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ORIGINAL RESEARCH article

Front. Phys., 30 April 2021
Sec. Statistical and Computational Physics
This article is part of the Research Topic Peregrine Soliton and Breathers in Wave Physics: Achievements and Perspectives View all 25 articles

On the Stabilization of Breather-type Solutions of the Damped Higher Order Nonlinear Schrödinger Equation

C. M. Schober
C. M. Schober*A. L. IslasA. L. Islas
  • Department of Mathematics, University of Central Florida, Orlando, FL, United States

Spatially periodic breather solutions (SPBs) of the nonlinear Schrödinger (NLS) equation are frequently used to model rogue waves and are typically unstable. In this paper we study the effects of dissipation and higher order nonlinearities on the stabilization of N-mode SPBs, 1N3, in the framework of a damped higher order NLS (HONLS) equation. We observe the onset of novel instabilities associated with the development of critical states resulting from symmetry breaking in the damped HONLS system. We develop a broadened Floquet characterization of instabilities of solutions of the NLS equation by showing that instabilities are associated with degenerate complex elements of not only the discrete, but also the continuous Floquet spectrum. As a result, the Floquet criteria for the stabilization of a solution of the damped HONLS centers around the elimination of all complex degenerate elements of the spectrum. For a given initial N-mode SPB, a short-time perturbation analysis shows that the complex double points associated with resonant modes split under the damped HONLS while those associated with nonresonant modes remain closed. The corresponding /damped HONLS numerical experiments corroborate that instabilities associated with nonresonant modes persist on a longer time scale than the instabilities associated with resonant modes.

1 Introduction

In one of his foundational studies, Stokes established the existence of traveling nonlinear periodic wave trains in deep water [1]. The stability of these waves was resolved when Benjamin and Feir proved that in sufficiently deep water the Stokes wave is modulationally unstable. Small perturbations of Stokes waves were found to lead to exponential growth of the side bands [2, 3]. More recently, modulational instability (MI) of the background state is considered to play a prominent role in the development of rogue waves in oceanic sea states, nonlinear optics, and plasmas [49].

The nonlinear Schrödinger (NLS) equation (when ε,γ=0 in Eq. 1) is one of the simplest models for studying phenomena related to MI; as such, special solutions of the NLS equation are regarded as prototypes of rogue waves. Among the more tractable “rogue wave” solutions of the NLS equation are the rational solutions (with the Peregrine breather being the lowest order) and the spatially periodic breathers (SPBs) which are constructed as heteroclinic orbits of modulationally unstable Stokes waves [1012]. In the case of the Stokes waves with N unstable modes (UMs), the associated SPBs can be of dimension MN and are referred to as M-mode SPBs; the single mode SPB is the Akhmediev breather [13]. For more realistic sea states with non-uniform backgrounds, heteroclinic orbits of unstable N-phase solutions have been used to describe rogue waves [1416].

For theoretical and practical purposes it is important to understand the stability of the SPBs with respect to small variations in initial data and small perturbations of the NLS equation. Using the squared eigenfunction connection between the Floquet spectrum of the NLS equation and the linear stability problem, the SPBs were shown to be typically unstable [17]. The effects of damping on deep water wave dynamics, even when weak, can be significant and in many instances must be included in models to enable accurate predictions of laboratory and field data [1822].

In a recent study the authors examined the stabilization of symmetric SPBs using the linear damped NLS equation (a near-integrable system that preserves even symmetry of solutions) [23]. The route to stability for these damped SPBs was determined by appealing to the Floquet spectral theory of the NLS equation. Degenerate complex elements of the periodic spectrum (referred to as complex double points) are associated with instabilities of the solution and may split under perturbation to the system. In the restricted subspace of even solutions complex double points can reform as time evolves. The damped solutions were found to be unstable as long as complex double points were present in the spectral decomposition of the data (either by persisting or reforming). A key issue in analyzing the route to stability is determining which complex double points in the spectrum of the SPB are split by damping. For an initial SPB with a given mode structure, perturbation analysis showed that only the complex double points associated with resonant modes split under damping while those associated with nonresonant modes remained closed [23].

The evolution of deep water waves is described only to leading order by the NLS equation. A more accurate description of the wave dynamics is provided by the Dysthe equation, obtained by extending the asymptotic analysis used to derived the NLS equation to fourth order. The Dysthe equation has been shown to accurately predict laboratory data for a wider range of wave parameters than the NLS equation [24, 25, 26]. Gramstad and Trulsen brought the Dysthe equation into Hamiltonian form obtaining a new higher order NLS (HONLS) equation (Eq. 1 with γ=0) [27]. Damped versions of the HONLS equation have successfully described ocean swell and frequency downshift of wave trains on deep water [28, 29].

In this paper we examine the competing effects of dissipation and higher order nonlinearities on the routes to stability of the N-mode SPBs in the framework of the linear damped HONLS equation over a spatially periodic domain:

iut+uxx+2|u|2u+iε(12uxxx8|u|2ux2iu[(|u|2)]x)+iγu=0(1)

where u(x,t) is the complex envelope of the wave train, {f(x)}=1πf(ξ)xξdξ is the Hilbert transform of f, and 0<ε,γ1. The initial data used in the numerical experiments is generated using exact SPB solutions of the integrable NLS equation. The SPBs are over Stokes waves with N unstable modes (referred to as the N-UM regime) for 1N3. We interpret the damped HONLS (near-integrable) dynamics by appealing to the NLS Floquet spectral theory.

The higher order nonlinearities in Eq. 1 break the even symmetry of both the initial data and the equation. This raises several interesting questions regarding the damped HONLS equation. Which integrable instabilities are excited by the damped HONLS flow and which elements of the Floquet spectrum are associated with these instabilities? What are the routes to stability under damping; i.e., what remnants of integrable NLS structures are detected in the damped HONLS evolution?

In the present study we observe the onset of novel instabilities as a result of symmetry breaking and the development of critical states in the damped HONLS flow which were nonexistent in the previously examined damped NLS system with even symmetry. Significantly, we determine these instabilities are associated with degenerate complex elements of both the periodic and continuous spectrum, i.e., with both complex “double points” and complex “critical points”, respectively. This association was not previously recognized. With regard to teminology, although double points are among the critical points of Δ, in this paper we exclusively call degenerate complex periodic spectrum where Δ=±2 “double points” and reserve the term “critical points” for degenerate complex spectrum where Δ±2.

The paper is organized as follows. In Section 2 we present elements of the NLS Floquet spectral theory which we use to distinguish instabilities in the numerical experiments. Whether higher phase solutions, such as the even 3-phase solution given in Eq. 8, are unstable with respect to general noneven perturbations and what the Floquet “signature” is of the possible instabilities, has been an open question. The closest stability results we are aware of are for the elliptic solutions of the focusing NLS equation [30]. We numerically show an even 3-phase solution of the NLS equation is unstable with respect to generic perturbations of initial data and find the relevant element of the Floquet spectrum associated with the instability in order to develop a broadened Floquet characterization of instabilities of the NLS equation.

A brief overview of the SPB solutions of the NLS is provided at the end of Section 2 before numerically examining their stabilization under the damped HONLS flow in Section 3. The Floquet decompositions of the numerical solutions are computed for 0t100. Complex double points are initially present in the spectrum. If one of the complex double points present initially splits due to the damped HONLS perturbation, the subsequent evolution involves repeated formation and splitting of complex critical points (not double points) which we correlate with the observed instabilities. The Floquet spectral analysis is complemented by an examination of the growth of small perturbations in the SPB initial data under the damped HONLS flow. We determine that the instabilities saturate and the solutions stabilize once all complex double points and complex critical points vanish in the spectral decomposition of the perturbed flow. Variations in the spectrum under the HONLS flow are correlated with deformations of certain NLS solutions to determine the routes to stability for the damped HONLS SPBs.

In Section 4, via perturbation analysis, we examine splitting of the complex double points, present in the SPB initial data, under the damped HONLS flow. We find that for short time, the complex double points associated with modes that resonate with the SPB structure split producing disjoint asymmetric bands, while the complex double points associated with nonresonant modes remain closed, substantiating the initial spectral evolutions observed in the numerical experiments. The nonresonant double points are observed to remain closed for the duration of the experiments, beyond the time-frame of the short time analysis, even though the solution evolves as a damped asymmetric multi-phase state. In this study resonances have a stabilizing effect; the instabilities of nonresonant modes persist on a longer time scale than the instabilities associated with resonant modes.

2 Analytical Framework

The nonlinear Schrödinger equation (when ε,γ=0 in Eq. 1) arises as the solvability condition of the Zakharov-Shabat (Z-S) pair of linear systems [31]:

(u)ϕ=λϕ,(u)=(ixuuix)(2)
ϕt=(2iλ2+i|u|22iλuux2iλu+ux2iλ2i|u|2)ϕ(3)

where λ is the spectral parameter, ϕ is a complex vector valued eigenfunction, and u(x,t) is a solution of the NLS equation itself. Associated with an Lperiodic NLS solution is it’s Floquet spectrum

σ(u):={λ|ϕ=λϕ,|ϕ| bounded x}(4)

Given a fundamental matrix solution of the Z-S system, Φ, one defines the Floquet discriminant as the trace of the transfer matrix across one period L, Δ(u,λ)=Trace[Φ(x+L,t;λ)Φ1(x,t;λ)]. The Floquet spectrum has an explicit representation in terms of the discriminant:

σ(u):={λ|Δ(u,λ),2Δ(u,λ)2}(5)

The Floquet discriminant Δ(λ) is analytic and is a conserved functional of the NLS equation. As such, the spectrum σ(u) of an NLS solution is invariant under the time evolution.

The spectrum consists of the entire real axis and curves or “bands of spectrum” in the complex λ plane ((u) is not self-adjoint). The periodic/antiperiodic points (abbreviated here as periodic points) of the Floquet spectrum are those at which Δ=±2. The endpoints of the bands of spectrum are given by the simple points of the periodic spectrum σs(u)={λjs|Δ(λj)=±2,Δ/λ0}. Located within the bands of spectrum are two important spectral elements:

1. Critical points of spectrum, λjc, determined by the condition Δ/λ=0.

2. Double points of periodic spectrum σd(u)={λjd|Δ(λjd)=±2,Δ/λ=0,2Δ/λ20}.

Double points are among the critical points of Δ. However, in this paper, we exclusively call the degenerate periodic spectrum where Δ=±2 “double points” and reserve the term “critical points” for degenerate elements of the spectrum where Δ±2.

The Floquet spectrum can be used to represent a solution in terms of a set of nonlinear modes where the structure and stability of the modes are determined by the band-gap structure of the spectrum. Simple periodic points are associated with stable active modes. The location of the double points is particularly important. Real double points correspond to zero amplitude inactive nonlinear modes. On the other hand, complex double points are associated with degenerate, potentially unstable, nonlinear modes with either positive or zero growth rate. When restricted to the subpace of even solutions, exponential instabilities of a solution are associated with complex double points in the spectrum [32].

A concrete example illustrating the correspondence between complex double points in the spectrum and linear instabilities is the Stokes wave solution ua(t)=aei(2a2t+ϕ). For small perturbations of the form u(x,t)=ua(t)[1+ε(x,t)], |ε|1, one finds ϵ satisfies the linearized NLS equation

iεt+εxx+2|a|2(ε+ε)=0(6)

Representing ϵ as a Fourier series with modes εjeiμjx+σjt, μj=2πj/L, gives σj2=μj2(4|a|2μj2). As a result, the jth mode is unstable if 0<(jπ/L)2<|a|2. The number of UMs is the largest integer M such that 0<M<|a|L/π.

The Floquet discriminant for the Stokes wave is Δ=2cos(a2+λ2L). The Floquet spectrum consists of continuous bands [ia,ia] and a discrete part containing λ0s=±i|a| and the infinte number of double points

(λjd)2=(jπL)2a2,j,j0(7)

as shown in Figure 1. Note that the condition for λjd to be complex is precisely the condition for the jth Fourier mode εj to be unstable. The remaining λjd for |j|>M are real double points.

FIGURE 1
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FIGURE 1. The Floquet spectrum of the Stokes wave with ⌊aL/π=2.

As the NLS spectrum is symmetric under complex conjugation, we subsequently only display the spectrum in the upper half λ-plane.

2.1 A Broadened Floquet Spectral Characterization of Instabilities

Earlier work on perturbations of the NLS equation dealt primarily with solutions with even symmetry whose instabilities were identified solely via complex double points [33, 34]. In general, imposing symmetry on a solution restricts it’s dynamical behavior and may suppress instabilities. In the current damped HONLS experiments we find instabilities arise due to the asymmetry of the system that are not captured by complex double points. Although complex double points, if present in the damped HONLS flow, still identify instabilities, we need to develop a broader Floquet spectral characterization of instabilities to capture the instabilities of generic solutions.

A clue as to which new spectral elements are associated with instabilities in the full solution space of the NLS equation is provided by considering generic perturbations of initial data for even solutions of the NLS equation. One of the simplest solutions to examine is the following even 3-phase solution of the NLS equation [13],

u0(x,t)=ae2ia2tκ1+κcn(axκ,1κ2)dn(a2tκ,κ)+iκsn(a2tκ,κ)2κ[1κ1+κcn(axκ,1κ2)cn(a2tκ,κ)](8)

With respect to t the solution has a double frequency; a frequency determined by the exponential function and a modulation frequency determined by the elliptic functions. Equation 8 describes an even standing wave, periodic in space and time, arising as the degeneration of a 3-phase solution due to symmetry in it’s spectrum. The spatial period L and temporal period T are functions of Kx(1κ2) and Kt(κ), respectively, where K is the complete elliptic integral of the first kind. As κ1 in Eq. 8, T and u0(x,t)U(1)(x,t), the SPB given in Eq. 10 associated with one complex double point.

The surface and Floquet spectrum for u0(x,t) are shown in Figures 2A,B. The spectrum forms an even “cross” state with two bands of spectrum in the upper half plane with endpoints given by the simple periodic spectrum λ0=0.5i and λ1±=0.35i±α. These two bands intersect transversally at λc on the imaginary axis. Since Δ/λ=0 at transverse intersections of bands of continuous spectrum, λc is a critical point. There are no complex double points in σ[u0(x,t)].

FIGURE 2
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FIGURE 2. NLS cross state: (A)|u0(x,t)| for 0t100, (B) it’s spectrum at t=0, (C)η(t) for uδ(x,0)=u0(x,0)+δg1(x).

To numerically address the stability of the cross state Eq. 8 with respect to initial data we consider small perturbations (both symmetric and asymmetric) of the following form:

uδ(x,0)=u0(x,0)+δgk(x),k=1,2

where g1(x)=eiϕcosμx, g2(x)=eiϕ1cosμx+reiϕ2sinμx, and δ=103,,5×103. We examine 1) the Floquet spectrum of uδ(x,t) as compared with u0(x,t) and 2) the growth of the perturbations as δ is varied. We consider a solution u(x,t) of the NLS equation to be stable if for every ε>0 there exists a δ>0 such that if uδ(x,0)u(x,0)H1<δ, then uδ(x,t)u(x,t)H1<ε, for all t. Therefore, to determine whether u and uδ stay close as time evolves, we monitor the evolution of the difference

η(t)=uδ(x,t)u(x,t)H1(9)

where fH12=L/2L/2(|fx|2+|f|2)dx.

Symmetric Perturbations of Initial Data

As ϕ and δ are varied, the surface and spectrum for uδ(x,t) for perturbation g1(x) are qualitatively the same as in Figures 2A,B. The endpoints of the band of spectrum, σs(uδ), are slightly shifted maintaining even symmetry. Due to analyticity of Δ, λc does not split under even perturbations and the spectrum is not topologically different. Figure 2C shows the evolution of η(t) for even perturbations uδ(x,0)=u0(x,0)+δg1(x). The small osciallations in η(t), typical in Hamiltonian systems, do not grow. The Floquet spectrum and the evolution of η(t) show that when restricted to the subspace of even solutions, u0(x,t) is stable.

Asymmetric Perturbations of Initial Data

The surface and spectrum of uδ(x,t) for uδ(x,0)=u0(x,0)+0.05sinμx are shown in Figures 3A,B. A topologically different spectral configuration is obtained and the waveform is a modulated right traveling wave. The critical point λc has split into λ±c and the two disjoint bands of spectrum form a “right” state: the upper band with endpoints λ0s and λ1,δ+ in the right quadrant and the lower band with endpoint λ1,δ extending to the real axis in the left quadrant. On the other hand, if e.g., uδ(x,0)=u0(x,0)+0.05(cosμx+eiπ/3sinμx), the waveform is a modulated left traveling wave. In this case the orientation of the bands of spectrum is reversed, forming a “left” state with the upper band in the left quadrant and the lower band in the right quadrant. As the parameters in g2(x) are varied these are the two possible noneven spectral configurations for uδ(x,t). The perturbation analysis in Section 4 is related and shows noneven perturbations of the SPB split complex double points into left and right states.

FIGURE 3
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FIGURE 3. NLS noneven 3-phase solution: (A)|uδ(x,t)| for 0t100, (B) it’s spectrum at t=0, (C)η(t) for uδ(x,0)=u0(x,0)+δg2(x).

Figure 3C shows η(t) grows to O(1) for asymmetric perturbations uδ(x,0)=u0(x,0)+δg2(x). Clearly uδ(x,t) does not remain close to u0(x,t) for these perturbations. We associate the transverse complex critical points with instabilities arising from symmetry breaking which are not excited when evenness is imposed. The exact nature of the instability associated with complex critical points is under investigation. The current damped HONLS experiments in Section 3 corroborate the significance of transverse critical points λc in identifying instabilities in the unrestricted solution space.

2.2 Spatially Periodic Breather Solutions of the NLS Equation

A variety of dressing methods can be used to derive new nontrivial solutions to integrable equations [see [35] for applications of the Darboux transformation to generate solutions of generalized NLS models]. Here we use the Bäcklund-gauge transformation (BT) for the NLS equation [36] to generate the heteroclinic orbits of a spatially periodic unstable NLS potential u(x,t) with complex double points, λjd, in its Floquet spectrum. Given a Stokes wave ua(t) with N complex double points, a single BT of ua(t) at λjd yields the one mode SPB, U(j)(x,t), associated with the jth UM, 1jN. Introducing μj=2πj/L, σj=2iμjλj, cospj=μj/2a, and τj=(ρσjt) one obtains [13, 33]:

U(j)(x,t)=ae2ia2t(isin2pjtanhτj+cos2pjsinpjcos(μjx+β)sechτj1+sinpjcos(μjx+β) sech τj)(10)

U(j)(x,t) exponentially approaches a phase shift of the Stokes wave, limt±U(j)(x,t)=ae2ia2t+α±, at a rate depending on λjd. Figures 4A,B show the amplitudes of two distinct single mode SPBs, U(1)(x,t) and U(2)(x,t) over a Stokes waves with N=2 UMs. U(1)(x,t) and U(2)(x,t) are both unstable as the BT based at λj saturates the instability of the jth UM while the other instabilities of the background persist.

FIGURE 4
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FIGURE 4. SPBs over a Stokes wave with N=2 UMs, a=0.5, L=42π: Amplitudes of (A–B) single mode SPBs |U(1)(x,t,ρ)| and |U(2)(x,t,ρ)| (ρ=β=0) and (C) 2-mode SPB |U(1,2)(ρ,τ)|.

When the Stokes wave posesses two or more UMs, the BT can be iterated to obtain multi mode SPBs. For example, the two-mode SPB with wavenumbers μi and μj, obtained by applying the BT successively at complex λid and λjd, is of the following form:

U(i,j)(x,t;ρ,τ)=ae2ia2tN(x,t;ρ,τ)D(x,t;ρ,τ)(11)

The exact formula is provided in [11]. The parameters ρ and τ determine the time at which the first and second modes become excited, respectively. Figure 4C shows the amplitude of the “coalesced” two mode SPB U(1,2)(x,t;ρ,τ) (ρ=2, τ=3) over a Stokes waves with N=2 UMs where the two modes are excited simultaneously.

An important property of the Bäcklund transformation: The BT is isospectral, i.e., σ[ua(t)]=σ[U(j)(x,t)]=σ[U(i,j)(x,t)]. For example, the Stokes wave with N=2 UMs (given in Figure 1) and each of the SPBs shown in Figure 4A–C share the same Floquet spectrum.

3 Numerical Investigation of Routes to Stability OF SPBS IN the Damped Higher Order NLS Equation

In our examination of the even 3-phase solution in Section 2 we found that when evenness is relaxed novel instabilities arise which are associated with complex critical points. Armed with this result, in this section we return to the questions posed at the outset of our study: 1) Which integrable instabilities are excited by the damped HONLS flow and what is the Floquet criteria for their saturation? 2) What remnants of integrable NLS structures are detected in the damped HONLS evolution?

The notation used in this section is as follows: 1) The “N UM regime” refers to the range of parameters a and L for which the underlying Stokes wave initially has N unstable modes. 2) The initial data used in the numerical experiments is generated using exact SPB solutions of the integrable NLS equation. The perturbed SPBs are indicated with subscripts: Uε,γ(j)(x,t) refers to the solution of the damped HONLS Eq. 1 for one-mode SPB initial data U(j)(x,0). Likewise Uε,γ(i,j)(x,t) is the solution to Eq. 1 for iterated SPB initial data.

The damped HONLS equation is solved numerically using a high-order spectral method due to Trefethen [37]. The integrator uses a Fourier-mode decomposition in space with a fourth-order Runge-Kutta discretization in time. The number of Fourier modes and the time step used depends on the complexity of the solution. For example, for initial data in the three UM regime, N=1024 Fourier modes are used with time step Δt=7.5×105. As a benchmark the first three global invariants of the HONLS equation, the energy E=0L|u|2dx, momentum P=i0L(uuxuux)dx, and Hamiltonian

H=0L{i|ux|2+i|u|4ε4(uxuxxuxuxx)+2ε|u|2(uuxuux)+iε|u|2[(|u|2)]x}dx

are preserved with an accuracy of O(1012) for 0t100. The invariant for the damped HONLS system, the spectral center km=P/2E, is preserved with an accuracy of at least O(1012) in the experiments.

Nonlinear mode decomposition of the damped HONLS flow: At each time t we compute the spectral decomposition of the damped HONLS data using the numerical procedure developed by Overman et. al. [34]. After solving system Eq. (2), the discriminant Δ is constructed. The zeros of Δ±2 are determined using a root solver based on Müller’s method and then the curves of spectrum filled in. The spectrum is calculated with an accuracy of O(106) which is sufficient given the perturbation parameters ϵ and γ used in the numerical experiments are O(102).

Notation used in the spectral plots: The periodic spectrum is indicated with a large × when Δ=2 and a large box when Δ=2. The continuous spectrum is indicated with small × when the Δ is negative and a small box when Δ is positive.

Interpreting the damped HONLS flow via the NLS spectral theory: A tractable example which illustrates the use of the Floquet spectrum to interpret near integrable dynamics is the spatially uniform solution (there is no depenence on ϵ) of the damped HONLS Eq. 1,

ua,γ(t)=aeγtei(|a|2(1e2γt)γ)(12)

At a given time t=t the nonlinear spectral decomposition of Eq. 12 can be explicitly determined by substituting ua,γ(t) into (u)ϕ=λϕ. We find the periodic Floquet spectrum consists of λ0s=±iaeγt and infinitely many double points

(λjd)2=(jπL)2e2γta2,j,j0(13)

where λjd is complex if (jπL)2<e2γta2. Under the damped HONLS evolution the endpoint of the band of spectrum λ0s and the complex double points λjd move down the imaginary axis and then onto the real axis. Similarly, at t=t, a linearized stability analysis about ua,γ(t) shows the growth rate of the jth mode σj2=μj2(4e2γta2μj2). Thus the number of complex double points and the number of unstable modes diminishes in time due to damping. As a result, ua,γ(t) stabilizes when the growth rate σ1=0, i.e., when λ1d=0, giving t=ln(aL/π)/γ.

Saturation time of the instabilities: Since the association of complex critical points with instabilities is a new result, we supplement the spectral analysis with an examination of the saturation time of the instabilities for the damped SPBs Uε,γ(j)(x,t) and Uε,γ(j,k)(x,t) as follows: we examine the growth of small asymmetric perturbations in the initial data of the following form,

Uε,γ,δ(j)(x,0)=U(j)(x,0)+δfk(x)

and

Uε,γ,δ(i,j)(x,0)=U(i,j)(x,0)+δfk(x)

where.

i.fk(x)=cosμkx+rkeiϕksinμkx, μk=2πk/L, 1k3,

ii.f4(x)=r(x), r(x)[0,1] is random noise.

To determine the closeness of Uε,γ(j)(x,t) and Uε,γ,δ(j)(x,t) as time evolves we monitor the evolution of η(t), as given by Eq. 9. We consider the solution to have stabilized under the damped HONLS flow once η(t) saturates.

In the damped HONLS numerical experiments we obtain a new criteria for the saturation of instabilities: η(t) saturates and the SPB stabilizes once damping eliminates all complex double points and complex critical points in the spectrum.

3.1 Damped HONLS SPB in the One Unstable Mode Regime

Uε,γ(1)(x,t) in the one UM regime

We begin by considering Uϵ,γ(1)(x,t) for ε=0.05 and γ=0.01 in the one UM regime with initial data generated using Eq. 10 with j=1, a=0.5 and L=22π. Figure 5A shows the surface |Uε,γ(1)(x,t)| for 0<t<100.

FIGURE 5
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FIGURE 5. One UM regime: (A)|Uε,γ(1)(x,t)| for 0t100 and the spectrum at (B)t=0, (C)t=6, (D)t=10.43, (E)t=24.1, (F)t=70 and (G)η(t) for f4(x,t), δ=105,,104 and ε=0.05, γ=0.01.

The evolution of the Floquet spectrum for Uε,γ(1)(x,t) is as follows: At t=0, the spectrum in the upper half plane consists of a single band of spectrum with end point at λ0s=0.5i, indicated by a large “box”, and one imaginary double point at λ1d=0.3535i, indicated by a large “×” (Figure 5B). Under the damped HONLS λ1d splits asymmetrically forming a right state, with the upper band of spectrum in the right quadrant and the lower band in the left quadrant, consistent with the short time perturbation analysis in Section 4. The right state is clearly visible at t=6 in Figure 5C with the waveform characterized by a single damped modulated mode traveling to the right. The spectrum persists in a right state with the separation distance between the two bands varying until t=10.43, Figure 5D, when a cross state forms with an embedded critical point indicating an instability. Subsequently the critical point splits into a right state. As damping continues the band in the right quadrant widens, Figure 5E, and the vertex of the loop eventually touches the origin at t27.5. The spectrum then has three bands emanating off the real axis, with endpoints λ1,λ0s,λ1+ which, as damping continues, diminish in amplitude and move away from the imaginary axis, Figure 5F.

Figure 5G shows the evolution of η(t) for Uε,γ(1)(x,t) using f4 and δ=105,,104. The saturation of η(t) at t16 is consistent with the Floquet criteria that the solution stabilizes after complex double points and complex critical points are eliminated in the damped HONLS flow.

From the spectral analysis of Uε,γ(1)(x,t) in the one UM regime, we find it may be characterized as a continuous deformation of a noneven generalization of the 3-phase solution, the right state, given by Eq. 8. The amplitude of the oscillations of Uε,γ(1)(x,t) decreases and the frequency increases until small fast oscillations about the damped Stokes wave, visible in Figure 5A, are obtained.

An alternate approach to studying the effect of small damping (or gain) on the one mode SPB is to consider the evolution of asymmetric initial data in the neighborhood of the SPB under the linearly damped NLS equation. Using the finite gap method of the periodic NLS equation [38], the solution is analytically approximated to leading order with a sum of SPBs shifted in space and time. The quantitative agreement between the leading order analytical formula and the corresponding numerical experiments was found to be good [21]. It is interesting to note that, although we use exact SPB initial data under the linearly damped HONLS equation, with different damping values, the asymmetric evolution of the Floquet spectral data is qualitatively consistent with the numerical experiments described in [21].

3.2 Damped HONLS SPBs in the Two Unstable Mode Regime

For the two UM regime we let a=0.5, L=42π and consider the two distinct perturbed single mode SPBs Uε,γ(1)(x,t) and Uε,γ(2)(x,t) and the perturbed iterated SPB Uε,γ(1,2)(x,t). The damped HONLS perturbation parameters are ε=0.05 and γ=0.01.

Uε,γ(1)(x,t) in the two UM regime

Figure 6A shows the surface |Uε,γ(1)(x,t)| for 0<t<100 for initial data given by Eq. 10 with j=1. The Floquet spectrum at t=0 is given in Figure 6B. The end point of the band of spectrum at λ0s=0.5i is indicated by a “box”. There are two complex double points at λ1d=0.4677i and λ2d=0.3535i, indicated by a “×” and “box”, respectively. For t>0 both double points split asymmetrically: λ1d, the complex double point at which U(1)(x,t) is constructed, splits at leading order into λ1± such that a right state forms with the first mode traveling to the right. The second double point λ2d splits at higher order into λ2± such that a left state forms with the second mode traveling to the left. These disjoint asymmetric bands of spectrum are consistent with the short time perturbation analysis in Section 4 for damped HONLS data of the form Eq. 18.

FIGURE 6
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FIGURE 6. Two UM regime: (A)|Uε,γ(1)(x,t)| for 0t100, and spectrum at (B)t=0, (C)t=11.5, (D)t=11.94, (E)t=12.5, (F)t=19.26, (G)t=19.32, (H)t=19.37, (I)t=42.4, and J)η(t) for f1(x,t), δ=105,,104.

This spectral configuration is representative of the spectrum during the initial stage of it’s evolution and is still observable in Figure 6C at t=11.5. A sequence of bifurcations occur at t11.94, Figure 6D, when two complex critical points emerge in rapid succession in the spectrum, indicating instabilities associated with both nonlinear modes. Subsequently both complex critical points split, Figure 6E, with the upper band in the left quadrant and the second band in the right quadrant corresponding to a damped waveform with the first mode traveling to the left and the second mode traveling to the right.

The bifurcation at t=11.94 corresponds to transitioning through the remnant of an unstable 5-phase solution (with two instabilities) of the NLS equation. The bands eventually become completely detached from the imaginary axis and a second complex critical point forms at t=19.32. The bifurcation sequence is shown in Figure 6F–H. The main band emanating from the real axis then reestablishes itself close to the imaginary axis, Figure 6I, and the spectrum settles into a configuration corresponding to a stable 5 phase solution. The bands move apart and downwards and hit the real axis with no further development of complex critical points. From the Floquet spectral perspective, once damping eliminates complex critical points in the spectrum at approximately t20, Uε,γ(1)(x,t) stabilizes.

Figure 6J shows the evolution of η(t) for Uε,γ(1)(x,t) with f2 for δ=105,,104. The perturbation f2 is chosen in the direction of the unstable mode associated with λ2d. η(t) stops growing by t20, confirming the instabilities associated with the complex critical ponts and time of stabilization obtained from the nonlinear spectral analysis.

For Uε,γ(1)(x,t) in the 2-UM regime, both λ1d and λ2d resonate with the perturbation. The route to stability is characterized by the appearance of the double cross state of the NLS and the proximity to this state is significant in organazing the damped HONLS dynamics. Once stabilized, Uε,γ(1)(x,t) may be characterized as a continuous deformation of a stable 5-phase solution.

Uε,γ(2)(x,t) in the two UM regime

We now consider Uε,γ(2)(x,t) whose initial data is given by Eq. 10 with j=2. Although U(1)(x,t) and U(2)(x,t) are both single mode SPBs over the same Stokes wave, their respective routes to stability under damping are quite different. Notice in Figure 7A the surface of |Uε,γ(2)(x,t)| for 0t100 is a damped modulated traveling state, exhibiting regular behavior, in contrast to the irregular behavior of |Uε,γ(1)(x,t)| in the two UM regime.

FIGURE 7
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FIGURE 7. Two UM regime: (A)|Uε,γ(2)(x,t)| for 0t100 and Spectrum at (B)t=6, (C)t=11, (D)t=23.4, (E)t=27.4, (F)t=71.9, and (G)η(t) for f1(x,t), δ=105,,104 and γ=0.01.

The spectrum of Uε,γ(2) at t=0 is the same as in Figure 6B. Under perturbation λ2d immediately splits asymmetrically into λ2± with the upper band in the right quadrant and the lower band in the left quadrant, while the first double point λ1d (indicated by the large “×”) does not split. Figure 7B clearly shows that at t=6 damping has only split λ2d, i.e., the double point at which the SPB U(2) is constructed. In fact λ1d does not split for the duration of the damped HONLS evolution, 0t100. In Figure 7C, by t=11, the two bands have aligned forming a cross state with a complex critical point at the transverse intersection of the bands while λ1d is still intact and has simply translated down the imaginary axis.

The complex critical point subsequently splits with the upper band of spectrum again in the right quadrant, Figure 7D. Complex critical points do not reappear in the spectrum. At t=27.4 the vertex of the upper band of spectrum touches the real axis Figure 7E. As damping continues λ1d moves down the imaginary axis and the two bands move away from the imaginary axis with diminishing amplitude. In Figure 7F the complex double point λ1d has moved almost all the way down the imaginary axis. At t72, λ1d=0 and complex double points do not arise in the subsequent spectral evolution.

We find η(t) grows until t=ts80, Figure 7G, consistent with the expectation that Uε,γ(2) will stabilize once all complex critical points and complex double points vanish in the spectrum. Until λ1d moves onto the real axis, perturbations to the initial data Uε,γ,δ(2) can excite the first mode associated with λ1d causing Uε,γ(2) and Uε,γ,δ(2) to grow apart.

Why doesn’t the HONLS perturbation split λ1d when given U(2)(x,0) initial data? In Section 4, for short time, a suitable linearization of the damped HONLS SPB data is found to be given by Eq. 19 for j=1,2, respectively where ε˜=ε˜(ε,γ). The perturbation analysis shows that at leading order damping asymmetrically splits only the double point λ2d associated with U(2)(x,0). The endpoint of spectrum, λ0s, decreases in amplitude and the rest of the double points simply move along curves of continuous spectrum without splitting. The resonant modes which correspond to λ2md split asymmetrically at higher order O(ε˜m). The splitting of λld, l2m, is zero and is termed “closed”.

The spectral evolution for Uε,γ(2)(x,t) in the 2 UM regime is reminiscent of the spectral evolution of Uε,γ(1)(x,t) in the one UM regime. There is an important difference though: The nearby cross state that appears in the spectral decomposition of Uε,γ(2)(x,t) has two different types of instabilites: the instability associated with the complex critical point (potentially a phase instability) and the exponential instability associated with the (nonresonant) complex double point λ1d. The numerical results suggest that when nonresonant modes are present in the damped HONLS, their instabilities persist and organize the dynamics on a longer time scale. As λ1d remains closed for t<ts, Uε,γ(2)(x,t) can be characterized as a continuous deformation of a noneven generalization of the degenerate 3 phase solution given by Eq. 8 (the parameter values change due to doubling the period L; i,e., a jth mode excitation with period L becomes a 2jth mode excitation with period 2L.)

Uε,γ(1,2)(x,t) in the two UM regime

Figure 8A shows the surface |Uε,γ(1,2)(x,t)| for 0<t<100 for initial data obtained from Eq. 11 by setting ρ=0 and τ=2, a=0.5. The spectrum of Uε,γ(1,2) at t=0 is given by Figure 6B. As we’ve seen in the previous examples, once complex double points split they do not reform in the perturbed system; if they are present in the spectral decomposition of the damped HONLS, it is because the modes corresponding to λjd didn’t resonate under perturbation. Here both double points λ1d and λ2d split at leading order under the damped HONLS perturbation. Figure 8B shows for short time (t=1.5) the spectrum has a band gap structure indicating the first mode travels to the right and the second travels to the left. As time evolves the bands shift and align and Figure 8C shows at t15.9 two bands in the right quadrant intersect with an embedded complex critical point. The critical point splits with one band moving back toward the imaginary axis. There are now two bands detached from the primary band and evolving towards the real axis, Figures 8D,E. Complex critical points do not appear for t>15.9.

FIGURE 8
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FIGURE 8. Two UM regime: (A)|Uε,γ(1,2)(x,t)| for 0t100 and spectrum at (B)t=1.5, (C)t=15.84, (D)t=17.4, (E)t=57.5, and η(t) for Uε,γ(1,2)(x,t) with f1(x,t) when (F)ρ=0,τ=2 (the uncoalesced SPB), (G)ρ=0.665,τ=1 (the coalesced SPB).

Figure 8F shows η(t) for the example under consideration (ρ=0,τ=2) when f1(x,t),δ=105,,104. We find η(t) saturates at ts20 indicating Uε,γ(1,2) has stabilized. For t>tsUε,γ(1,2) is characterized as a continuous deformation of a stable NLS five-phase solution. The numerically observed initial splitting of complex double points λ1d and λ2d is consistent with the perturbation analysis of damped HONLS SPB data (19).

Among the two mode SPBs in the 2 UM regime, the one of highest amplitude due to coalescence of the modes appeared to be more robust [17]. An interesting observation is obtained if we examine the evolution of spectrum and of η(t) using the initial data for the special coalesced two mode SPB, generated from Eq. 11 by setting ρ=0.665, τ=1, and a=0.5. For the coalesced case, complex critical points form 4 times for 0<t<45 in the damped HONLS system. Figure 8G shows η(t) saturates for t50. A comparison with the results of the non-coalesced two mode SPB given above indicate that remnants of the coalesced U(1,2) and it’s instabilities influence the damped HONLS dynamics over a longer time period, suggesting enhanced robustness with respect to perturbations of the NLS equation.

3.3 Damped SPBs in the Three Unstable Mode Regime

The parameters used for the three UM regime are a=0.7 and L=42π. The damped HONLS perturbation parameters are ε=0.05 and γ=0.01. We present the results of two damped HONLS SPBs, Uε,γ(2)(x,t) and Uε,γ(2,3)(x,t), which exhibit an interesting or new feature. The evolutions of the other damped HONLS SPBs in the three UM regime in the three UM regime are discussed in relation to these cases.

Uε,γ(2)(x,t) in the three UM regime

The surface |Uε,γ(2)(x,t)| for initial data given by Eq. 10 with j=2 is shown in Figure 9A for 0<t<100. Notice in the 3 UM regime Uε,γ(2)(x,t) exhibits regular behavior and is a damped modulated right traveling wave as was Uε,γ(2)(x,t) in the 2 UM regime. The spectrum at t=0 is given in Figure 9B. The end point of the band of spectrum, λ0s=0.7i is indicated by a “box”. There are three complex double points at λ1d=0.677i, λ2d=0.604i, and λ3d=0.456i indicated by an “×”, “box” and “×”, respectively. Under the damped HONLS perturbation the complex double point at which Uε,γ(2)(x,t) is constructed, λ2d, splits into a right state as shown in Figure 9C. The complex double points λ1d and λ3d remain closed; λ1d lies on the upper band in the right quadrant and λ3d lies on the lower band. Transverse cross states with embedded complex critical points form frequently in the spectrum until t68, e.g., a cross state is shown at t=36.6 in Figure 9D. Figures 9E,F show the complex double points persist on the bands of spectrum until damping sufficiently diminishes the amplitude of the background and the complex double points reach the real axis at t85. In Figure 9Gη(t) saturates at t=ts90. Due to the presence of the complex double points for t<ts, Uε,γ(2)(x,t) can be viewed as a continuous deformation of an unstable 3 phase solution (with two instabilities). As discussed previously, the instabilities associated with the nonresonant modes persist longer than for the resonant modes.

FIGURE 9
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FIGURE 9. Three UM regime: (A)|Uε,γ(2)(x,t)| for 0t100 and Spectrum at (B)t=0, (C)t=4.7, (D)t=36.6, (E)t=42.8, (F)t=85 and (G)η(t) for f1(x,t), δ=106,,105.

In the 3 UM regime the behavior of the SPB Uε,γ(3)(x,t) is similar to Uε,γ(2)(x,t). In this case λ3d initially splits asymmetrically into the right state (which we’ve now seen frequently in the initial damped HONLS system when only one mode is activated). The double points λ1d and λ2d do not split, they move along the band of spectrum created by λ0s and λ3+. As a result Uε,γ(3)(x,t) stabilizes only when λ1d and λ2d become real, at t140. As in the previous cases, it is striking that the prediction from a short time perturbation analysis that certain double points remain closed, holds for the duration of the experiments (even while the solution evolves as a perturbed degenerate 3-phase state (with two instabilities). In contrast, for Uε,γ(1)(x,t) the higher order nonlinearities and damping excite all the modes. The solution is characterized by the formation of complex critical points and irregular behavior before stabilizing at t40.

Uε,γ(2,3)(x,t) in the three UM regime

Figure 10A shows the surface |Uε,γ(2,3)(x,t)| for 0<t<100 for initial data given by Eq. 11 with i,j=2,3.

FIGURE 10
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FIGURE 10. Three UM regime: (A)|Uε,γ(2,3)(x,t)| and Spectrum at (B)t=10.5, (C)t=14.78, (D)t=24.96, (E)t=68.38, (F)t=69.9, and (G)η(t) for f1(x,t), δ=105,,104 and γ=0.01.

The spectrum at t=0 is as in Figure 9B. The perturbation initially splits the double points λ2d and λ3d into λ2± and λ3± that correspond to a left and right modulated traveling modes, respectively. The new feature here is that the first complex double point, λ1d, splits at higher order into λ1± (see the analysis in Section 4 showing that a multi-mode perturbation in a higher UM regime introduces new resonances not seen with single mode perturbations). The higher order splitting is visible in Figure 10B at t=10.5.

Complex double points are not observed in the spectral evolution for t>0.The formation of complex critical points in the spectrum occurs frequently as shown, for example, in Figure 10C and Figure 10D. Since the amplitude of the background state at a=0.7 is initially very close to the 4 UM regime, in this example we observe that nearby real double points are noticeably split by the perturbation. Figure 10E shows the spectrum at t=68.4 when the last complex critical point forms. This is reflected in Figure 10G which shows η(t) saturates at ts68. Each of the bursts of growth in η(t) can be correlated with a complex critical point crossing. As time evolves disspiation diminshes the strength of the instability captured by the complex critical points or complex double points. Uε,γ(2,3)(x,t) exhibits quite rich and compex dynamics before damping saturates the instabilities and it’s behavior is not easy to characterize as when dealing with the perturbed SPBs in the N=1,2 UM regimes. For t>ts the evolution of Uε,γ(2,3)(x,t) may be characterized as a continuous deformation of a stable 7-phase solution (Figure 10F).

As a comparison, Uε,γ(1,2)(x,t) and Uε,γ(1,3)(x,t) exhibit shorter term irregular behavior with all the dominant modes excited and they stabilize at t15,18. respectively. Uε,γ(2,3)(x,t) was observed to take longer to stabilize due to the higher order splitting in λ1d.

The exact nature of the instability associated with complex critical points in under investigation. They may be weaker than the exponential instabilities associated with complex double points but the evolution of Uε,γ(2,3)(x,t) illustrates their cumulative impact can be significant.

4 Perturbation Analysis

While examining the route to stability of the SPBs under the damped HONLS several novel results arose. One feature was that the instabilities of nonresonant modes persist longer than the instabilities of the resonant modes. We are interested in the fate of complex double points under noneven perturbations induced by HONLS as they characterize the SPB. Following the perturbation analysis in [39] used to determine the O(ε) splitting of double points for single mode perturbations, we carry the analysis to higher order for noneven multi mode perturbations of the SPBs. We find 1) additional modes resonate with the perturbation and 2) complex double points associated with nonresonant modes remain closed.

To obtain linearized initial conditions for the one and two mode SPBs we use the Hirota formulation of the SPBs [40]. For example for the one mode SPB one obtains,

u(j)(x,t)=ae2ia2t1+2e2iθj+Ωjt+γcosμjx+A12e2(2iθj+Ωjt+γ)1+2eΩjt+γcosμjx+A12e2(Ωjt+γ)(14)

where μj=2πj/L, Ωj=μj4a2μj2, sinθj=μj/2a, A12=sec2θj, and γ is an arbitrary phase.

The appropriate linearized initial conditions for the one and two mode SPBs, u(i)(x,0) and u(i,j)(x,0) respectively, are obtained by choosing t and γ such that ε˜s=4isinθseΩst+γ, s=i,j, are small. After neglecting second-order terms we obtain:

u(j)(x,0)=a(1+ε˜jeiθjcosμjx)(15)
u(i,j)(x,0)=a(1+ε˜ieiθicosμix+ε˜jeiθjcosμjx)(16)

The damped HONLS yields the following noneven first order approximation for small time,

u(j)(x,h)=a[1+ε˜j(eiθjcosμjx+rjeiϕjsinμjx)](17)
u(i,j)(x,h)=a[1+ε˜i(eiθicosμix+rieiϕisinμix)+ε˜j(eiθjcosμjx+rjeiϕjsinμjx)](18)

where θsϕs and a,ε˜s,rs are functions of h and the damped HONLS parameters ϵ and γ, for s=i,j. For simplicity we set ε=ε˜s and suppress their explicit dependence on ε,γ:

u=a+ε[eiθicosμix+rieiϕisinμix+Q(eiθjcosμjx+rjeiϕjsinμjx)]=a+εu(1)(19)

where rs0 and Q can be 0 or 1, depending on whether a one or two mode SPB is under consideration.

Since Δ(λ,u) and the eigenfunctions vn=[vn1vn2] are analytic functions of their arguments, at the double points λn we assume the following expansions:

vn=vn(0)+εvn(1)+ε2vn(2)+(20)
λn=λn(0)+ελn(1)+ε2λn(2)+(21)

Substituting these expansions into Eq. 25 we obtain the following:

O(ε0):vn(0)=0(22)
O(ε1):vn(1)=[iλn(1)vn1(0)+u(1)vn2(0)iλn(1)vn2(0)+u(1)vn1(0)]F(23)
O(ε2):vn(2)=[iλn(1)vn1(1)iλn(2)vn1(0)+u(1)vn2(1)iλn(1)vn2(1)iλn(2)vn2(0)+u(1)vn1(1)]G(24)

where

=[/x+iλn(0)aa/x+iλn(0)](25)

The leading order Eq. 22 provides the spectrum for the Stokes wave. At the double points λn(0) the two dimensional eigenspace is spanned by the eigenfunctions

ϕn±=e±iknx[1ia(±kn+λn)](26)

where (λn(0))2=kn2a2,kn=nπ/L, and the general solution is given by

vn(0)=A+ϕn++Aϕn

4.1 First Order Results

For periodic v, the solvability condition for the system v=F=[F1F2] is given by the orthogonality condition

0L(F1w1+F2w2)=0

for all w in the nullspace of the Hermitian operator

H=[/xiλnaa/xiλn]

At the double points the nullspace of H is spanned by the eigenfunctions [ϕn2±ϕn1±] and the orthogonality condition becomes

0L(F1ϕn2±+F2ϕn1±)dx=0(27)

Applying this orthogonality condition to Eq. 23 yields the system of equations

[T+TTT][A+A]=0

where

T=2λn(0)λn(1)/a(28)
T±=12{(±kn+λna)2(eiθn±irneiϕn)(eiθn±irneiϕn)n=i,j0ni,j(29)

Non trivial solutions A± are obtained only at the complex double points λn, n=i,j at which the SPB was constructed providing the first order correction

(λn(1))2={a24λn2[sin(ωn+θn)sin(ωnθn)+rn2sin(ωn+ϕn)sin(ωnϕn)n=i,j+irnsin(ϕnθn)sin2ωn]0ni,j(30)

where tanωn=Im(λn(0))/kn and θsϕs±nπ for s=i,j. As a result λn(1)=±r1/2eip/2 where 0<p<2π and the double point splits asymmetrically in any direction. Examining Δ in a neighborhood of u(0) we find that when u(1) resonates with a particular mode, the band of continuous spectrum along the imaginary axis splits asymmetrically into two disjoint bands in the upper half plane. The other double points do not experience an O(ε) correction.

The spectral configuration is determind by the location of λn(±)=λ(0)+ελ(1). λn+ determines the speed and direction of the associated phase. For example, in the one complex double point regime there are only two spectral configurations associated with noneven perturbation: 1) For 0<p<π, Re λ+>0 and the upper band of spctrum lies in the first quadrant. The wave form is characterized by a single modulated mode traveling to the right. 2) For π<p<2π, Re λ+<0, the upper band of spectrum is in the second quadrant, and the wave form is characterized by a single modulated mode traveling to the left.

As observed in the damped HONLS numerical experiments, the spectrum evolves between two distinct configurations when the continuous spectrum develops a complex critical point (not double point) due to the formation of transverse bands which then splits.

4.2 Second Order Results

Determining the O(ε2) corrections to the double points λn for ni,j, requires determining the eigenfunctions at O(ε). When λn(1)=0 the right hand side of Eq. 23 simplifies to

vn(1)==[0u(1)u(1)0]vn(0)=[u(1)(A+ϕn2++Aϕn2)u(1)(A+ϕn1++Aϕn1)]

where ϕn± is given by Eq. 26. We assume vn(1)=vn(0)+n(1) where

n(1)=Aiei(kn+μi)x+Biei(kn+μi)x+Ciei(knμi)x+Diei(kn+μi)x+Ajei(kn+μj)x+Bjei(knμj)x+Cjei(knμj)x+Djei(kn+μj)x(31)

Substituting n(1) into Eq. 23 we find the coefficient vectors to be (with ns, s=i,j)

As=A+/2μs2+2knμs[1a[2(cosθssinϕs)a2+(eiθs+ieiϕs)(kn+λn)μs]i[2(cosθssinϕs)(kn+λn)+(eiθsieiϕs)μs]]
Bs=A+/2μs22knμs[1a[2(cosθssinϕs)a2(eiθs+ieiϕs)(kn+λn)μs]i[2(cosθssinϕs)(kn+λn)(eiθsieiϕs)μs]]
Cs=A/2μs22knμs[1a[2(cosθssinϕs)a2+(eiθs+ieiϕs)(kn+λn)μs]i[2(cosθssinϕs)(kn+λn)+(eiθsieiϕs)μs]]
Ds=A/2μs2+2knμs[1a[2(cosθssinϕs)a2(eiθs+ieiϕs)(kn+λn)μs]i[2(cosθssinϕs)(kn+λn)(eiθsieiϕs)μs]]

With vn(1) in hand, applying the orthogonality condition to Eq. 24 yields the system

[αn+λn(2)βnλn(2)βnαn][A+A]=0(32)

giving an O(ε2) correction of the form

(λn(2)βn)2={αn+αnn=2i,2j,i+j,ji0for all other cases(33)

Consequently only the double points λn(0) with n=2i,2j,i+j, or ji experience an O(ε2) splitting. All other double points experience an O(ε2) translation. This calculation can be carried to higher order O(εm). In the simpler case of a damped single mode SPB, Uε,γ(j)(x,t), only λn(0) corresponding to the resonant mode n=mj will split at order O(εm) whereas the splitting is zero for λn(0), nmj [41].

For the two mode damped SPB Uε,γ(2,3) in the 3 UM regime we find λ2d and λ3d will split at O(ε). The mode associated with λ1(0) resonates also with u(1) at O(ε2). All 3 complex double points split, in contrast with the one mode Uε,γ(2) where λ1(0) and λ3(0) do not split.

5 Conclusions

In this paper we investigated the route to stability for even N-mode SPB solutions of the NLS equation in the framework of a damped HONLS equation using the Floquet spectral theory of the NLS equation. We found novel instabilities emerging in the symmetry broken solution space of the damped HONLS which are not captured by complex double points in the Floquet spectrum. We developed a broadened Floquet characterization of instabilities by examining the stability of an even 3-phase solution of the NLS equation with respect to noneven perturbations. We found the transverse complex critical point in its spectrum is associated with an instability which is not excited when evenness is imposed.

The association of instabilities excited by symmetry breaking with complex critical points of the Floquet spectrum was corroborated by the numerical experiments. If one of the complex double points present at t=0 splits in the damped HONLS system, the subsequent spectral evolution involves repeated formation and splitting of complex critical points (not double points) which we correlated with the observed instabilities.

In the numerical study we presented experiments using fixed values of the perturbation parameters ϵ and γ. As these parameters are varied fewer or more critical points may form and the time the damped HONLS solution stabilizes may vary but the following interesting results are independent of their specific value: 2) Instabilities excited by symmetry breaking are associated with complex critical points. 2) Solutions stabilize once damping eliminates all the complex critical points and complex double points in the spectral deomposition of the damped HONLS data. 3) Only certain modes resonate with the damped HONLS perturbation. Resonant modes aid in stabilizing the solution. If nonresonant modes are present, their instabilities persist and appear to organize the dynamics on a longer timescale.

Each burst of growth in η(t) can be correlated with the emergence of a complex critical point. The numerics suggest the instabilities associated with complex critical points may be weaker than those associated with complex double points. Even so the exact nature of the instability warrants further investigation. As demonstrated by the evolution of Uε,γ(2,3)(x,t) their cumulative impact can be significant.

Via a short time perturbation analysis we find that resonant complex double points split producing disjoint asymmetric bands, while the nonresonant complex double points remain closed as they move along the bands of spectrum, corroborating the initial spectral evolutions observed in the numerical experiments. Further, the nonresonant double points remain closed for the duration of the experiments, beyond the time-frame of the short time analysis, even though the solution evolves as a damped asymmetric multi-phase state.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author Contributions

CS was responsible for the theoretical framework, calculations, and the writing of the manuscript. CS and AI performed the numerical simulations. Both authors approve the manuscript.

Funding

This work was partially supported by Simons Foundation, Grant No. #527565.

Conflict of Interest

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

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Keywords: spatially periodic breathers, rogue waves, modulational instability, higher order nonlinear Schrödinger, Floquet spectrum

Citation: Schober CM and Islas AL (2021) On the Stabilization of Breather-type Solutions of the Damped Higher Order Nonlinear Schrödinger Equation. Front. Phys. 9:633890. doi: 10.3389/fphy.2021.633890

Received: 26 November 2020; Accepted: 08 February 2021;
Published: 30 April 2021.

Edited by:

Amin Chabchoub, The University of Sydney, Australia

Reviewed by:

Petr Grinevich, Steklov Mathematical Institute, Russia
Xing Lu, Beijing Jiaotong University, China

Copyright © 2021 Schober and Islas. 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: C. M. Schober, cschober@ucf.edu

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