Existence, uniqueness, and analyticity of space-periodic solutions to the regularized long-wave equation. (English) Zbl 1292.35227

Chertovskih, R.; Chian, A. C.-L.; Podvigina, O.; Rempel, E. L.; Zheligovsky, V.

Adv. Differ. Equ. 19, No. 7-8, 725-754 (2014).

Summary: We consider space-periodic evolutionary and travelling-wave solutions to the regularized long-wave equation (RLWE) with damping and forcing. We establish existence, uniqueness and smoothness of the evolutionary solutions for smooth initial conditions, and global in time spatial analyticity of such solutions for analytical initial conditions. The width of the analyticity strip decays at most polynomially. We prove existence of travelling-wave solutions and uniqueness of travelling waves of a sufficiently small norm. The importance of damping is demonstrated by showing that the problem of finding travelling-wave solutions to the undamped RLWE is not well-posed. Finally, we demonstrate the asymptotic convergence of the power series expansion of travelling waves for a weak forcing.

MSC:

35Q35	PDEs in connection with fluid mechanics
35Q51	Soliton equations
35C07	Traveling wave solutions
35A01	Existence problems for PDEs: global existence, local existence, non-existence
35A02	Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35B40	Asymptotic behavior of solutions to PDEs