# Breather Solutions to a Two-dimensional
Nonlinear Schrödinger Equation with Non-local Derivatives

Alexander Hrabski  Yulin Pan<EMAIL_ADDRESS>Department of Naval
Architecture and Marine Engineering, University of Michigan, Ann Arbor,
Michigan 48109, USA

###### Abstract

We consider the nonlinear Schrödinger equation with non-local derivatives in a
two-dimensional periodic domain. For certain orders of derivatives, we find a
new type of breather solution dominating the field evolution at low
nonlinearity levels. With the increase of nonlinearity, the breathers break
down, giving way to wave turbulence (or Rayleigh-Jeans) spectra. Phase-space
trajectories associated with the breather solutions are found to be close to
that of the linear system, revealing a connection between the breather
solution and Kolmogorov-Arnold-Moser (KAM) theory.

††preprint: APS/123-QED

##  I Introduction

Breathers are a broadly-defined class of features that arise in nonlinear
dynamical systems, generally describing a family of solutions with strong
spatial localization and oscillations in time. Together with solitons,
breathers are considered as prototypes for rogue waves that can occur across
many fields, such as water waves [1, 2], optics [3], and plasma physics [4].
Mathematically, breathers are fundamental solutions to both continuous field
equations and discrete lattice problems. In the former case, breather
solutions have been found primarily in one-dimensional (1D) nonlinear partial
differential equations, including the Sine-Gordon equation [5], the nonlinear
Schrödinger equation (NLS) [6, 1, 2, 3, 4], and the Korteweg–De Vries (KdV)
equation [7, 8]. In the latter case, discrete breathers [9] (as counterparts
to breathers in continuous fields) have been constructed as solutions to a
wide variety of systems, including Josephson Junctions [10, 11] and the Fermi-
Pasta-Ulam-Tsingou (FPUT) problem [12]. Unlike their continuous counterparts,
discrete breathers haven been studied extensively in problems with more than
one dimension.

Another important category of studies regards the spontaneous emergence of
breathers (and other types of coherent structures) under the free evolution of
a system. These coherent structures include the quasi-solitons [13, 14],
quasi-breathers [15], and wave collapses [13, 16, 17] identified in the 1D
Majda-McLaughlin-Tabak (MMT) model, as well as the discrete breathers in the
FPUT problem [18] and the discrete nonlinear Schrödinger equation [19, 20]. As
in the case of constructing exact breather solutions, when considering
continuous fields, these studies are predominantly performed for 1D
situations. The only exception, to our knowledge, is [21], which identifies a
breather solution to the NLS with a potential on a two-dimensional (2D)
domain, but the mechanism associated with the breather remains unexplained. In
general, very little is known about 2D breathers unless one considers discrete
lattice problems.

In this letter, we demonstrate the existence of breather solutions to a family
of (non-local) derivative NLS without a potential, realized in a 2D periodic
domain (we call such solutions generally as breathers instead of quasi-
breathers since the latter, as defined in [15] for periodic domains, are
associated with very different physics). In addition to being a novel 2D
breather in a continuous field, other remarkable and distinguishing features
of the solution include: (1) the breather spontaneously emerges from a
stochastic wave field after long-time evolution, not relying on specific
initial conditions; (2) the breather appears equivalently for both the
focusing and defocusing cases, but exists only in the weak nonlinearity
regime. As the nonlinearity of the system increases, we find a breakdown of
the breather state with the field relaxing to the Rayleigh-Jeans spectrum (or
wave turbulence spectrum if dissipation is included). Further analysis shows
that the state trajectory of the breathers is associated with a Kolmogorov-
Arnold-Moser (KAM) torus, which is a distorted trajectory of the linear
(integrable) system that survives when the nonlinearity level is sufficiently
low [22].

Figure 1:  The time series of $H$ (dashed) and $H_{2}$ (solid) for (a)
$\varepsilon=0.00071$ and (b) $\varepsilon=0.20$, as well as the corresponding
$H_{4}$ for (c) $\varepsilon=0.00071$ and (d) $\varepsilon=0.20$. Note that a
low sampling frequency (1 point per $T_{f}$) is used to plot the figure to
improve its readability, leading to aliasing. Therefore, only a small portion
of all $O(1600)$ peaks are visible.

##  II Setup of Numerical Experiments

The Majda-McLaughlin-Tabak (MMT) model is a family of nonlinear dispersive
wave equations that have been widely used to study wave turbulence [23, 24,
13] and coherent structures [13, 14, 15, 16, 17], due to its effectiveness in
representing nonlinear waves in different physical contexts. In the present
work, we consider the MMT model in two spatial dimensions, constructed as

$i\frac{\partial\psi}{\partial
t}=|\partial_{\bm{x}}|^{2}\psi+\lambda|\partial_{\bm{x}}|^{\beta/4}\left(\left||\partial_{\bm{x}}|^{\beta/4}\psi\right|^{2}|\partial_{\bm{x}}|^{\beta/4}\psi\right),$
(1)

where $\psi\equiv\psi(\bm{x},t)$ is a complex scalar, $\bm{x}$ is the spatial
coordinates, and $t$ the time. The non-local derivative operator
$|\partial_{\bm{x}}|^{\alpha}$ denotes a multiplication by $k^{\alpha}$ on
each spectral component in wave number domain, with $k=|\bm{k}|$. The free
parameter $\beta$ controls the order of derivatives, and $\lambda=\pm 1$
generates a defocusing/focusing nonlinearity, respectively. Equation (1) is
equivalent to a non-local derivative NLS, which can be shown more explicitly
after a transformation $\phi=|\partial_{\bm{x}}|^{\beta/4}\psi$, leading to

$i\frac{\partial\phi}{\partial
t}=|\partial_{\bm{x}}|^{2}\phi+\lambda|\partial_{\bm{x}}|^{\beta/2}\left(\left|\phi\right|^{2}\phi\right).$
(2)

The MMT model (1) can be derived from a Hamiltonian $H=H_{2}+H_{4}$, with

$\begin{split}H_{2}&=\int\big{|}|\partial_{\bm{x}}|\psi\big{|}^{2}d\bm{x},\\\
H_{4}&=\frac{1}{2}\lambda\int\left||\partial_{\bm{x}}|^{\beta/4}\psi\right|^{4}d\bm{x}.\end{split}$
(3)

The nonlinearity level of the system can be quantified via a parameter
$\varepsilon\equiv H_{4}/H_{2}$.

We solve (1) on a 2D periodic domain, starting from an initial field
$\psi_{0}\equiv\psi(\bm{x},t=0)$, via a pseudospectral method [24, 25, 26]
with $128\times 128$ modes (with higher resolution results available in
supplemental material [27]). Our numerical method treats the linear term via
an integrating factor, and the nonlinear term via an explicit 4th order Runge-
Kutta scheme. The initial field $\psi_{0}$ is set as an exponential form in
Fourier space as
$\hat{\psi}_{0}(\bm{k})=A\text{exp}[-0.1|k-k_{p}|+i\theta_{\bm{k}}]$, where
$k_{p}=4$, and $\theta_{\bm{k}}$ is a random phase that is decorrelated for
all $\bm{k}$. In order to investigate dynamics at different nonlinearity
levels, we choose a range of $A$ (about 10 values) leading to approximately
$\varepsilon\in[0.0005,0.1]$ for each $\beta$ value of interest.

##  III Results

We start by describing a typical simulation leading to a breather state, with
parameters $\beta=3$ and $A=35$ (corresponding to $\varepsilon=0.00071$).
Figures 1a and 1c show the long-time evolution of $H_{2}$ and $H_{4}$ from
$t=0$ to $2000T_{f}$, with $T_{f}=2\pi$ the period of the fundamental wave
mode. The total Hamiltonian $H$, as shown in figure 1a, is well conserved over
$2000T_{f}$. After an initial evolution of about $400T_{f}$ with smooth
profiles of $H_{2}$ and $H_{4}$, we observe that $H_{4}$ undergoes strong
periodic jumps with corresponding dips in $H_{2}$. These jumps are associated
with bursts of coherent structures, which are only present at low nonlinearity
levels. In contrast, as demonstrated in figures 1b and 1d, the field evolution
at a higher nonlinearity ($\varepsilon=0.20$) exhibits smooth profiles of
$H_{2}$ and $H_{4}$ over the same time interval.

Figure 2:  Contour plots of Re$[\psi]$ at $\beta=3$ for $\varepsilon=0.00071$
at various stages of the cycle of the breather (a/b/c/d), corresponding
chronologically to the times marked by the blue circles in (e) the time series
of $H_{4}$. Note that this plot of $H_{4}$ has sufficient sampling to resolve
all features.

The oscillations in $H_{4}$ (and $H_{2}$) correspond to oscillations of a
breather. To better visualize this breather state, we plot in figure 2 the
real part of $\psi$ at different phases of its oscillations (i.e., different
stages of the oscillation pattern in $H_{4}$). Fig. 2a shows the field right
before the first jump of $H_{4}$, where a concentric wave appears and later
converges into a breather peak seen in Fig. 2b. This peak then collapses, with
a second one emerging after about $T_{f}/2$ (according to fig. 2e) at the
maximally distant location in the periodic domain, shown in 2c. The cycle then
repeats itself with a peak emerging in Fig. 2d (at the same location as in
2a), forming an (oscillating) breather solution coexisting with a stochastic
wave background. The smaller peaks of $H_{4}$ seen in Fig. 2e correspond to
groups of secondary peaks in $\psi$. We encourage the reader to watch the
animation of the full breather cycle (including the secondary peaks) in the
supplemental material [27]. From figure 2 we see that the breather oscillates
with a period (very close to) $T_{f}$. Therefore, the simulation in figure
1a/c covers $O(1600)$ cycles of the breather, demonstrating a very long
(perhaps infinite) time of existence. We note that only a small number of
jumps are visible in figure 1a/c simply due to aliasing associated with
limited resolution of plotting (see caption of figure 1).

Figure 3:  The quantities (a) $\rho$ and (b) $\gamma$ as functions of
$\varepsilon$ for $\beta=0$ (orange $+$), $\beta=1$ (green $\bullet$),
$\beta=2$ (magenta $\times$), $\beta=3$ (blue $\blacksquare$), and $\beta=4$
(red $\blacktriangle$). The inset of (b) shows fully-developed, angle-averaged
wave action spectra at a few nonlinearity levels (for $\beta=3$), with the
Rayleigh-Jeans spectral slope of $\gamma=-2$ indicated (dashed).

We next investigate the existence and intensity of the breathers for varying
values of $\beta$ and $\epsilon$. To measure the intensity of the breather
(relative to the background wave field), we define the peak-to-background
ratio as

$\rho=\frac{\text{avgmax}[|\psi|]}{4\sigma_{|\psi|}},$ (4)

where the avgmax operator returns the average of the maximum height of primary
peaks (as in Fig. 2b/c/d) over many cycles of the breather, and
$\sigma_{|\psi|}$ is the total standard deviation of the field $|\psi|$ over
space and time. By definition (4), $\rho=2$ corresponds to the typical rogue
wave criterion used in many fields [3].

Figure 3a shows the value of $\rho$ obtained for $\beta=0,1,2,3,4$ and
$\varepsilon$ across three orders of magnitude. In general, we see that the
breather state is present for smaller $\varepsilon$ (i.e., weak nonlinearity)
and becomes stronger when $\beta$ is closer to 3. The case with $\beta=0$
(corresponding to NLS) leads to no breathers, indicating a non-local
derivative in NLS is necessary for their emergence. We note that when the
breathers are not present, the value of $\rho$ is evaluated by taking the
average of the maximum of the field $|\psi|$ every $T_{f}/2$ as the numerator
in (4).

Furthermore, we examine in figure 3b the slope $\gamma$ of the stationary wave
action spectrum
$n(\bm{k})\equiv\langle\hat{\psi}(\bm{k})\hat{\psi}^{*}(\bm{k})\rangle$ across
all values of $\beta$ and $\varepsilon$. The inset of figure 3b shows a
typical example of fully-developed, angle-averaged $n(k)$ for $\beta=2$ and
several values of $\varepsilon$. We see that the Rayleigh-Jeans spectrum with
$\gamma=-2$ is only achieved at higher nonlinearity when the breather is not
present. This trend is generally true for all values of $\beta$ as shown in
Figure 3b.

A few additional remarks are in order. First, we note that the breather also
emerges for the focusing equation (1) with $\lambda=-1$ under the same
conditions. Second, the breather can also be observed under a
forced/dissipated system [28], but with the Rayleigh-Jeans spectrum replaced
by the wave turbulence spectrum at high nonlinearity levels. Last but not
least, we have performed extensive numerical analysis to verify that the
breather we observe is not a numerical artifact. This includes the
verification of the robustness of our results under symplectic integration,
higher resolution, and different dealiasing schemes. Details of all of the
above points can be found in the supplemental material [27].

##  IV Discussion

In this section, we discuss the physical mechanism associated with the
breather solution. We start by stating that there exists an exact breather
solution to the linear system of (1), i.e., $H_{4}=0$. What we mean precisely
is that, starting from an initial condition with a breather peak (say figure
2b), the field propagated by the linear equation returns to the same state
after exactly $T_{f}$. This is because the linear system only contains integer
frequencies due to the NLS dispersion relation $\omega=k^{2}$, so that $T_{f}$
is the period of the linear system. This fact suggests that the breather
solution to the nonlinear system arises from a deformed trajectory of the
linear system.

Since visualizing the high-dimensional trajectory is very difficult, we define
a projection of the trajectory to some physically meaningful reference field:

$I_{n}=\left|\frac{\int\psi^{*}_{R}(\bm{x},t_{n})\psi(\bm{x},t)d\bm{x}}{\int\psi^{*}_{R}(\bm{x},t_{n})\psi_{R}(\bm{x},t_{n})d\bm{x}}\right|$
(5)

where $\psi^{*}_{R}(\bm{x},t_{n})$ is the reference field where a breather
peak is present, e.g., taken from $t_{1}$ in figure 2e, and $\psi(\bm{x},t)$
is the solution of either the linear or nonlinear system propagated from
$\psi^{*}_{R}(\bm{x},t_{n})$. Figure 4 shows the evolution of $I_{1}$ from
both linear and nonlinear systems for a range of four nonlinearity levels (at
high nonlinearity in Fig. 4d, $t_{1}$ is taken from an arbitrary time in the
quasi-stationary state). It is clear that the linear system evolution exhibits
a period of $T_{f}$ in all sub-figures as expected. When the nonlinearity
level is low, the trajectory identified by $I_{1}$ shows a small deformation
from the linear trajectory, as seen in fig. 4a. Such deformation leads to a
high-dimensional quasi-periodic trajectory that is evident from not only the
gradual time shift of the $I_{1}$ peak, but also the deviation of $I_{1}$ peak
from 1 (indicating completely different background waves). As nonlinearity
level increases, we observe an increased deformation of the trajectory, until
the linear trajectory is entirely destroyed at high nonlinearity in fig 4d.
This trajectory deformation can also be observed in the $(I_{1},I_{2})$ plane
in figure 5 (on which the linear trajectory shows an “L” shape due to the
orthogonality between $\psi^{*}_{R}(\bm{x},t_{1})$ and
$\psi^{*}_{R}(\bm{x},t_{2})$), as well as in animations included the
supplemental material [27].

Figure 4:  $I_{1}$ evaluated for the nonlinear system $\beta=3$ (blue) and the
linear system (red) for (a) $\varepsilon=0.00071$, (b) $\varepsilon=0.0084$,
(c) $\varepsilon=0.013$, and (d) $\varepsilon=0.20$. Figure 5:  The trajectory
of $\psi$ projected on to $I_{1}$ and $I_{2}$ for several fundamental periods
of $\psi$ computed via the nonlinear system $\beta=3$ (blue) and the linear
system (red) for (a) $\varepsilon=0.00071$, (b) $\varepsilon=0.0084$, (c)
$\varepsilon=0.013$, and (d) $\varepsilon=0.20$.

The trajectory deformation visualized above can be connected to Kolmogorov-
Arnold-Moser (KAM) theory. If one considers the linear system as the base
integrable system, then the nonlinear term is the perturbation added to the
system to form a nearly integrable system. According to KAM theory, if the
(nonlinear) perturbation is sufficiently small, some trajectories of linear
system can be preserved with small deformation to a KAM torus [22]. In our
case, these preserved trajectories (i.e., KAM tori) correspond to those
associated with the breather solution observed in figure 2. We also note that
equation (1) (or more generally the NLS) is a special case on which KAM theory
has not been fully understood mathematically, mainly because the base linear
system does not contain irrational frequencies (or quasi-periodicity) that are
required by the traditional KAM theorem. While most mathematical work of KAM
on NLS relies on some way to introduce irrational frequencies, e.g., by
including a potential term as in [29], recent progress [30] does show that it
is possible to prove the existence of quasi-periodic trajectories for the 2D
NLS (without a potential) for small nonlinearity. Our results therefore
demonstrate a breather solution associated with the quasi-periodic trajectory
if a non-local derivative is included in the NLS. Furthermore, if a different
dispersion relation, say $\omega=k^{1/2}$, is prescribed for (1), the KAM
theorem is more straightforward to apply (due to quasi-periodicity of the
linear system). However, in this case, the breather state we observe is no
longer a solution to the linear system, and one would not expect to observe it
in the nonlinear system as well. We have confirmed this argument through
numerical experiments.

Finally, the current analysis clearly does not resolve all the questions
regarding the new 2D breather solution. One critical question is why it arises
more strongly for $\beta$ close to 3. For these values of $\beta$, it is clear
not only that the trajectories associated with breathers are stable, but also
that they lie on some type of quasi-periodic “attractor” [31], such that a
variety of initial conditions lead to the breather state. The reason for such
dynamical behavior needs to be investigated in future work. In addition, our
work provides a clear demonstration of the discrete wave turbulence regime
governed by the KAM theory that has long been hypothesized in the literature
[32].

##  V Conclusion

In this paper, we present results regarding a novel breather which
spontaneously emerges from a 2D non-local derivative NLS. We show that the
breather emerges at low nonlinearity with parameter $\beta$ close to 3. An
analysis of phase space trajectory reveals that the trajectory associated with
the breather solution is close to that of the linear system, but with quasi-
periodicity introduced by the nonlinearity. The numerical findings support an
explanation of the breather solution by the KAM theory, in the sense that a
trajectory with the breather solution of the linear system is deformed but
preserved when a small nonlinearity is introduced.

###### Acknowledgements.

We thank Benno Rumpf, Bobby Wilson, Peter Miller, Zaher Hani, Gigliola
Staffilani, and Ricardo Grande for their thoughtful comments and suggestions.
This material is based upon work supported by the National Science Foundation
Graduate Research Fellowship under Grant No. DGE 1841052. Any opinions,
findings, and conclusions or recommendations expressed in this material are
those of the authors and do not necessarily reflect the views of the National
Science Foundation. This work used the Extreme Science and Engineering
Discovery Environment (XSEDE), which is supported by National Science
Foundation grant number ACI-1548562. Computation was performed on XSEDE
Bridges-2 at the Pittsburgh Supercomputing Center through allocation
PHY200041. This research was supported in part through computational resources
and services provided by Advanced Research Computing (ARC), a division of
Information and Technology Services (ITS) at the University of Michigan, Ann
Arbor.

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