The stability spectrum for elliptic solutions to the sine-Gordon equation

Deconinck, Bernard; McGill, Peter; Segal, Benjamin L.

doi:10.1016/j.physd.2017.08.010

Cited by 24 publications

(35 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This paper is part of an ongoing research program of analyzing the stability of periodic solutions of integrable equations ( [5,6,12,14,15,16,35]). The present work is the first in the program to establish a nonlinear stability result for periodic solutions for which the underlying Lax pair is not self adjoint.…”

Section: Introductionmentioning

confidence: 99%

The Orbital Stability of Elliptic Solutions of the Focusing Nonlinear Schrödinger Equation

Deconinck¹,

Upsal²

2020

SIAM J. Math. Anal.

Self Cite

View full text Add to dashboard Cite

We examine the stability of the elliptic solutions of the focusing nonlinear Schrödinger equation (NLS) with respect to subharmonic perturbations. Using the integrability of NLS, we discuss the spectral stability of the elliptic solutions, establishing that solutions of smaller amplitude are stable with respect to larger classes of perturbations. We show that spectrally stable solutions are orbitally stable by constructing a Lyapunov functional using higher-order conserved quantities of NLS.1. Spectral stability is considered in Section 3. This is motivated by considering the simpler case of the well-known Stokes waves in Section 3.2. For these solutions, all operators involved have constant coefficients, and all calculations are explicit. We get to the spectrum of the operator obtained by linearizing about a solution through its connection with the Lax spectrum. To this end, we introduce the Lax pair and its spectrum in Section 3.3. The results in Section 3.3.1 are from [16] while the results in Section 3.3.2 and all subsequent sections are new. Section 3.4 contains our main spectral stability result: solutions are spectrally stable with respect to subharmonic perturbations if the solution 1

show abstract

Section: Introductionmentioning

confidence: 99%

The Orbital Stability of Elliptic Solutions of the Focusing Nonlinear Schrödinger Equation

Deconinck¹,

Upsal²

2020

SIAM J. Math. Anal.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Some of these conditions are new and some have been used in Ref. , 4, 6–10. This is the first time all are written down in full generality.Remark The conditions () are generalizations of the Lax condition () for members of the AKNS hierarchy.…”

Section: Generalization Of the Akns Resultsmentioning

confidence: 99%

“…To determine the subset of

σ_{L}

off the real line, one may use the integral condition () (used in Ref. 7 to find

σ_{L}

) or the Floquet discriminant (Section ).…”

Section: Akns Examplesmentioning

confidence: 99%

“…An important feature of equations with a Lax pair is the Lax spectrum: the set of all Lax parameter values for which the solution of the Lax pair is bounded. For our purposes, this is important for determining the stability of solutions of a given integrable equation –13 . Until recently, the Lax spectrum has only been determined explicitly for decaying potentials on the whole line or for self‐adjoint problems with periodic coefficients.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Real Lax spectrum implies spectral stability

Upsal

Deconinck

2020

Stud Appl Math

Self Cite

View full text Add to dashboard Cite

We consider the dynamical stability of periodic and quasiperiodic stationary solutions of integrable equations with 2 × 2 Lax pairs. We construct the eigenfunctions and hence the Floquet discriminant for such Lax pairs. The boundedness of the eigenfunctions determines the Lax spectrum. We use the squared eigenfunction connection between the Lax spectrum and the stability spectrum to show that the subset of the real line that gives rise to stable eigenvalues is contained in the Lax spectrum. For non-self-adjoint members of the AKNS hierarchy admitting a common reduction, the real line is always part of the Lax spectrum and it maps to stable eigenvalues of the stability problem. We demonstrate our methods work for a variety of examples, both in and not in the AKNS hierarchy.

show abstract

“…In [21] the authors present an analysis of the stability spectrum for all stationary periodic solutions to the sine-Gordon equation. Yousif an Mahmood [22] used the variational homotopy perturbation method for solving the Klein-Gordon and sine-Gordon equations.…”

Section: Introductionmentioning

confidence: 99%

Legendre spectral element method for solving sine-Gordon equation

Lotfi

Alipanah

2019

Adv Differ Equ

View full text Add to dashboard Cite

In this paper, we study the Legendre spectral element method for solving the sine-Gordon equation in one dimension. Firstly, we discretize the equation by Legendre spectral element in space and then discretize the time by the second-order leap-frog method. We study the stability and convergence of the method and show the convergence of our method. Finally, we show the results with numerical examples. MSC: 65M70; 65M06; 74G15

show abstract

The stability spectrum for elliptic solutions to the sine-Gordon equation

Cited by 24 publications

References 33 publications

The Orbital Stability of Elliptic Solutions of the Focusing Nonlinear Schrödinger Equation

The Orbital Stability of Elliptic Solutions of the Focusing Nonlinear Schrödinger Equation

Real Lax spectrum implies spectral stability

Legendre spectral element method for solving sine-Gordon equation

Contact Info

Product

Resources

About