de Brito LFC, Kamchatnov AM. Undular bore theory for the modified Korteweg-de Vries-Burgers equation.
Phys Rev E 2024;
109:015102. [PMID:
38366500 DOI:
10.1103/physreve.109.015102]
[Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2023] [Accepted: 12/19/2023] [Indexed: 02/18/2024]
Abstract
We consider nonlinear wave structures described by the modified Korteweg-de Vries equation, taking into account a small Burgers viscosity for the case of steplike initial conditions. The Whitham modulation equations are derived, which include the small viscosity as a perturbation. It is shown that for a long enough time of evolution, this small perturbation leads to the stabilization of cnoidal bores, and their main characteristics are obtained. The applicability conditions of this approach are discussed. Analytical theory is compared with numerical solutions and good agreement is found.
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