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Instability of two-pulse periodic waves with long wavelength in some Hamiltonian PDEs

Thomas Courant

TL;DR

This work analyzes the spectral stability of two-pulse periodic waves in quasilinear Hamiltonian PDEs, where each period features a bright and a dark soliton in the large-wavelength limit. It develops a dual methodology: (i) asymptotics of the action Hessian to predict instability from the soliton pair, and (ii) a renormalized Evans-function framework showing convergence to the product of the individual soliton Evans functions as the period grows. The main finding is that large-period two-pulse waves are spectrally unstable whenever the sum of the second derivatives of the Boussinesq momentum for the two solitons is positive, or when either soliton is unstable, with modulational instability ruled out when the sum is negative. This bridges the soliton limit with periodic-wave stability, providing a rigorous mechanism by which two well-separated solitons fail to maintain stable synchronization, and it broadens the understanding of stability for quasilinear dispersive PDEs like generalized KdV and Euler–Korteweg systems.

Abstract

We consider quasilinear generalizations of the Korteweg-de Vries equation and dispersive perturbations of the Euler equations for compressible fluids, either in Lagrangian or in Eulerian coordinates. In particular, our framework includes hydrodynamic formulation of the nonlinear Schrödinger equations. The periodic waves we study exhibit on each period two pulses, one converging to a bright soliton and one converging to a dark soliton, when wavelength goes to infinity. We show that such waves, for sufficiently large periods, are spectrally unstable. To do so, we combine two approaches. The first one is to calculate the asymptotic expansion of the Hessian matrix of the action integral and concludes using arXiv:1505.01382 as in arXiv:1710.03936 . This shows instability when both limiting solitary waves are stable. The second approach studies the convergence of the spectrum when the period goes to infinity and is applied in remaining cases, when one of the solitary waves is unstable. To carry out the latter, we prove the convergence of an appropriate renormalization of the periodic Evans function as in arXiv:1802.02830 .

Instability of two-pulse periodic waves with long wavelength in some Hamiltonian PDEs

TL;DR

This work analyzes the spectral stability of two-pulse periodic waves in quasilinear Hamiltonian PDEs, where each period features a bright and a dark soliton in the large-wavelength limit. It develops a dual methodology: (i) asymptotics of the action Hessian to predict instability from the soliton pair, and (ii) a renormalized Evans-function framework showing convergence to the product of the individual soliton Evans functions as the period grows. The main finding is that large-period two-pulse waves are spectrally unstable whenever the sum of the second derivatives of the Boussinesq momentum for the two solitons is positive, or when either soliton is unstable, with modulational instability ruled out when the sum is negative. This bridges the soliton limit with periodic-wave stability, providing a rigorous mechanism by which two well-separated solitons fail to maintain stable synchronization, and it broadens the understanding of stability for quasilinear dispersive PDEs like generalized KdV and Euler–Korteweg systems.

Abstract

We consider quasilinear generalizations of the Korteweg-de Vries equation and dispersive perturbations of the Euler equations for compressible fluids, either in Lagrangian or in Eulerian coordinates. In particular, our framework includes hydrodynamic formulation of the nonlinear Schrödinger equations. The periodic waves we study exhibit on each period two pulses, one converging to a bright soliton and one converging to a dark soliton, when wavelength goes to infinity. We show that such waves, for sufficiently large periods, are spectrally unstable. To do so, we combine two approaches. The first one is to calculate the asymptotic expansion of the Hessian matrix of the action integral and concludes using arXiv:1505.01382 as in arXiv:1710.03936 . This shows instability when both limiting solitary waves are stable. The second approach studies the convergence of the spectrum when the period goes to infinity and is applied in remaining cases, when one of the solitary waves is unstable. To carry out the latter, we prove the convergence of an appropriate renormalization of the periodic Evans function as in arXiv:1802.02830 .
Paper Structure (24 sections, 29 theorems, 251 equations, 4 figures, 1 table)

This paper contains 24 sections, 29 theorems, 251 equations, 4 figures, 1 table.

Table of Contents

  1. Keywords:
  2. AMS Subject Classifications:
  3. Introduction
  4. Acknowledgment
  5. Framework and main results
  6. A class of Hamiltonian PDEs
  7. Traveling waves
  8. Main result and strategy of proof
  9. Asymptotic expansion of the action
  10. Preparation for the asymptotic
  11. Asymptotic behavior of the action and the period
  12. Asymptotic behavior of the Hessian of the actions
  13. Conclusion on the action of the periodic waves
  14. Asymptotic expansion of the Evans function
  15. Behavior of the periodic waves for a large period
  16. ...and 9 more sections

Key Result

Theorem 2.3.1

Under Assumption hypothèse_Hm, for a wave locally parametrized by $(c,\bm{\lambda},\mu) \in \mathbb{R}\times \mathbb{R}^N \times \mathbb{R}$ such that $\partial_\mu^2 \Theta$ is nonzero and $\nabla_{(\mu,\bm{\lambda},c)}^2 \Theta$ is non singular,

Figures (4)

  • Figure 1: "Generic" potential $\mathcal{W}$ that we consider
  • Figure 2: Phase portrait for a "generic" potential $\mathcal{W}$
  • Figure 3: Period of a "generic" two-pulse periodic wave
  • Figure 4: "Generic" periodic wave that we are looking at

Theorems & Definitions (55)

  • Theorem 2.3.1
  • Theorem 2.3.2
  • Theorem 2.3.3
  • Remark 2.3.1
  • Proposition 3.2.1
  • proof
  • Proposition 3.2.2
  • proof
  • Proposition 3.3.1
  • proof
  • ...and 45 more