The Analysis of Bifurcation Solutions for the Camassa-Holm Equation
This paper studies the Camassa-Holm equation’s bifurcation solutions by using the local method of Lyapunov-Schmidt. The Camassa-Holm equation has been studied with the formula ODE. We have found the key function corresponding to the functional related to this equation. The bifurcation analysis of this function has been investigated by the boundary singularities. We have found the parametric equation of the bifurcation set (caustic) with the geometric description of this caustic. Also, the critical points’ bifurcation spreading has been found.
M.A. Abdul Hussain, Corner Singularities of Smooth Functions in the Analysis of Bifurcations Balance of the Elastic Beams and Periodic Waves, PhD Thesis, Voronezh Univ. – Russia (2005) (in Russian).
V. I. Arnold, Singularities of Differential Maps, M. Science (1989), DOI: 10.1007/978-1-4612-3940-6.
G. Berczi, Lectures on Singularities of Maps, Trinity Term, Oxford (2010).
B.M. Darinskii, C.L. Tcarev and Yu. I. Sapronov, Bifurcations of extremals of Fredholm functionals, Journal of Mathematical Sciences 145(6) (2007), 5311 – 5453, DOI: 10.1007/s10958-007-0356-2.
M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory: Vol I (Applied Mathematical Sciences 51), Springer, New York (1985), DOI: 10.1007/978-1-4612-5034-0 .
J. Li and Z. Qiao, Bifurcation and exact traveling wave solutions for a generalized Camassa-Holm equation, International Journal of Bifurcation and Chaos 23, Article ID 1350057 (2013), DOI: 10.1142/S0218127413500570.
J. Li, Singular Nonlinear Travelling Wave Equations: Bifurcations and Exact Solutions, Science Press, Beijing (2013).
B. V. Loginov, Theory of Branching Nonlinear Equations in Theconditions of Invariance Group, Tashkent, Fan (1985).
Yu. I. Sapronov, Regular Perturbation of Fredholm Maps and Theorem About Odd Field, Works Dept. of Math., Voronezh Univ., 10, 82 – 88 (1973).
Yu. I. Sapronov, Finite dimensional reduction in the smooth extremely problems, Uspehi Math., Science 51(1) (1996), 101 – 132.
Yu. I. Sapronov, B. M. Darinskii and C. L. Tcarev, Bifurcation of Extremely of Fredholm Functionals, Voronezh Univ. (2004), DOI: 10.1007/s10958-007-0356-2.
O. V. Shveriova, Caustic and bif-spreadings for a boundary extremal with a triple degeneration along the boundary, Tr. Mat.Fak. Voronezh. Gos. Univ. (N.S.) 7 (2002), 149 – 160 (in Russian).
M. M. Vainberg and V. A. Trenogin, Theory of Branching Solutions of Non-linear Equations, M. Science (1969), DOI: 10.1090/S0002-9904-1975-13871-7.
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