Abstract
We classify the weak traveling wave solutions for a class of one-dimensional non-linear shallow water wave models. The equations are shown to admit smooth, peaked, and cusped solutions, as well as more exotic waves such as stumpons and composite waves. We also explain how some previously studied traveling wave solutions of the models fit into this classification.
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References
Burgers J. (1974). Nonlinear Diffusion Equation. Reidel, Dordrecht, Netherlands
Camassa R., Holm D. (1993). An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664
Camassa R., Holm D., Hyman J. (1994). A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33
Cao C., Holm D., Titi E. (2004). Traveling wave solutions for a class of one-dimensional nonlinear shallow water wave models. J. Dyn. Diff. Equat. 16, 1661–1664
Constantin A., McKean H.P. (1999). A shallow water equation on the circle. Comm. Pure Appl. Math. 52, 949–982
Constantin A., Molinet L.(2000). Global weak solutions for a shallow water equation. Comm. Math. Phys. 211, 45–61
Constantin A., Strauss W. (2000). Stability of peakons. Comm. Pure Appl. Math. 53, 603–610
Constantin A., Strauss W. (2002). Stability of the Camassa-Holm solitons. J. Nonlinear Sci. 12, 415–422
Dullin H., Gottwald G., Holm D. (2001). An integrable shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett. 87, 4501–4504
Holm D., Staley M. (2003). Wave structure and nonlinear balances in a family of evolutionary PDEs. SIAM J. Appl. Dyn. Syst. 2, 323–380
Johnson R. (2002). Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 455, 63–82
Kalisch H., Lenells J. (2005). Numerical study of traveling-wave solutions for the Camassa-Holm equation. Chaos, Solitons Fractals 25, 287–298
Lenells J. (2004). Stability of periodic peakons. Int. Math. Res. Not. 10, 485–499
Lenells J. (2004). A variational approach to the stability of periodic peakons. J. Nonlinear Math. Phys. 11, 151–163
Lenells J. (2005). Traveling wave solutions of the Camassa-Holm equation. J. Diff. Equat. 217, 393–430
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Lenells, J. Classification of Traveling Waves for a Class of Nonlinear Wave Equations. J Dyn Diff Equat 18, 381–391 (2006). https://doi.org/10.1007/s10884-006-9009-2
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DOI: https://doi.org/10.1007/s10884-006-9009-2