Smooth and non-smooth traveling waves in a nonlinearly dispersive equation. (English) Zbl 0985.37072

Summary: The method of the phase plane is employed to investigate the solitary and periodic traveling waves in a nonlinear dispersive integrable partial differential equation. It is shown that the existence of a singular straight line in the corresponding ordinary differential equation for traveling wave solutions is the reason that smooth solitary wave solutions converge to solitary cusp wave solutions when the parameters are varied. The different parameter conditions for the existence of different kinds of solitary and periodic wave solutions are rigorously determined.


37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q58 Other completely integrable PDE (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
76B25 Solitary waves for incompressible inviscid fluids
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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[1] Andronov, A.A.; Leontovich, E.A.; Gordon, J.I.; Maier, A.G., Theory of bifurcations of dynamical systems on a plane, (1973), Wiley New York · Zbl 0282.34022
[2] Byrd, P.F.; Friedman, M.D., Handbook of elliptic integrals for engineers and scientists, (1971), Springer New York · Zbl 0213.16602
[3] Chow, S.N.; Hale, J.K., Methods of bifurcation theory, (1982), Springer New York
[4] Grasman, J., Asymptotic methods for relaxation ocillations and applications, (1987), Springer New York
[5] Guckenheimer, J.; Holmes, P., Dynamical systems and bifurcations of vector fields, (1983), Springer New York · Zbl 0515.34001
[6] M. Hirsch, C. Pugh, M. Shub, Invariant manifolds, Lecture Notes in Mathematics, vol. 583, Springer, New York, 1976 · Zbl 0355.58009
[7] A.Li, Y.; Olver, P.J., Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system I: compactons and peakons, Discrete and continuous dynamical systems, 3, 419-432, (1997) · Zbl 0949.35118
[8] Li, Y.A.; Olver, P.J., Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system II: complex analytic behaviour and convergence to non-analytic solutions, Discrete and continuous dynamical systems, 4, 159-191, (1998) · Zbl 0959.35157
[9] O’Malley, R., Singular perturbation methods for ordinary differential equations, (1991), Springer New York · Zbl 0743.34059
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