# Instability of Double-Periodic Waves in the Nonlinear Schrödinger Equation

- Department of Mathematics, McMaster University, Hamilton, ON, Canada

It is shown how to compute the instability rates for the double-periodic solutions to the cubic NLS (nonlinear Schrödinger) equation by using the Lax linear equations. The wave function modulus of the double-periodic solutions is periodic both in space and time coordinates; such solutions generalize the standing waves which have the time-independent and space-periodic wave function modulus. Similar to other waves in the NLS equation, the double-periodic solutions are spectrally unstable and this instability is related to the bands of the Lax spectrum outside the imaginary axis. A simple numerical method is used to compute the unstable spectrum and to compare the instability rates of the double-periodic solutions with those of the standing periodic waves.

## 1 Introduction

Peregrine breather is a rogue wave arising on the background of the constant-amplitude wave due to its modulational instability [1, 2]. The focusing cubic NLS (nonlinear Schrödinger) equation is the canonical model which describes both the modulational instability and the formation of rogue waves. Formation of rogue waves on the constant-amplitude background have been modeled from different initial data such as local condensates [3], multi-soliton gases [4–6], and periodic perturbations [7, 8]. Rogue waves have been experimentally observed both in hydrodynamical and optical laboratories [9] (see recent reviews in [10, 11]).

Mathematical theory of rogue waves on the constant-amplitude background has seen many recent developments. universal behavior of the modulationally unstable constant-amplitude background was studied asymptotically in [12, 13]. The finite-gap method was employed to relate the unstable modes on the constant-amplitude background with the occurrence of rogue waves [14, 15]. Rogue waves of infinite order were constructed in [16] based on recent developments in the inverse scattering method [17]. Rogue waves of the soliton superposition were studied asymptotically in the limit of many solitons [18, 19].

At the same time, rogue waves were also investigated on the background of standing periodic waves expressed by the Jacobian elliptic functions. Such exact solutions to the NLS equation were constructed first in [20] (see also early numerical work in [21] and the recent generalization in [22]). It was confirmed in [23] that these rogue waves arise due to the modulational instability of the standing periodic waves [24] (see also [25, 26]). Instability of the periodic standing waves can be characterized by the separation of variables in the Lax system of linear equations [27] (see also [28, 29]), compatibility of which gives the NLS equation. Instability and rogue waves on the background of standing periodic waves have been experimentally observed in [30].

*The main goal of this paper is to compute the instability rates for the double-periodic solutions to the NLS equation, for which the wave function modulus is periodic with respect to both space and time coordinates.* In particular, we consider two families of double-periodic solutions expressed as rational functions of the Jacobian elliptic functions which were constructed in the pioneering work [31]. These solutions represent perturbations of the Akhmediev breathers and describe generation of either phase-repeated or phase-alternating wave patterns [32, 33]. Rogue waves on the background of the double-periodic solutions were studied in [34] (see also numerical work in [35, 36]). Experimental observation of the double-periodic solutions in optical fibers was reported in [37].

The double-periodic solutions constructed in [31] are particular cases of the quasi-periodic solutions of the NLS equation given by the Riemann Theta functions of genus two [38–40]. Rogue waves for general quasi-periodic solutions of any genus were considered in [41–43].

Instability of the double-periodic solutions is studied using the Floquet theory for the Lax system of linear equations both in space and time coordinates. We compute the instability rates of the double-periodic solutions and compare them with those for the standing periodic waves. In order to provide a fair comparison, we normalize the amplitude of all solutions to unity. *As a main outcome of this work, we show that the instability rates are larger for the constant-amplitude waves and smaller for the double-periodic waves.*

The article is organized as follows. The explicit solutions to the NLS equation are reviewed in Section 2. Instability rates for the standing periodic waves and the double-periodic solutions are computed in Sections 3 and 4 respectively. Further directions are discussed in Section 5.

## 2 Explicit Solutions to the NLS Equation

The nonlinear Schrödinger (NLS) equation is a fundamental model for nonlinear wave dynamics [44, 45]. We take the NLS equation in the standard form:

This model has several physical symmetries which are checked directly:

translation:

scaling:

Lorentz transformation:

In what follows, we use the scaling symmetry Eq. 2.3 to normalize the amplitude of periodic and double-periodic solutions to unity and the Lorentz symmetry Eq. 2.4 to set the wave speed to zero. We also neglect the translational parameters

A solution

and

where *ψ* and

The algebraic method developed in [34] allows us to construct the stationary (Lax–Novikov) equations which admit a large class of bounded periodic and quasi-periodic solutions to the NLS Eq. 2.1 The simplest first-order Lax–Novikov equation is given by

where *c* is arbitrary real parameter. A general solution of this equation is given by *A* is the integration constant. This solution determines the constant-amplitude waves of the NLS Eq. 2.1 in the form:

where *c* can be set to 0 due to the Lorentz transformation. Indeed, transformation Eq. 2, 4 with *A* can be set to unity, which yields the normalized solution

The second-order Lax–Novikov equation is given by

where *u* to the second-order Eq. 2.9 determines the standing traveling waves of the NLS Eq. 2.1 in the form:

Without loss of generality, we set

and

where the parameter *k* for Eq. 2 12 In order to normalize the amplitude to unity for the cnoidal wave Eq. 2.12 we can use the scaling transformation Eq. 2.3 with

Due to the well-known expansion formulas

both the periodic waves Eqs. 2.11, 2.12 approaches the NLS soliton

The third-order Lax–Novikov equation is given by

where

where *t* compared to

and

where

The double-periodic solutions Eqs. 2.15, 2.16 can be written in the form:

where

Figure 1 shows surface plots of

**FIGURE 1**. Amplitude-normalized double-periodic waves Eq. 2.15**(left)** and Eq. 2.16**(right)** with

As

As

whereas the solution Eq. 2.16 approaches the scaled cnoidal wave

These limits are useful to control accuracy of numerical computations of the modulational instability rate for the double-periodic solutions in comparison with the similar numerical computations for the standing waves.

## 3 Instability of Standing Waves

Here we review how to use the linear Eq. 2.5, 2.6 in order to compute the instability rates for the standing periodic waves (Eq. 2.10) (see [23, 27]). Due to the separation of variables in Eq. 2.10, one can write

where

and

We say that λ belongs to *the Lax spectrum* of the spectral problem Eq. 3.2 if

where

Since the spectral problem Eq. (3.3) is a linear algebraic system, it admits a nonzero solution if and only if the determinant of the coefficient matrix is zero. The latter condition yields the *x*-independent relation between

with parameters *a* and *d* being the conserved quantities of the second-order Eq. 2.9:

and

Polynomial

where the turning points *b* and *d* in the form:

Roots of

so that the polynomial

By adding a perturbation *v* to the standing wave *u* in the form

and dropping the quadratic terms in *v*, we obtain the linearized system of equations which describe linear stability of the standing waves (Eq. 2.10):

The variables can be separated in the form:

where

where

We say that *the stability spectrum* of the spectral problem Eq. 3.15 if

Validity of Eq. 3.16 can be checked directly from Eqs. 3.2, 3.3, 3.14, and 3.15. If *spectrally unstable*. It is called *modulationally unstable* if the unstable spectrum with

The importance of distinguishing between spectral and modulational instability of the periodic standing waves appears in the existence of rogue waves on their background. It was shown in [46] that if the periodic standing waves are spectrally unstable but modulationally stable, the rogue waves are not fully localized and degenerate into propagating algebraic solitons. Similarly, it was shown in [23] that if the unstable spectrum with

Next, we compute the instability rates for the standing periodic waves (2.10) of the trivial phase with

For the

Since the *k*, and vanishes for the soliton limit

**FIGURE 2**. Lax spectrum on the λ-plane **(left)** and stability spectrum on the **(right)** for the

For the *k*. Hence, we use the scaling transformation (2.3) with

**FIGURE 3**. The same as Figure 2 but for the amplitude-normalized **(top)** and **(bottom)**.

Figure 4 compares the instability rates for different standing waves of the same unit amplitude. *k* is increased in

Due to the scaling transformation Eq. 2.3 and the expansion Eq. 3.17, the maximal growth rate in the limit *k* increases in

**FIGURE 4**. Instability rate **(left)** and **(right)** waves. The values of *k* in the elliptic functions are given in the plots.

## 4 Instability of Double-Periodic Waves

Here we describe the main result on how to compute the instability rates for the double-periodic waves by using the linear Eqs. 2.5, 2.6. We write the solutions Eqs. 2.15, 2.16 in the form Eq. 2.17. We represent solution *φ* to the linear Eqs. 2.5, 2.6 in the form:

where

and

Parameters

By Floquet theorem, spectral parameters *x* and the time coordinate *t* in order to consider stability of the double-periodic waves Eq. 2.17 in the time evolution of the NLS Eq. 2.1.

*The Lax spectrum* is defined by the condition that λ belongs to an admissible set for which the solution Eq. 4.1 is bounded in *x*. Hence

With λ defined in the Lax spectrum, the spectral problem Eq. 4.3 can be solved for the spectral parameter

If *spectrally unstable*. The amplitude-normalized double-periodic waves are taken by using the scaling transformation (2.3). We observe again that the unstable spectrum with

For the amplitude-normalized double-periodic wave Eq. 2.15 with

**FIGURE 5**. Lax spectrum on the λ-plane **(left)** and stability spectrum on the **(right)** for the amplitude-normalized double-periodic wave Eq. 2.15 with **(top)** and **(bottom)**.

Figure 6 shows the same as Figure 5 but for the amplitude-normalized double-periodic wave Eq. 2.16 with

**FIGURE 6**. The same as Figure 5 but for the amplitude-normalized double-periodic wave (2.16) with **(top)**, **(middle)**, and **(bottom)**.

Figure 7 compares the instability rates for different double-periodic waves of the same unit amplitude.

**FIGURE 7**. Instability rate **(left)** and the double-periodic wave Eq. 2.16**(right)**. The values of *k* are given in the plots.

For the amplitude-normalized double-periodic wave Eq. 2.15 (left), the instability rate is maximal as

For the amplitude-normalized double-periodic wave Eq. 2.16 (right), the instability rate is large in the limit *k* is increased, however, the rates increase again and reach the maximal values as

## 5 Conclusion

We have computed the instability rates for the double-periodic waves of the NLS equation. By using the Lax pair of linear equations, we obtain the Lax spectrum with the Floquet theory in the spatial coordinate at fixed *t* and the stability spectrum with the Floquet theory in the temporal coordinate at fixed *x*. This separation of variables is computationally simpler than solving the full two-dimensional system of linearized NLS equations on the double-periodic solutions.

As the main outcome of the method, we have shown instability of the double-periodic solutions and have computed their instability rates, which are generally smaller compared to those for the standing periodic waves.

The concept can be extended to other double-periodic solutions of the NLS equation which satisfy the higher-order Lax–Novikov equations. Unfortunately, the other double-periodic solutions are only available in Riemann theta functions of genus

## Author Contributions

The author confirms being the sole contributor of this work and has approved it for publication.

## Conflict of Interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (No. 11971103).

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Keywords: modulational instability, double-periodic solutions, Floquet spectrum, nonlinear Schrödinger equation, standing waves

Citation: Pelinovsky DE (2021) Instability of Double-Periodic Waves in the Nonlinear Schrödinger Equation. *Front. Phys.* **9**:599146. doi: 10.3389/fphy.2021.599146

Received: 26 August 2020; Accepted: 04 January 2021;

Published: 22 February 2021.

Edited by:

Heremba Bailung, Ministry of Science and Technology, IndiaReviewed by:

Maximo Aguero, Universidad Autónoma del Estado de México, MexicoConstance Schober, University of Central Florida, United States

Copyright © 2021 Pelinovsky. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Dmitry E. Pelinovsky, dmpeli@math.mcmaster.ca