Advances in Differential Equations

Existence, uniqueness, and analyticity of space-periodic solutions to the regularized long-wave equation

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

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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.

Article information

Adv. Differential Equations, Volume 19, Number 7/8 (2014), 725-754.

First available in Project Euclid: 6 May 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q35: PDEs in connection with fluid mechanics 35Q51: Soliton-like equations [See also 37K40] 35C07: Traveling wave solutions 35A01: Existence problems: global existence, local existence, non-existence 35A02: Uniqueness problems: global uniqueness, local uniqueness, non- uniqueness


Chertovskih, R.; Chian, A.C.L.; Podvigina, O.; Rempel, E.L.; Zheligovsky, V. Existence, uniqueness, and analyticity of space-periodic solutions to the regularized long-wave equation. Adv. Differential Equations 19 (2014), no. 7/8, 725--754.

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