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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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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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http://mi.mathnet.ru/eng/tmf426https://doi.org/10.4213/tmf426 http://mi.mathnet.ru/eng/tmf/v126/i2/p214
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