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 TMF, 2001, Volume 126, Number 2, Pages 214–227 (Mi tmf426)

Asymptotic behavior in the trailing edge domain of the solution of the KdV equation with an initial condition of the “threshold type”

V. B. Baranetskii, V. P. Kotlyarov

B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine

Abstract: We derive a new integral equation that linearizes the Cauchy problem for the Korteweg–de Vries equation for the initial condition of the threshold type, where the initial function vanishes as $x\to-\infty$ and tends to some periodic function as $x\to+\infty$. We also expand the solution of the Cauchy problem into a radiation component determined by the reflection coefficient and a component determined by the nonvanishing initial condition. For the second component, we derive an approximate determinant formula that is valid for any $t\ge 0$ and $x\in(-\infty,X_N)$, where $X_N\to\infty$ with the unboundedly increasing parameter $N$ that determines the finite-dimensional approximation to the integral equation. We prove that as $t\to\infty$, the solution of the Cauchy problem in the neighborhood of the trailing edge decays into asymptotic solitons, whose phases can be explicitly evaluated in terms of the reflection coefficient and other parameters of the problem.

DOI: https://doi.org/10.4213/tmf426

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English version:
Theoretical and Mathematical Physics, 2001, 126:2, 175–186

Bibliographic databases:

Citation: V. B. Baranetskii, V. P. Kotlyarov, “Asymptotic behavior in the trailing edge domain of the solution of the KdV equation with an initial condition of the “threshold type””, TMF, 126:2 (2001), 214–227; Theoret. and Math. Phys., 126:2 (2001), 175–186

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf426
• https://doi.org/10.4213/tmf426
• http://mi.mathnet.ru/eng/tmf/v126/i2/p214

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. de Monvel, AB, “Soliton asymptotics of rear part of non-localized solutions of the Kadomtsev-Petviashvili equation”, Journal of Nonlinear Mathematical Physics, 9:1 (2002), 58
2. Egorova, I, “On the Cauchy problem for the Korteweg-de Vries equation with steplike finite-gap initial data: I. Schwartz-type perturbations”, Nonlinearity, 22:6 (2009), 1431
3. Kotlyarov V., Minakov A., “Riemann–Hilbert problem to the modified Korteveg-de Vries equation: Long-time dynamics of the steplike initial data”, J Math Phys, 51:9 (2010), 093506
4. A. Minakov, “Asymptotics of rarefaction wave solution to the mKdV equation”, Zhurn. matem. fiz., anal., geom., 7:1 (2011), 59–86
5. Minakov A., “Long-time behavior of the solution to the mKdV equation with step-like initial data”, J. Phys. A: Math. Theor., 44:8 (2011), 085206
6. Egorova I., Teschl G., “On the Cauchy Problem for the Kortewegde Vries Equation With Steplike Finite-Gap Initial Data II. Perturbations With Finite Moments”, J Anal Math, 115 (2011), 71–101
7. V. Kotlyarov, A. Minakov, “Step-initial function to the mKdV equation: hyper-elliptic long-time asymptotics of the solution”, Zhurn. matem. fiz., anal., geom., 8:1 (2012), 38–62
8. Samoilenko V.H., Samoilenko Yu.I., “Two-Phase Solitonlike Solutions of the Cauchy Problem For a Singularly Perturbed Korteweg-de-Vries Equation With Variable Coefficients”, Ukr. Math. J., 65:11 (2014), 1681–1697
9. Alexander E. Elbert, Sergey V. Zakharov, “Dispersive rarefaction wave with a large initial gradient”, Ural Math. J., 3:1 (2017), 33–43
10. Zhu J., Wang L., Qiao Zh., “Inverse Spectral Transform For the Ragnisco-Tu Equation With Heaviside Initial Condition”, J. Math. Anal. Appl., 474:1 (2019), 452–466
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