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Soliton-like solutions of higher order wave equations of the Korteweg-de Vries type. (English) Zbl 1060.35127
Summary: In this work we study second and third order approximations of water wave equations of the Korteweg–de Vries (KdV) type. First we derive analytical expressions for solitary wave solutions for some special sets of parameters of the equations. Remarkably enough, in all these approximations, the form of the solitary wave and its amplitude-velocity dependence are identical to the \(\text{sech}^2\) formula of the one-soliton solution of the KdV. Next we carry out a detailed numerical study of these solutions using a Fourier pseudospectral method combined with a finite-difference scheme, in parameter regions where soliton-like behavior is observed. In these regions, we find solitary waves which are stable and behave like solitons in the sense that they remain virtually unchanged under time evolution and mutual interaction. In general, these solutions sustain small oscillations in the form of radiation waves (trailing the solitary wave) and may still be regarded as stable, provided these radiation waves do not exceed a numerical stability threshold. Instability occurs at high enough wave speeds, when these oscillations exceed the stability threshold already at the outset, and manifests itself as a sudden increase of these oscillations followed by a blowup of the wave after relatively short time intervals.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35B35 Stability in context of PDEs
35Q51 Soliton equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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[1] DOI: 10.1103/PhysRevLett.77.2347 · Zbl 0982.76511 · doi:10.1103/PhysRevLett.77.2347
[2] DOI: 10.1103/PhysRevLett.77.2347 · Zbl 0982.76511 · doi:10.1103/PhysRevLett.77.2347
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