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Instability bands for periodic traveling waves in the modified Korteweg-de Vries equation

Shikun Cui, Dmitry E. Pelinovsky

TL;DR

This work studies spectral stability of periodic traveling waves in the focusing mKdV equation by leveraging a squared-eigenfunction relation between the Lax pair and the linearized stability problem. It confirms that the dnoidal-like wave (dn-generalization) is spectrally stable for all parameters, while the cn-like wave (cn-generalization) is spectrally unstable for all parameters; crucially, the onset of co-periodic instability for the cn-family transforms instability bands from a figure-8 pattern (crossing the imaginary axis) to a figure-infinity pattern (crossing the real axis). Analytical criteria, grounded in the Lax spectrum and the root structure of the associated quartic invariant, are complemented by robust Fourier-collocation numerics that illustrate both stable and unstable regimes and the 8→∞ transition. The results enhance understanding of modulational-type instabilities in integrable and near-integrable wave models and have potential relevance for Stokes-wave-type phenomena in fluids and optical systems, with open questions about explicit Lax-spectrum locations for cn-waves and extensions to nonintegrable settings.

Abstract

Two families of periodic traveling waves exist in the focusing mKdV (modified Korteweg-de Vries) equation. Spectral stability of these waveforms with respect to co-periodic perturbations of the same period has been previously explored by using spectral analysis and variational formulation. By using tools of integrability such as a relation between squared eigenfunctions of the Lax pair and eigenfunctions of the linearized stability problem, we revisit the spectral stability of these waveforms with respect to perturbations of arbitrary periods. In agreement with previous works, we find that one family is spectrally stable for all parameter configurations, whereas the other family is spectrally unstable for all parameter configurations. We show that the onset of the co-periodic instability for the latter family changes the instability bands from figure-$8$ (crossing at the imaginary axis) into figure-$\infty$ (crossing at the real axis).

Instability bands for periodic traveling waves in the modified Korteweg-de Vries equation

TL;DR

This work studies spectral stability of periodic traveling waves in the focusing mKdV equation by leveraging a squared-eigenfunction relation between the Lax pair and the linearized stability problem. It confirms that the dnoidal-like wave (dn-generalization) is spectrally stable for all parameters, while the cn-like wave (cn-generalization) is spectrally unstable for all parameters; crucially, the onset of co-periodic instability for the cn-family transforms instability bands from a figure-8 pattern (crossing the imaginary axis) to a figure-infinity pattern (crossing the real axis). Analytical criteria, grounded in the Lax spectrum and the root structure of the associated quartic invariant, are complemented by robust Fourier-collocation numerics that illustrate both stable and unstable regimes and the 8→∞ transition. The results enhance understanding of modulational-type instabilities in integrable and near-integrable wave models and have potential relevance for Stokes-wave-type phenomena in fluids and optical systems, with open questions about explicit Lax-spectrum locations for cn-waves and extensions to nonintegrable settings.

Abstract

Two families of periodic traveling waves exist in the focusing mKdV (modified Korteweg-de Vries) equation. Spectral stability of these waveforms with respect to co-periodic perturbations of the same period has been previously explored by using spectral analysis and variational formulation. By using tools of integrability such as a relation between squared eigenfunctions of the Lax pair and eigenfunctions of the linearized stability problem, we revisit the spectral stability of these waveforms with respect to perturbations of arbitrary periods. In agreement with previous works, we find that one family is spectrally stable for all parameter configurations, whereas the other family is spectrally unstable for all parameter configurations. We show that the onset of the co-periodic instability for the latter family changes the instability bands from figure- (crossing at the imaginary axis) into figure- (crossing at the real axis).
Paper Structure (8 sections, 6 theorems, 51 equations, 7 figures)

This paper contains 8 sections, 6 theorems, 51 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Waveforms for the periodic traveling waves
  3. Lax spectrum for the periodic traveling waves
  4. Lax and stability spectra: analytical results
  5. Lax and stability spectra: numerical results
  6. Waveform (\ref{['solution_1']})
  7. Waveform (\ref{['solution_2']})
  8. Conclusion

Key Result

Proposition 1

If the roots $\{ u_1,u_2,u_3.u_4 \}$ are real and ordered as $u_4\leq u_3\leq u_2\leq u_1$, then the first-order invariant (ini_5) is satisfied by where If $\{ u_1,u_2 \}$ are real and $\{u_3.u_4 \}$ are complex-conjugate such that $u_2\leq u_1$ and $u_3=\bar{u}_4=\gamma+i\eta$ with $\eta > 0$, then the first-order invariant (ini_5) is satisfied by where and

Figures (7)

  • Figure 1: The Lax and stability spectra for the cnoidal wave (\ref{['cn-wave']}) with different values of $k$. (a)-(c) and (g)-(i): Lax spectrum in $\lambda$-plane. (d)-(f) and (j)-(l): stability spectrum in $\Lambda$-plane.
  • Figure 2: (a) Phase portrait in the phase plane $(U,U')$ for $b = 0.8$ and $c = 4$. (b) Levels of $d$ at the plot of $Q(U)$.
  • Figure 3: Numerically computed Lax and stability spectra for the periodic solution with the profile (\ref{['solution_1']}) for $u_1=1$, $u_2=0.5$, $u_3=0$, and $u_4 = -1.5$.
  • Figure 4: Numerically computed Lax and stability spectra for the periodic solution with the profile (\ref{['solution_2']}) for $u_1=1$, $u_2=0.2$, and $u_3 = \bar{u}_4 = -0.6+0.6i$.
  • Figure 5: The same as in Figure \ref{['fig_cn2']} but for $u_1=1$, $u_2=-0.2$, and $u_3 = \bar{u}_4 = -0.4+0.2i$.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Proposition 1
  • Remark 1
  • Remark 2
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Theorem 1
  • ...and 4 more