Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aceves-Sánchez, P., Minzoni, A.A., Panayotaros, P.: Numerical study of a nonlocal model for water-waves with variable depth. Wave Motion 50, 80–93 (2013)
Bruell, G., Ehrnström, M., Pei, L.: Symmetry and decay of traveling wave solutions to the Whitham equation. J. Differ. Equ. 262, 4232–4254 (2017)
Buffoni, B.: Existence and conditional energetic stability of capillary–gravity solitary water waves by minimisation. Arch. Ration. Mech. Anal. 173, 25–68 (2004)
Carter, J.D.: Bidirectional Whitham equations as models of waves on shallow water. Wave Motion 82, 51–61 (2018)
Claassen, K.M., Johnson, M.A.: Numerical bifurcation and spectral stability of wavetrains in bidirectional Whitham models. Stud. Appl. Math. 141, 205–246 (2018)
Dinvay, E.: On well-posedness of a dispersive system of the Whitham–Boussinesq type. Appl. Math. Lett. 88, 13–20 (2018)
Dinvay, E., Dutykh, D., Kalisch, H.: A comparative study of bi-directional Whitham systems (2018) (submitted for publication)
Duchêne, V., Israwi, S., Talhouk, R.: A new class of two-layer Green–Naghdi systems with improved frequency dispersion. Stud. Appl. Math. 137, 356–415 (2016)
Duchêne, V., Nilsson, D., Wahlén, E.: Solitary wave solutions to a class of modified Green–Naghdi systems. J. Math. Fluid Mech. 20, 1059–1091 (2018)
Ehrnström, M., Groves, M.D., Wahlén, E.: On the existence and stability of solitary-wave solutions to a class of evolution equations of Whitham type. Nonlinearity 25, 2903–2936 (2012)
Ehrnström, M., Johnson, M.A., Claassen, K.M.: Existence of a highest wave in a fully dispersive two-way shallow water model. Arch. Ration. Mech. Anal. 231, 1635–1673 (2018)
Ehrnström, M., Kalisch, H.: Traveling waves for the Whitham equation. Differ. Integral Equ. 22, 1193–1210 (2009)
Ehrnström, M., Pei, L., Wang, Y.: A conditional well-posedness result for the bidirectional Whitham equation (2017). arXiv:1708.04551
Ehrnström, M., Wahlén, E.: On Whitham’s conjecture of a highest cusped wave for a nonlocal dispersive equation (2016). arXiv:1602.05384
Groves, M.D., Wahlén, E.: On the existence and conditional energetic stability of solitary gravity–capillary surface waves on deep water. J. Math. Fluid Mech. 13, 593–627 (2011)
Hur, V.M., Tao, L.: Wave breaking in a shallow water model. SIAM J. Math. Anal. 50, 354–380 (2018)
Kalisch, H., Pilod, D.: On the local well-posedness for a full dispersion Boussinesq system with surface tension (2018). arXiv:1805.04372
Klein, C., Linares, F., Pilod, D., Saut, J.-C.: On Whitham and related equations. Stud. Appl. Math. 140, 133–177 (2018)
Lannes, D.: The Water Waves Problem: Mathematical Analysis and Asymptotics. Mathematical Surveys and Monographs, vol. 188. American Mathematical Society, Providence (2013)
Lions, P.-L.: The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. Henri Poincaré Anal. Non Linéaire 1, 223–283 (1984)
Moldabayev, D., Kalisch, H., Dutykh, D.: The Whitham equation as a model for surface water waves. Phys. D 309, 99–107 (2015)
Saut, J.-C., Wang, C., Xu, L.: The Cauchy problem on large time for surface-waves-type Boussinesq systems II. SIAM J. Math. Anal. 49, 2321–2386 (2017)
Stefanov, A., Wright, D.: Small amplitude traveling waves in the full-dispersion Whitham equation. arXiv:1802.10040
Trillo, S., Klein, M., Clauss, G.F., Onorato, M.: Observation of dispersive shock waves developing from initial depressions in shallow water. Phys. D 333, 276–284 (2016)
Whitham, G.B.: Variational methods and applications to water waves. In: Hyperbolic equations and waves (Rencontres, Battelle Res. Inst., Seattle, Wash., 1968), pp. 153–172. Springer, Berlin (1970)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Dag Nilsson was supported by an ERCIM ‘Alain Bensoussan’ Fellowship. Yuexun Wang acknowledges the support by Grants Nos. 231668 and 250070 from the Research Council of Norway.
Rights and permissions
About this article
Cite this article
Nilsson, D., Wang, Y. Solitary wave solutions to a class of Whitham–Boussinesq systems. Z. Angew. Math. Phys. 70, 70 (2019). https://doi.org/10.1007/s00033-019-1116-0
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00033-019-1116-0