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Traveling wave solutions of the doubly regularized nonlinear Boussinesq equation

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  • Erbay, H.A.
  • Erbay, S.

Abstract

In this study we consider traveling wave solutions of a nonlinear dispersive wave equation involving the fourth-order time derivative term. We first discuss existence of traveling wave solutions to the dispersive wave equation with a quadratic nonlinearity and report sech-type solitary wave solutions. Using asymptotic expansion techniques we derive the well-known unidirectional nonlinear dispersive wave equations for small amplitude waves. The KdV equation models the propagation of long acoustic waves, while the NLS equation models the evolution of the envelope of short optic waves. We also show that when a long-wave–short-wave resonance condition is satisfied, a coupled system of equations describes the nonlinear interaction between long acoustic waves and short optic waves. We study traveling wave solutions of the asymptotic models derived to assess the relative importance of nonlocality in time with respect to nonlocality in space.

Suggested Citation

  • Erbay, H.A. & Erbay, S., 2026. "Traveling wave solutions of the doubly regularized nonlinear Boussinesq equation," Chaos, Solitons & Fractals, Elsevier, vol. 206(C).
    • Handle: RePEc:eee:chsofr:v:206:y:2026:i:c:s0960077926001086
    DOI: 10.1016/j.chaos.2026.117967
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