Abstract
AbstractThis paper investigates the problem of the orbital stability of dn periodic wave solutions of the Boussinesq equation with quadratic-cubic nonlinear terms. First, the dn periodic wave solution of the studied equation is solved by using the integral method and the knowledge of elliptic functions, and the existence of smooth curves of dn periodic wave solutions with fixed period L is proved. Then the Floquet theory and Wely’s essential spectrum theorem are applied to the spectral analysis of the operator, and obtain its spectral properties. Finally, according to the ideas for proving the stability of solitary wave solutions from Benjamin and Bona et al., by overcoming the complexity caused by the quadratic-cubic nonlinear terms in the studied equation, we prove the dn periodic wave solution of the studied equation is orbitally stable under small perturbations of the $$L^{2}$$
L
2
norm.
Funder
National Natural Science Foundation of China
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
Reference37 articles.
1. Boussinesq, M.J.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 2(17), 55–108 (1872)
2. Whitham, G.B.: Linear and Nonlinear Wave. Springer, New York (1974)
3. Zhakarov, V.E.: On stochasization of one-dimensional chains of nonlinear oscillation. J. Soviet Phys. Jetp. 38(1), 108–110 (1974)
4. McKean, H. P.: Boussinesq’s equation on the circle. J. Commun. Pure Appl. Math. 34(1), 599–691 (1981)
5. Ursell, F.: The long-wave paradox in the theory of gravity waves. J. Math. Proc. Cambridge Philos. Soc. 49, 685-694 (1953)