)$
$\displaystyle=u_{1}.$ (76)

An Ansatz for $u(x,t)$ is expressed as a perturbation series, where again
$0<\epsilon\ll 1$ is a small parameter

$u(x,t)=\hat{u}_{0}+\epsilon\hat{u}_{1}+\epsilon^{2}\hat{u}_{2}+\dots,$ (77)

with $u_{n}(x,X,t,\tau_{1},\tau_{2})$, where $X=\epsilon x$,
$\tau_{1}=\epsilon t$, and $\tau_{2}=\epsilon^{2}t$ act as slower variables.
Substituting (77) to the IVP (75) and (76) yields a series progressing in the
order of $\epsilon$. The vanishing of the lowest-order term simply reduces to
a linear Klein-Gordon IVP:

$\displaystyle\partial_{t}^{2}\hat{u}_{0}-a\partial_{x}^{2}\hat{u}_{0}+b\hat{u}_{0}$
$\displaystyle=0$ (78) $\displaystyle\hat{u}_{0}(x,X,0,0,0;\epsilon)$
$\displaystyle=u_{0}/\epsilon\qquad\qquad\partial_{t}\hat{u}_{0}(x,X,0,0,0;\epsilon)=u_{1}/\epsilon.$
(79)

We seek the solution in the form

$\hat{u}_{0}(x,X,t,\tau_{1},\tau_{2})=A(X,\tau_{1},\tau_{2})e^{i(kx-\omega
t)}+\text{c.c.},\qquad A\in\mathbb{C}$ (80)

where c.c. denotes the complex conjugation of the preceding term and
$(k,\omega)$ satisfies the dispersion relationship
$\omega=\Omega(k)=\sqrt{ak^{2}+b}$. The vanishing of the first-order terms
reads

$\displaystyle\partial_{t}^{2}\hat{u}_{1}-a\partial_{x}^{2}\hat{u}_{1}+b\hat{u}_{1}$
$\displaystyle=-2\partial_{\tau_{1}}\partial_{t}\hat{u}_{0}+2a\partial_{X}\partial_{x}\hat{u}_{0}$
$\displaystyle=2i\left(\omega\partial_{\tau_{1}}A+ak\partial_{X}A\right)e^{i(kx-\omega
t)}+\text{c.c.}$ (81) $\displaystyle\hat{u}_{1}(x,X,0,0,0;\epsilon)$
$\displaystyle=0$ (82)
$\displaystyle\partial_{t}\hat{u}_{1}(x,X,0,0,0;\epsilon)$
$\displaystyle=-\partial_{\tau_{1}}\hat{u}_{0}(x,X,0,0,0;\epsilon).$ (83)

Since the right-hand side represents secular terms that would lead to
unbounded growth in $\hat{u}_{1}$ over a long period of time, we need to
eliminate them by taking $\omega\partial_{\tau_{1}}A+ak\partial_{X}A=0$. This
condition is equivalent to the group velocity
$\Omega^{\prime}(k)=ak/\Omega(k)$. The solution of the linear Klein-Gordon
equation for $\hat{u}_{1}$ is similar to the one for $\hat{u}_{0}$:

$\hat{u}_{1}(x,X,t,\tau_{1},\tau_{2})=B(X,\tau_{1},\tau_{2})e^{i(kx-\omega
t)}+\text{c.c.},\qquad B\in\mathbb{C}.$ (84)

Collecting the second-order term and requiring it to vanish gives

$\displaystyle\partial_{t}^{2}\hat{u}_{2}-\alpha\partial_{x}^{2}\hat{u}_{2}+\beta\hat{u}_{2}$
$\displaystyle=-2\partial_{\tau_{2}}\partial_{t}\hat{u}_{0}-\partial_{\tau_{1}}^{2}\hat{u}_{0}+a\partial_{X}^{2}\hat{u}_{0}+\lambda\hat{u}_{0}^{3}+2\partial_{\tau_{1}}\partial_{t}\hat{u}_{1}-2a\partial_{X}\partial_{x}\hat{u}_{1}$
$\displaystyle=\left\\{2i\Omega(k)\partial_{\tau_{2}}A+(a-\Omega^{\prime}(k)^{2})\partial_{\xi}^{2}A+3\lambda|A|^{2}A\right.$
$\displaystyle\qquad\left.+2i\left(\omega\partial_{\tau_{1}}B+ak\partial_{X}B\right)\right\\}e^{i(kx-\omega
t)}+\lambda A^{3}e^{3i(kx-\omega t)}+\text{c.c.}$ (85)
$\displaystyle\hat{u}_{2}(x,X,0,0,0;\epsilon)$ $\displaystyle=0$ (86)
$\displaystyle\partial_{t}\hat{u}_{2}(x,X,0,0,0;\epsilon)$
$\displaystyle=-\partial_{\tau_{1}}\hat{u}_{0}(x,X,0,0,0;\epsilon)-\partial_{\tau_{2}}\hat{u}_{0}(x,X,0,0,0;\epsilon).$
(87)

Here, we have used $\xi=\epsilon(x-\Omega^{\prime}(k)t)$. Similar to previous
method, we would like to remove secular terms by requiring $\hat{u}_{2}$ to be
bounded and $B$ must satisfy $\omega\partial_{\tau_{1}}B+ak\partial_{X}B=0$.
For $A(\xi,\tau_{2})$, it satisfies the temporal NLSE

$i\partial_{\tau}A+\beta\partial_{\xi}^{2}A+\gamma|A|^{2}A=0$ (88)

where we drop the subscript 2 from the variable $\tau_{2}$ and the dispersive
and the nonlinear coefficients are given as follows, respectively

$\beta=\frac{1}{2}\Omega^{\prime\prime}(k)\qquad\qquad\text{and}\qquad\qquad\gamma=\frac{3}{2}\frac{\lambda}{\Omega(k)}.$
(89)

The additional term $\lambda A^{3}e^{3i(kx-\omega t)}+\text{c.c.}$ is not a
resonant term and thus, is not problematic since generally $\Omega(3k)\neq
3\Omega(k)$.

###  3.2 Applications of sine-Gordon model

Both the nonlinear Klein-Gordon and the sine-Gordon equations have been
studied analytically using the Zakharov-Shabat method [Zakharov and Shabat
1972, Grundland and Infeld 1992] and the sine-cosine and tanh methods [Wazwaz
2005], as well as numerically using finite difference method [Strauss and
Vázquez 1978, Jiménez and Vázques 1990, Vu-Quoc and Li 1993, Duncan 1997, Li
and Vu-Quoc 1995] and using thin plate splines–radial basis functions [Dehghan
and Shokri 2009].

These evolution equations find many applications in physical sciences and
different nonlinear phenomena [Dodd et al 1982, 389–494, Drazin and Johnson
1989, 198–199]. Historically, the sine-Gordon equation arises from the field
of differential geometry where it describes surfaces with a constant negative
Gaussian curvature [Enneper 1870]. The propagation of a crystal dislocation
with sinusoidal periodicity is governed by the sine-Gordon equation [Frenkel
and Kontorova 1939]. The behavior and interaction of elementary particles of
mesons and baryons using an identical model has been proposed by [Perring and
Skyrme 1962]. The real-valued amplitude of wave packet envelope nonlinear
evolution equations governing modulationally weakly unstable baroclinic shear
flow can be transformed to the sine-Gordon equation [Gibbon et al 1979].

Another important application is in the field of superconductivity, which is
known as Josephson effect across a Josephson junction. The latter is a quantum
mechanical device composed of superconductor electrodes separated by a barrier
and coupled by a weak link. The former is a macroscopic quantum phenomenon
where a current could flow for a long period of time without any voltage
supply. It was British physicist Brian David Josephson who investigated the
relationship between the current and voltage across that weak link [Josephson
1962, Josephson 1974]. Reviews on Josephson junctions and superconducting
soliton oscillators in the context of the sine-Gordon model are given by
[Parmentier 1993, Pnevmatikos and Pedersen 1993].

The derivation of the NLSE from the sine-Gordon system for a small-amplitude
limit was considered by [Kaup and Newell 1978] who discovered a phase-locked
breather with an applied alternating current field by means of perturbation
theory. Nonlinear breather dynamics of an alternating current parametric force
in the presence of loss in a sine-Gordon system has been analyzed by
[Grønbech-Jensen, Kivshar and Samuelsen 1993]. In the case of a small-
amplitude limit where the system can be described by an effective NLSE, a
correct threshold value for the driving force amplitude was obtained when the
breather frequency was identically one.

Inspired by the recent progress in quantum graph theory and its applications
[Gnutzmann and Smilansky 2006, Kuchment 2008, Berkolaiko and Kuchment 2013],
interactions of traveling localized wave solutions with a vertex in a star
graph from a tricrystal Josephson junction has been investigated recently by
[Susanto et al 2019]. Other applications of the sine-Gordon and nonlinear
Klein-Gordon models include a mechanical model with springs, wires and
bearings, and Bloch wall dynamics in magnetic crystals [Barone et al 1971].
For a summary in contemporary developments of the sine-Gordon model and its
wide range of applications, please consult [Cuevas-Maraver, Kevrekidis and
Williams 2014].

##  4 Nonlinear optics

###  4.1 NLSE derivation from Maxwell’s and Helmholtz’ equations

The derivation in this subsection follows the argument presented by [Agrawal
2012, Kivshar 2003, Sulem and Sulem 1999]. See also [Moloney and Newell 2019,
Banerjee 2004, Butcher and Cotter 1990]. Maxwell’s equations govern the
propagation of electromagnetic waves and optical fields in fibers, given as
follows in the International System of Units:

$\displaystyle\nabla\times\mathbf{E}$
$\displaystyle=-\partial_{t}\mathbf{B}\qquad\qquad\quad\text{(Faraday's law)}$
(90) $\displaystyle\nabla\times\mathbf{H}$
$\displaystyle=\mathbf{J}+\partial_{t}\mathbf{D}\qquad\qquad\text{(Ampere's
law)}$ (91) $\displaystyle\nabla\cdot\mathbf{D}$ $\displaystyle=\rho$ (92)
$\displaystyle\nabla\cdot\mathbf{B}$ $\displaystyle=0.$ (93)

Here, $\mathbf{E}$ and $\mathbf{H}$ denote electric and magnetic vector
fields, respectively; $\mathbf{D}$ and $\mathbf{B}$ denote the corresponding
electric and magnetic flux densities; $\rho$ is the charge density and
$\mathbf{J}$ is the corresponding current density of free charges and both
represent the sources for the electromagnetic field. We also have the
following constitutive relations for electric and magnetic flux densities:

$\displaystyle\mathbf{D}$ $\displaystyle=\epsilon_{0}\mathbf{E}+\mathbf{P}$
(94) $\displaystyle\mathbf{B}$ $\displaystyle=\mu_{0}\mathbf{H}+\mathbf{M}$
(95)

where $\epsilon_{0}$ is the vacuum permittivity, $\mu_{0}$ is the vacuum
permeability, and $\mathbf{P}$ and $\mathbf{M}$ are the induced electric and
magnetic polarizations. We will adopt the following common assumptions in
nonlinear fiber optics: the absence of free charges ($\rho=0$ and
$\mathbf{J}=\mathbf{0}$) and a nonmagnetic medium like fiber optics
($\mathbf{M}=\mathbf{0}$). By taking the curl of Faraday’s law (90), using
(95), (91), and (94), we can eliminate $\mathbf{B}$ and $\mathbf{D}$ to obtain
an expression in $\mathbf{E}$ and $\mathbf{P}$

$\nabla\times\nabla\times\mathbf{E}+\frac{1}{c^{2}}\partial_{t}^{2}\mathbf{E}=-\mu_{0}\partial_{t}^{2}\mathbf{P},$
(96)

where $1/c^{2}=\mu_{0}\epsilon_{0}$ is the speed of light in a vacuum.

We adopt a common relationship between the induced polarization $\mathbf{P}$
and the electric field $\mathbf{E}$ which is valid in the electric-dipole
approximation and under the assumption of local medium response. The induced
polarization $\mathbf{P}$ is written as the combination of the linear and the
nonlinear parts, $\mathbf{P}_{L}$ and $\mathbf{P}_{NL}$, respectively:
$\mathbf{P}(\mathbf{r},t)=\mathbf{P}_{L}(\mathbf{r},t)+\mathbf{P}_{NL}(\mathbf{r},t)$,
where

$\displaystyle\mathbf{P}_{L}(\mathbf{r},t)$
$\displaystyle=\epsilon_{0}\int_{-\infty}^{t}\chi^{(1)}(t-t_{0})\mathbf{E}(\mathbf{r},t_{0})\,\mathrm{d}t_{0}$
(97) $\displaystyle\mathbf{P}_{NL}(\mathbf{r},t)$
$\displaystyle=\epsilon_{0}\int_{-\infty}^{t}\int_{-\infty}^{t}\int_{-\infty}^{t}\chi^{(3)}(t-t_{1},t-t_{2},t-t_{3})\mathbf{E}(\mathbf{r},t_{1})\mathbf{E}(\mathbf{r},t_{2})\mathbf{E}(\mathbf{r},t_{3})\,\mathrm{d}t_{1}\,\mathrm{d}t_{2}\,\mathrm{d}t_{3}$
(98)

where $\chi^{(j)}$ is a tensor of rank $j+1$, the $j$-th order of
susceptibility. Consider the case where $\mathbf{P}_{NL}=\mathbf{0}$. Let
$\mathbf{\hat{E}}(\mathbf{r},\omega)$ be the Fourier transform of
$\mathbf{E}(\mathbf{r},t)$, defined as

$\mathbf{\hat{E}}(\mathbf{r},\omega)=\int_{-\infty}^{\infty}\mathbf{E}(\mathbf{r},t)\,e^{i\omega
t}\,\mathrm{d}t$ (99)

and let $\hat{\chi}^{(1)}(\omega)$ be the Fourier transform of the linear,
first-order susceptibility of $\chi^{(1)}(t)$, then (96) can be written in the
frequency domain

$\nabla\times\nabla\times\hat{\mathbf{E}}=\epsilon(\omega)\frac{\omega^{2}}{c^{2}}\mathbf{\hat{E}}(\mathbf{r},\omega)$
(100)

where $\epsilon(\omega)=1+\hat{\chi}^{(1)}(\omega)=\left(n+i\alpha
c/(2\omega)\right)^{2}$ is the frequency-dependent dielectric constant and its
real and imaginary parts are related to the refractive index $n(\omega)$ and
the absorption coefficient $\alpha(\omega)$.

Due to low optical loses in fibers within the wavelength region of interest,
Im$\\{\chi^{(1)}(\omega)\\}\ll\text{Im}\\{\chi^{(1)}(\omega)\\}$ and hence,
$\epsilon(\omega)\approx n^{2}(\omega)$. In addition, since usually
$n(\omega)$ is independent of spatial coordinates, then
$\nabla\cdot\mathbf{D}=\epsilon\,\nabla\cdot\mathbf{E}=0$ and hence,
$\nabla\times\nabla\times\mathbf{E}=\nabla(\nabla\cdot\mathbf{E})-\nabla^{2}\mathbf{E}=-\nabla^{2}\mathbf{E}$.
Finally, we arrive at the Helmholtz equation

$\nabla^{2}\mathbf{\hat{E}}+n^{2}(\omega)\frac{\omega^{2}}{c^{2}}\mathbf{\hat{E}}=0.$
(101)

By including the nonlinear effect of the induced polarization, the Helmholtz
equation can be written as

$\nabla^{2}\mathbf{\hat{E}}+\epsilon(\omega)k_{0}^{2}\mathbf{\hat{E}}=0$ (102)

where $k_{0}=\omega/c$ and the dielectric constant
$\epsilon(\omega)=1+\hat{\chi}^{(1)}(\omega)+\frac{3}{4}\frac{d^{4}\chi^{(3)}}{dx^{4}}|E(\mathbf{r},t)|^{2}$.
The Helmholtz equation (102) can be solved using the method of separation of
variables. We assume an Ansatz in the following form:

$\hat{E}(\mathbf{r},\omega-\omega_{0})=\hat{A}(z,\omega-\omega_{0})\,B(x,y)\,e^{i\beta_{0}z}$
(103)

where $\hat{A}(z,\omega)$ is a slowly varying function of $z$ and $\beta_{0}$
is a wavenumber that needs to be determined. The Helmholtz equation (102)
leads to the following equations for $\hat{A}(z,\omega)$ and for $B(x,y)$,
where we have neglected $\partial_{z}^{2}\hat{A}$ due to an assumption of
slowly varying function in $z$:

$\displaystyle
2i\beta_{0}\partial_{z}\hat{A}+(\hat{\beta}^{2}-\beta_{0}^{2})^{2}\hat{A}$
$\displaystyle=0$ (104)
$\displaystyle\nabla^{2}B+\left[\epsilon(\omega)k_{0}^{2}-\hat{\beta}^{2}\right]B$
$\displaystyle=0.$ (105)

The wavenumber $\hat{\beta}$ is determined by solving the eigenvalue equation
(105) using the first-order perturbation theory. We obtain

$\hat{\beta}(\omega)=\beta(\omega)+\Delta\beta$ (106)

where

$\Delta\beta=\frac{\omega^{2}n(\omega)}{c^{2}\beta(\omega)}\frac{\displaystyle\int_{\infty}^{\infty}\int_{-\infty}^{\infty}\Delta
n(\omega)|B(x,y)|^{2}\,\mathrm{d}x\,\mathrm{d}y}{\displaystyle\int_{\infty}^{\infty}\int_{-\infty}^{\infty}|B(x,y)|^{2}\,\mathrm{d}x\,\mathrm{d}y}.$
(107)

Using (106) and approximating $\hat{\beta}^{2}-\beta_{0}^{2}\approx
2\beta_{0}(\hat{\beta}-\beta_{0})$, the Fourier transform
$\hat{A}(z,\omega-\omega_{0})$ satisfying (104) can be written as follows:

$\partial_{z}\hat{A}=i\left[\beta(\omega)+\Delta\beta(\omega)-\beta_{0}\right]\hat{A}.$
(108)

Since an exact form of the propagation constant $\beta(\omega)$ is rarely
known, it is beneficial to expand both $\beta(\omega)$ and
$\Delta\beta(\omega)$ in a Taylor series about the carrier frequency
$\omega_{0}$

$\displaystyle\beta(\omega)$
$\displaystyle=\beta(\omega_{0})+\beta^{\prime}(\omega_{0})(\omega-\omega_{0})+\frac{1}{2}\beta^{\prime\prime}(\omega_{0})(\omega-\omega_{0})^{2}+\dots$
(109) $\displaystyle\Delta\beta(\omega)$
$\displaystyle=\Delta\beta(\omega_{0})+\Delta\beta^{\prime}(\omega_{0})(\omega-\omega_{0})+\frac{1}{2}\Delta\beta^{\prime\prime}(\omega_{0})(\omega-\omega_{0})^{2}+\dots.$
(110)

Replacing $\omega-\omega_{0}$ with the differential operator $i\partial_{t}$
and taking back the inverse Fourier transform of
$\hat{A}(z,\omega-\omega_{0})$, we obtain the following equation for $A(z,t)$:

$i\partial_{z}A+i\beta^{\prime}(\omega_{0})\partial_{t}A-\frac{1}{2}\beta^{\prime\prime}(\omega_{0})\partial_{t}^{2}A+\Delta\beta(\omega_{0})A=0.$
(111)

Using the transformation of a moving frame of reference
$T=t-\beta^{\prime}(\omega_{0})z$ and considering that the last term contains
the fiber loss and nonlinearity effects, we obtain the NLSE

$i\partial_{z}A-\frac{1}{2}\beta^{\prime\prime}(\omega_{0})\partial_{T}^{2}A+\gamma|A|^{2}A=0.$
(112)

Here, the nonlinear coefficient $\gamma$ is given by

$\gamma(\omega_{0})=-\frac{\omega_{0}}{c}n_{2}(\omega_{0})\frac{\displaystyle\int_{\infty}^{\infty}\int_{-\infty}^{\infty}|B(x,y)|^{4}\,\mathrm{d}x\,\mathrm{d}y}{\displaystyle\left(\int_{\infty}^{\infty}\int_{-\infty}^{\infty}|B(x,y)|^{2}\,\mathrm{d}x\,\mathrm{d}y\right)^{2}}.$
(113)

For a single-mode fiber, the modal distribution $B(x,y)$ corresponds to the
fundamental fiber mode, given by one of the following expressions:

$\displaystyle B(x,y)$
$\displaystyle=\left\\{\begin{array}[]{ll}J_{0}(p\sqrt{x^{2}+y^{2}}),&\qquad\sqrt{x^{2}+y^{2}}\leq
a\\\
\frac{\sqrt{a}}{\sqrt[4]{x^{2}+y^{2}}}J_{0}(pa)e^{-q(\sqrt{x^{2}+y^{2}}-a)},&\qquad\sqrt{x^{2}+y^{2}}\geq
a\end{array}\right.$ (116) $\displaystyle\text{or}\quad B(x,y)$
$\displaystyle=e^{-\frac{(x^{2}+y^{2})}{w^{2}}}.$ (117)

Here, $J_{0}$ denotes the Bessel function of the first kind of order zero, $a$
is the radius of the fiber core, $w$ is a width parameter, and the quantities
$p=\sqrt{n_{1}^{2}k_{0}^{2}-\beta^{2}}$ and
$q=\sqrt{\beta^{2}-n_{c}^{2}k_{0}^{2}}$.

###  4.2 Applications in nonlinear optics

In the context of nonlinear optics, an of interest object of study is
“solitary waves,” also known as “solitons.” Depending on whether the light
confinement occurs in time or space during wave propagation, solitons can be
classified as either temporal or spatial. The NLSE (112) governs the time-
dependent pulse envelope propagation in optical fibers, also known as temporal
soliton. On the other hand, the spatial soliton, a continuous wave beam
propagation inside a nonlinear optical medium with Kerr (or cubic)
nonlinearity, is governed by the following $(1+1)$D NLSE

$i\partial_{z}A+\beta\partial_{X}^{2}A+\gamma|A|^{2}A=0.$ (118)

A classical example for the latter is a bell-shaped spatial wavepacket with
the self-induced lensing effect, a phenomenon of self-trapping in dielectric
waveguide modes discovered by [Chiao, Garmire and Townes 1964]. A stable
spatial soliton was also observed experimentally with self-trapping laser
beams propagating through homogeneous transparent dielectrics [Barthelemy,
Maneuf and Froehly 1985]. For an extensive coverage on spatial solitons,
please consult [Trillo and Torruellas 2001].

In the following, we will discuss temporal solitons in optical fibers. An
optical fiber is a flexible and transparent material fiber made by drawing a
pure silica glass or plastic fiber. The center core is surrounded by an outer
layer, known as cladding, with lower refractive index optical material than
the core. The fiber is then coated with buffer and jacket for protection from
moisture and physical damage. Optical fibers are widely used in fiber-optics
communications as a means to transmit light between the two ends of the fiber.
The light remains in the core due to total internal reflection so that the
fiber acts as a waveguide.

The result of an interaction between fiber dispersion and nonliearity is fiber
optic solitons. The self-phase modulation phenomenon, where an ultrashort
pulse of light inducing a varying refractive index when traveling in a medium
due to the optical Kerr effect, can balance the anomalous group velocity
dispersion with a nonlinearity effect to create an optical soliton. An
existence of temporal optical solitons in the context of optical fibers was
predicted theoretically by [Hasegawa and Tappert 1973a, Hasegawa and Tappert
1973b] and experimentally confirmed by [Mollenauer, Stolen and Gordon 1980].
These solitons represent optical pulses that maintain their shape during
propagation and belong to families of exact solution of the NLSE.

The field of nonlinear fiber optics continues to progress with the development
of Erbium-doped fiber amplifiers [Becker, Olsson and Simpson 1999, Desurvire
et al 2002]. With the advent of this century, new types of fiber optic
amplifiers indicating nonlinear effects were developed, including stimulated
Raman scattering and four-wave mixing [Headley and Agrawal 2005, Pal 2010].
This further leads to other types of solitons such as dispersion-managed
solitons and dissipative solitons [Hasegawa and Matsumoto 2003, Kivshar 2003,
Akhmediev and Ankiewicz 2005, Mollenauer and Gordon 2006, Akhmediev and
Ankiewicz 2008]. For feature articles on theoretical and experimental
challenges in optical solitons, the readers may consult a volume edited by
[Porsezian and Kuriakose 2003].

In addition to the well-known bright, dark and gray solitons, “optical rogue
waves” have gained popularity during the past decade in the field of nonlinear
optics. The experimental result on rogue waves in an optical system is
supported by numerical simulation based on probabilistic supercontinuum
generation in a highly nonlinear microstructured optical fiber and a
generalized NLSE as a mathematical model [Solli et al 2007, Dudley, Genty and
Eggleton 2008, Bonatto et al 2011]. An overview on the research dynamics in
optical rogue waves and the state of the art on the subject has been covered
by [Akhmediev et al 2013]. For discussion and debate on whether the science of
rogue waves is moving towards a unifying concept, please consult the papers
published by various authors in The European Physical Journal Special Topics,
Volume 185, pages 1–266, July 2010, published by Springer-Verlag.

##  5 Bose-Einstein condensation

###  5.1 NLSE derivation from Bose-Einstein condensed state

A Bose-Einstein condensate (BEC) is a state of matter of a low density dilute
gas, also known as bosons, for which cooling down to a nearly absolute zero
temperature would cause them to condense into the lowest accessible quantum
state. The phenomenon was predicted nearly a century ago by [Bose 1924] and
[Einstein 1924]. In the context of BEC, the NLSE is known as Gross-Pitaevskii
equation (GPE), where the model was derived independently by [Gross 1961] and
[Pitaevskii 1961]. The derivation presented in this section follows an
argument presented in [Pitaevskii and Stringari 2003, Pethick and Smith 2008,
Dalfovo et al 1999]. For a rigorous treatment of the modeling, see [Erdős,
Schlein and Yau 2007, Erdős, Schlein and Yau 2010].

Consider a system of a weakly-interacting Bose gas where its Hamiltonian can
be written in terms of the field operator $\psi$

$\hat{H}=\int\left(\frac{\hbar}{2m}\nabla\hat{\Psi}^{\dagger}\nabla\hat{\Psi}\right)\mathrm{d}\mathbf{r}+\frac{1}{2}\int\hat{\Psi}^{\dagger}\hat{\Psi}^{\dagger\prime}V(\mathbf{r}^{\prime}-\mathbf{r})\hat{\Psi}\hat{\Psi}^{\prime}\,\mathrm{d}\mathbf{r}^{\prime}\,\mathrm{d}\mathbf{r}.$
(119)

Here, $\hbar$ is the Planck constant, $m$ is particle mass,
$\hat{\Psi}^{\dagger}(\mathbf{r})$ and $\hat{\Psi}(\mathbf{r})$ are the field
operator creating and annihilating a particle at the point $\mathbf{r}$, and
$V(\mathbf{r})$ is the two-body potential. The field operators satisfy the
following commutation relationship

$\left[\hat{\Psi}(\mathbf{r}),\hat{\Psi}^{\dagger}(\mathbf{r})\right]=\delta(\mathbf{r}-\mathbf{r}^{\prime})\quad\quad\text{and}\quad\quad\left[\hat{\Psi}(\mathbf{r}),\hat{\Psi}(\mathbf{r}^{\prime})\right]=0.$
(120)

In the Heisenberg representation, the field operator
$\hat{\Psi}(\mathbf{r},t)$ satisfies

$\displaystyle i\hbar\partial_{t}\hat{\Psi}(\mathbf{r},t)$
$\displaystyle=\left[\hat{\Psi}(\mathbf{r},t),\hat{H}\right]$ (121)
$\displaystyle=\left[-\frac{\hbar^{2}\nabla^{2}}{2m}+V_{\text{ext}}(\mathbf{r},t)+\int\hat{\Psi}^{\dagger}(\mathbf{r}^{\prime},t)V(\mathbf{r}^{\prime}-\mathbf{r})\hat{\Psi}(\mathbf{r}^{\prime},t)\,\mathrm{d}\mathbf{r}^{\prime}\right]\,\hat{\Psi}(\mathbf{r},t)$
(122)

where $V_{\text{ext}}$ is an external potential and we have utilized the
Hamiltonian (119) and the commutation relationship (120). If we apply an
effective potential $V_{\text{eff}}$ where the Born approximation is
applicable [Born 1926], then we can replace the field operator
$\hat{\Psi}(\mathbf{r},t)$ with a classical field or the condensation wave
function $\Psi_{0}(\mathbf{r},t)$ at a very low temperature and up to the
lowest-order approximation. Under an assumption of the slowly varying function
$\Psi_{0}(\mathbf{r},t)$ on distances of the order of interatomic force range,
$\mathbf{r}^{\prime}$ can be replaced with $\mathbf{r}$. We arrive at the GPE

$i\hbar\partial_{t}\Psi_{0}(\mathbf{r},t)=\left(-\frac{\hbar^{2}\nabla^{2}}{2m}+V_{\text{ext}}(\mathbf{r},t)+g|\Psi_{0}(\mathbf{r},t)|^{2}\right)\Psi_{0}(\mathbf{r},t),$
(123)

where ${\displaystyle g=\int
V_{\text{eff}}(\mathbf{r})\,\mathrm{d}\mathbf{r}}$. This GPE governs the
ground state of a quantum system of identical bosons where it is used as a
model equation for the single-particle wavefunction in a BEC. In particular,
the presence of external potential $V_{\text{ext}}$ allows us to model various
situations of the external world action on the condensate.

###  5.2 Applications in BEC

Although the low-temperature and high-density state of BEC was predicted in
the 1920s, it was not until the 1990s that the phenomenon has been
successfully implemented and experimentally tested in laboratories, by
confining in magnetic traps and cooling down to extremely low temperatures,
vapors of rubidium 87Rb [Anderson et al 1995], lithium 7Li [Bradley et al
1995], and sodium 23Na [Davis et al 1995] atoms. An intensive effort on other
atomic species has also produced fruitful experimental results, such as on
dilute gasses of atomic hydrogen [Fried et al 1998, Greytak et al 2000] and
helium in the $2^{3}\,S_{1}$ metastable state 4He∗ [Dos Santos et al 2001,
Robert et al 2001], as well as samples of potassium 41K [Modugno et al 2001],
cesium 133Cs [Weber et al 2003], and another isotope of rubidium 85Rb [Cornish
et al 2000]. The group at the Joint Institute for Laboratory Astrophysics in
Colorado successfully measured, for the first time, the collective excitations
of a Bose condensed dilute gas in a trap [Jin et al 1996].

For the modeling aspect, the dynamics of a dilute trapped BEC has been
successfully constructed using the mean-field theory, within the self-
consistent [Hartree 1928]-[Fock 1930]-[Bogoliubov 1947] approximation, where
indeed the GPE can also be derived [Griffin 1996]. The majority of theoretical
approach in solving the GPE is centered around the Thomas-Fermi approximation
[Thomas 1927, Fermi 1927], where the nonlinear atomic and interactions are
much larger than the kinetic energy pressure, and hence the latter is
neglected [Baym and Pethick 1996, Stringari 1996, Dalfovo, Pitaevskii and
Stringari 1996]. An analytical attempt in solving the GPE to model the
dynamics of dilute ultracold atom clouds in the BEC phase by applying a
variational technique is proposed by [Pérez-García et al 1996, Pérez-García et
al 1997].

The GPE for BEC has also been studied and solved numerically, by direct
numerical integration for time-independent [Edwards and Burnett 1995] and
time-dependent GPEs [Ruprecht et al 1996]. Other techniques for the latter
include, but are not limited to, the semi-implicit Crank-Nicholson scheme
[Ruprecht et al 1995], an eigenfunction basis expansion method [Edwards et al
1996], an explicit finite-difference scheme [Cerimele et al 2000], and the
time-splitting spectral method [Bao, Jaksch and Markowich 2003].

Another application of the GPE to a 1D cloud boson is covered by [Rogel-
Salazar 2013]. This 1D bosonic gas with a uniform potential can be obtained by
getting a condensate in an elongated trap. In particular, the phenomena
occurring at the center of the trap where the density is relatively uniform
and the condensate behaves as a fluid is interesting. For repulsive
interactions, the solution of the GPE is in the form of a hyperbolic tangent
profile while for attractive interactions, it becomes a hyperbolic secant
profile.

An extensive review of the mean-field theory applied to BEC is given by
[Dalfovo et al 1999]. See also other reviews by [Burnett 1996] and [Parkins
and Walls 1998]. For an excellent symbiosis between theoretical and
experimental contributions on BEC, the readers are encouraged to consult
[Kevrekidis, Frantzeskakis and Carretero-Gonzáles 2008].

##  6 Conclusion

In this chapter, we have provided an overview of modeling and application
aspects of the NLSE in various physical settings. Indeed, the subject of NLSE
is an active and dynamic research area not only in Mathematical Physics, but
also in other areas of Science as well as in Engineering. We have derived the
NLSE heuristically as well as by implementing the method of multiple-scales.
We also covered the applications of NLSE in surface water waves,
superconductivity, nonlinear optics and BEC, including solitons and rogue
waves. The evolution equation admittedly has some limitations, yet, it is
remarkable that the NLSE provides a rather universal model in several areas
that do not seem to be closely connected according to a general point of view.
We hope that this chapter will stimulate further research on these exciting
topics.

## Acknowledgements

The author would like to acknowledge Professor E. (Brenny) W. C. van Groesen
(University of Twente, the Netherlands and LabMath Indonesia), Professor Hadi
Susanto (The University of Essex, UK), Professor Nail N. Akhmediev (Australian
National University), Professor Panayotis G. Kevrekidis (University of
Amherst, Massachusetts), Professor Mason A. Porter (University of California,
Los Angeles), Professor Guido Schneider (University of Stuttgart, Germany),
Dr. Gert Klopman (Witteveen$+$Bos, the Netherlands), Dr. Agung Trisetyarso
(Bina Nusantara University, Indonesia), Drs. Andonowati, Alexander A. P.
Iskandar, and Rudy Kusdiantara (Bandung Institute of Technology, Indonesia),
Professor Frédéric Dias (University College Dublin, Ireland), Professor
Kristian B. Dysthe (University of Bergen, Norway), Professor Karsten Trulsen
(University of Oslo, Norway), Professors Miguel Onorato and Alfred R. Osborne
(Universitá di Torino, Italy), Dr. Ardhasena Sopaheluwakan (Meteorology,
Climatology, and Geophysical Agency, Indonesia), Professor Christian Kharif
(Institut de Recherche sur les Phénomènes Hors Equilibre, Research Institute
for Non-Equilibrium Phenomena, Aix- Marseille University, France), Professors
René H. M. Huijsmans and Jurjen A. Battjes (Delft University of Technology,
the Netherlands), Professor Arthur E. Mynett (International Institute for
Hydraulic and Environmental Engineering IHE Delft Institute for Water
Education and Deltares, formerly WL|Delft Hydraulics, the Netherlands), and
Dr. Pearu Peterson (Tallinn University of Technology, Estonia) for advices,
suggestions, and fruitful discussions. This research is supported by the Dutch
Organization of Scientific Research NWO (Nederlandse Organisatie voor
Wetenschappelijk Onderzoek), subdivision Applied Sciences STW (Stichting
Technische Wetenschappen) through Mathematics and Innovation Transfer Point
(Transferpunt Wiskunde en Innovatie) Grant No. TWI-5374, the New Researcher
Fund from the University of Nottingham, University Park Campus, UK, through
Grant No. NRF 5035-A2RL20, the SKKU Samsung Intramural Research Fund No.
2016-1299-000, and the Beginning Independent Researcher Program (Saengae-Cheot
Yeongu) from the National Research Foundation of Korea through Grant No.
NRF-2017-R1C1B5-017743 under the Basic Research Program in Science and
Engineering.

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