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arXiv · 1203.3228

On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type

Mats Ehrnström·Mark D. Groves·Erik Wahlén
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Abstract

We consider a class of pseudodifferential evolution equations of the form $$u_t + (n(u) + Lu)_x = 0,$$ in which $L$ is a linear smoothing operator and $n$ is at least quadratic near the origin; this class includes in particular the Whitham equation. A family of solitary-wave solutions is found using a constrained minimisation principle and concentration-compactness methods for noncoercive functionals. The solitary waves are approximated by (scalings of) the corresponding solutions to partial differential equations arising as weakly nonlinear approximations; in the case of the Whitham equation the approximation is the Korteweg-deVries equation. We also demonstrate that the family of solitary-wave solutions is conditionally energetically stable.

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Mats Ehrnström, Mark D. Groves, Erik Wahlén. 2012-03-14. On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type. https://doi.org/10.1088/0951-7715%2F25%2F10%2F2903

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