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Bifurcations of travelling wave solutions for a two-component Camassa-Holm equation. (English) Zbl 1182.34064

The following two-component generalization of the Camassa-Holm equation is considered

m t +um x +2mu x +eρρ x =0,
ρ t +(ρu) x =0,

where e=±1, m=u-α 2 u xx -k, for α,k real parameters.

The existence of solitary wave solutions (homoclinic orbits), kink and anti-kink wave solutions (heteroclinic orbits) and periodic wave solutions is investigated by using a dynamical systems approach. Some exact explicit parametric representations of travelling wave solutions are provided too.

Finally, the existence of uncountably infinite many breaking wave solutions (whose maximal existence interval is bounded) is proved for e=-1.

MSC:
34C37Homoclinic and heteroclinic solutions of ODE
34C23Bifurcation (ODE)
35Q53KdV-like (Korteweg-de Vries) equations
35Q51Soliton-like equations
37K40Soliton theory, asymptotic behavior of solutions
37L05General theory, nonlinear semigroups, evolution equations
References:
[1]Chen, M., Liu, S. Q., Zhang, Y. J.: A 2-component generalization of the Cammassa-Holm equation and its solution. Letters in Math. Phys., 75, 1–15 (2006) · Zbl 1105.35102 · doi:10.1007/s11005-005-0041-7
[2]Cammasa, R., Holm, D. D.: An integrable shallow water equation with peaked solution. Phys. Rev. Lett., 71, 1161–1164 (1993)
[3]Cammasa, R., Holm, D. D., Hyman, J. M.: A new integrable shallow water equation. Adv. Appl. Mech., 31, 1–33 (1994) · doi:10.1016/S0065-2156(08)70254-0