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<p>A periodic boundary-value problem for the Kuramoto–Sivashinsky equation is considered. We prove that there exists a two-parameter family of traveling-wave solutions and obtain asymptotic formulas for them. We also prove that the set of such solutions is a two-dimensional invariant manifold, which is a local attractor. The solutions indicated have different periods in the variable&#xa0;<i>t</i>, are unstable in the Lyapunov sense, but are stable in the Perron, Poincaré, and Zhukovsky senses. The study is based on the theory of invariant manifolds and Poincaré–Dulac normal forms.</p>

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PERIODIC TRAVELING WAVES OF THE KURAMOTO-SIVASHINSKY EQUATION

  • 期刊论文

  • 发表日期：

    2025年10月1日

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  • A. N. Kulikov,
  • D. A. Kulikov,
  • D. G. Frolov

摘要

A periodic boundary-value problem for the Kuramoto–Sivashinsky equation is considered. We prove that there exists a two-parameter family of traveling-wave solutions and obtain asymptotic formulas for them. We also prove that the set of such solutions is a two-dimensional invariant manifold, which is a local attractor. The solutions indicated have different periods in the variable t, are unstable in the Lyapunov sense, but are stable in the Perron, Poincaré, and Zhukovsky senses. The study is based on the theory of invariant manifolds and Poincaré–Dulac normal forms.