Uniqueness of the solution of the Riemann — Hilbert problem for a rarefaction wave of the Korteweg — de Vries equation

Abstract

Единственность решения задачи Римана -Гильберта для волны разрежения уравнения Кортевега -де Фриза Представлено академиком НАН Украины Е.Я. Хрусловым Существенным моментом в асимптотическом анализе решений нелинейных вполне интегрируемых уравнений методом наискорейшего спуска является исследование единственности решения соответствующей задачи Римана -Гильберта. В работе устанавливается единственность решения задачи Римана-Гильберта, построенной по левым данным рассеяния для уравнения Кортевега -де Фриза с начальными данными типа ступеньки, соответствующими волне разрежения. Такая задача позволяет исследовать асимптотическое поведение решения позади заднего волнового фронта. Доказательство единственности проведено как для нерезонансного, так и для резонансного случаев.

On the Long-Time Asymptotics for the Korteweg-de Vries Equation with Steplike Initial Data Associated with Rarefaction Waves

“…c K. Andreiev and I. Egorova, 2017 was studied. The initial profile of type (2) corresponds to the rarefaction wave.…”

“…Egorova, 2017 was studied. The initial profile of type (2) corresponds to the rarefaction wave. Its asymptotics is well understood on a physical level of rigor (see [10,12,15]).…”

“…Its asymptotics is well understood on a physical level of rigor (see [10,12,15]). In [1], the asymptotics of the solution for (1), (2) is studied mathematically rigorously for the regions ahead of the back wave front by using the nonlinear steepest descent method [6]. As for the region behind the back wave front, in [1], the respective Riemann-Hilbert (RH) problem was reduced to a model RH problem in the nonresonant case.…”

“…As for the region behind the back wave front, in [1], the respective Riemann-Hilbert (RH) problem was reduced to a model RH problem in the nonresonant case. The structure of the transformations which were performed to get this model problem led to an assumption that the solution of (1), (2) should be asymptotically close to the respective background constant c 2 , plus a decaying "radiation part" of order O(t −1/2 ). Moreover, for this second term of the asymptotical expansion a formula was given which had the same form as for the decaying initial data, that is, when q 0 (x) → 0 as x → ±∞.…”

“…The objectives of the present paper are: (a) to justify the asymptotical expansion for the solution of (1), (2) with respect to large t in the region x < (−6c 2 − )t; (b) to check a possible influence of the resonance on the asymptotical expansion; (c) to clarify the formula for the second term.…”