Abstract
Our present study is devoted to the constructive study of the modulational instability for the Korteweg-de Vries (KdV)-family of equations u t + s u p u x + u x x x (here s = ± 1 and p > 0 is an arbitrary integer). For deducing the conditions of the instability, we first computed the nonlinear corrections to the frequency of the Stokes wave and then explored the coefficients of the corresponding modified nonlinear Schrödinger equations, thus deducing explicit expressions for the instability growth rate, maximum of the increment and the boundaries of the instability interval. A brief discussion of the results, open questions and further research directions completes the paper.
Highlights
Our present study is devoted to the constructive study of the modulational instability for the class of the generalized Korteweg-de Vries equations of the general form ut + su p u x + u xxx = 0, s = ±1, (1)
The log-Korteweg-de Vries (KdV) equation used for solitary waves in Fermi-Pasta-Ulam lattices can be mentioned, e.g., [5,6,7]
An alternative way to investigate the modulational instability of wave packets is based on the Whitham’s modulational theory and is presented e.g., in [24,25,26,27]. This approach is valid for weakly modulated waves of arbitrary shape, for quasi-monochromatic waves, and can be applied directly to the generalized KdV equation; see [25] and references
Summary
Where p > 0 is an arbitrary integer They describe weakly nonlinear dispersive waves met in various physical problems of fluid mechanics, plasma, astrophysics, oceanography and studied in numerous books and papers, only a few of them can be cited further on. An alternative way to investigate the modulational instability of wave packets is based on the Whitham’s modulational theory and is presented e.g., in [24,25,26,27] This approach is valid for weakly modulated waves of arbitrary shape, for quasi-monochromatic waves, and can be applied directly to the generalized KdV equation; see [25] and references . We will demonstrate constructively that for p > 2 the weakly modulated wave train is described by the high-order NLS equation whose solutions are stable for s = −1.
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