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Solitons and the Inverse Scattering Transform

In: Nonlinear Partial Differential Equations for Scientists and Engineers

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  • Lokenath Debnath

    (University of Texas, Pan American, Department of Mathematics)

Abstract

Dispersion and nonlinearity play a fundamental role in wave motions in nature. The nonlinear shallow water equations that neglect dispersion altogether lead to breaking phenomena of the typical hyperbolic kind with the development of a vertical profile. In particular, the linear dispersive term in the Korteweg–de Vries equation prevents this from ever happening in its solution. In general, breaking can be prevented by including dispersive effects in the shallow water theory. The nonlinear theory provides some insight into the question of how nonlinearity affects dispersive wave motions. Another interesting feature is the instability and subsequent modulation of an initially uniform wave profile.

Suggested Citation

  • Lokenath Debnath, 2012. "Solitons and the Inverse Scattering Transform," Springer Books, in: Nonlinear Partial Differential Equations for Scientists and Engineers, edition 0, chapter 9, pages 425-533, Springer.
    • Handle: RePEc:spr:sprchp:978-0-8176-8265-1_9
    DOI: 10.1007/978-0-8176-8265-1_9
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