Nonlinear stability of periodic-wave solutions for systems of dispersive equations

We prove the orbital stability of periodic traveling-wave solutions for systems of dispersive equations with coupled nonlinear terms. Our method is basically developed under two assumptions: one concerning the spectrum of the linearized operator around the traveling wave and another one concerning the existence of a conserved quantity with suitable properties. The method can be applied to several systems such as the Liu-Kubota-Ko system, the modified KdV system and a log-KdV type system.

• Publication year

  2020

• Venue

  Communications on Pure and Applied Analysis

• Publication date

  2020-02-08

• Fields of study

  Mathematics, Physics

• Identifiers
  DOI 10.3934/cpaa.2020225arXiv 2002.03245
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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  2017influential reference
• Stability of Solitary-Wave Solutions of Systems of Dispersive Equations

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• Dynamics Of Heterogeneous Materials

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  2008influential reference
• Lectures on the Orbital Stability of Standing Waves and Application to the Nonlinear Schrödinger Equation

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• Stability and instability of solitary waves of Korteweg-de Vries type

  1987influential reference
• Weak and strong interactions between internal solitary waves

  1984influential reference
• Resonant Transfer of Energy between Nonlinear Waves in Neighboring Pycnoclines

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• The spectral theory of periodic differential equations

  1973cited by this paper
• Nonlinear-evolution equations of physical significance

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  1971cited by this paper
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  1966cited by this paper

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