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

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

2012-03-14

arxiv.org/abs/1203.3228v1

Mathematics

Analysis of PDEs

37 pages

Scientific paper

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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