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

This article is cited in 10 scientific papers (total in 10 papers)

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

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Received: 23.06.2000

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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    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  crossref  mathscinet  zmath  adsnasa  isi
    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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    4. A. Minakov, “Asymptotics of rarefaction wave solution to the mKdV equation”, Zhurn. matem. fiz., anal., geom., 7:1 (2011), 59–86  mathnet  mathscinet  zmath  elib
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    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  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    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  mathnet  mathscinet  zmath
    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  crossref  mathscinet  zmath  isi  scopus
    9. Alexander E. Elbert, Sergey V. Zakharov, “Dispersive rarefaction wave with a large initial gradient”, Ural Math. J., 3:1 (2017), 33–43  mathnet  crossref
    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  crossref  mathscinet  zmath  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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