The initial boundary problem for the Korteweg-de Vries equation on the negative quarter-plane

The initial boundary problem for the Korteweg-de Vries equation on the negative quarter-plane

The initial boundary-value problem for the Korteweg-de Vries (KdV) equation on the negative quarter-plane, x < 0 and t > 0, is considered. The formulation of this problem is different to the usual initial boundary-value problem on the positive quarter-plane, for which x > 0 and t > 0. Tw...

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Veröffentlicht in:Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Mathematical, physical, and engineering sciences, 2002-04, Vol.458 (2020), p.857-871
Hauptverfasser: Marchant, T. R., Smyth, N. F.
Format: Artikel
Sprache:eng
Schlagworte:
Amplitude
Approximate values
Boundary conditions
Boundary value problems
Cnoidal waves
Initial Boundary-Value Problems
Inverse scattering
KdV equation
Mathematics
Modulation Theory
Solitons
Trailing edges
Waves
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container_title Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences
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creator Marchant, T. R.
Smyth, N. F.
description The initial boundary-value problem for the Korteweg-de Vries (KdV) equation on the negative quarter-plane, x < 0 and t > 0, is considered. The formulation of this problem is different to the usual initial boundary-value problem on the positive quarter-plane, for which x > 0 and t > 0. Two boundary conditions are required at x = 0 for the negative quarter-plane problem, in contrast to the one boundary condition needed at x = 0 for the positive quarter-plane problem. Solutions of the KdV equation on the infinite line, such as the soliton, cnoidal wave, mean height variation and undular bore solution, are used to find approximate solutions to the negative quarter-plane problem. Five qualitatively different types of solution are found, depending on the relation between the initial and boundary values. Excellent comparisons are obtained between these solutions and full numerical solutions of the KdV equation.
doi_str_mv 10.1098/rspa.2001.0868
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subjects Amplitude
Approximate values
Boundary conditions
Boundary value problems
Cnoidal waves
Initial Boundary-Value Problems
Inverse scattering
KdV equation
Mathematics
Modulation Theory
Solitons
Trailing edges
Waves
title The initial boundary problem for the Korteweg-de Vries equation on the negative quarter-plane
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