Solitary wave solutions to a class of Whitham–Boussinesq systems
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In this note, we study solitary wave solutions of a class of Whitham–Boussinesq systems which include the bidirectional Whitham system as a special example. The travelling wave version of the evolution system can be reduced to a single evolution equation, similar to a class of equations studied by Ehrnström et al. (Nonlinearity 25:2903–2936, 2012). In that paper, the authors prove the existence of solitary wave solutions using a constrained minimization argument adapted to noncoercive functionals, developed by Buffoni (Arch Ration Mech Anal 173:25–68, 2004), Groves and Wahlén (J Math Fluid Mech 13:593–627, 2011), together with the concentration–compactness principle.
KeywordsWhitham-type equations Dispersive equations Solitary wave
Mathematics Subject Classification76B15 76B25 35S30 35A20
Both authors thank M. Ehrnström and E. Wahlén for suggesting this topic. We would also like to thank the referees for their careful reading, helpful suggestions and valuable comments, which helped us a lot to improve the presentation of this manuscript.
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