Abstract
A differential equation that models the motion of a domain wall is considered. For this equation, in the case of constant coefficients, there exists a solution in the form of a travelling wave which describes the transition from one equilibrium to another. For an equation with slowly varying coefficients, we construct an asymptotic one-phase solution. The phase is found from the Hamilton–Jacobi equation whose coefficients are taken from the asymptotics of the unperturbed wave at infinity. The asymptotic construction is based on the requirement that the first correction is small as compared to the leading term uniformly over a wide range of independent variables.
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Notes
The same situation occurs in the case of \(\omega^2-\Omega^2>0\) if the unperturbed wave has a too high stabilization rate with \(\lambda_+=-\lambda_2>-\lambda_1\) near the equilibrium \(\Phi=\pi\) of the knot type.
The continuity requirement for the derivatives leads to the unsolvability of the problem in the general case: the first derivatives on the transition line must be coordinated with the differential equation, [10, p. 326 (Russian transl.)]. These requirements are not used or discussed here.
Another situation in which the asymptotics as \(s\to\infty\) is determined by the exponent \(-\lambda_2\) is not considered in the present paper.
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Translated from Matematicheskie Zametki, 2022, Vol. 112, pp. 553–566 https://doi.org/10.4213/mzm13730.
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Kalyakin, L.A. Perturbation of a Simple Wave in a System with Dissipation. Math Notes 112, 549–560 (2022). https://doi.org/10.1134/S0001434622090243
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DOI: https://doi.org/10.1134/S0001434622090243