Volume 14, Issue 1, February 2007, Pages 28 - 43
The Initial-Boundary Value Poblem for the Korteweg-de Vries Equation on the Positive Quarter-Plane
Pham Loi Vu
Pham Loi Vu
Received 6 March 2006, Accepted 24 August 2006, Available Online 1 February 2007.
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- The paper deals with a problem of developing an inverse-scattering transform for solving the initial-boundary value problem (IBVP) for the Korteweg-de Vries equation on the positive quarter-plane: pt - 6ppx + pxxx = 0, x 0, t 0, (a) with the given initial and boundary conditions: p(x, 0) = p(x), p(x) is a real-valued rapidly decreasing function, (b) p(0, t) = f(t), f(t) is a real-valued continuous function. (c) The Sturm-Liouville scattering problem (SP) in the interval (0, b) (b is a large positive number) generated by the linear Schrödinger equation (LSEq) with the zero boundary conditions (BCs) at x = 0 and at x = b is regarded as the linear problem associated with the IBVP (a)-(c). The time dependencies of the scattering data set of the SP are determined by the unknown boundary values (BVs) evaluated at x = 0 of the Jost solution of the LSEq. To overcome the difficulty we derive the asymptotic equation for the normalization eigenfunction of the Sturm-Liouville SP. This allows one to show the approximate time-independence of the scattering phase. Then, from the evolution equation for the scattering phase we deduce the asymptotic formulas for caculating the unknown BVs. We prove that the potential p(x, t) in the LSEq is uniquely found from the solution of the inverse SP in terms of the given data (b) and (c) and therefore, p(x, t) is a solution of the IBVP (a)-(c). Every solution of the problem (a)-(c) corresponds to an unique scattering data set and evolves from the continuous and discrete spectrum of the SP.
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Cite this article
TY - JOUR AU - Pham Loi Vu PY - 2007 DA - 2007/02 TI - The Initial-Boundary Value Poblem for the Korteweg-de Vries Equation on the Positive Quarter-Plane JO - Journal of Nonlinear Mathematical Physics SP - 28 EP - 43 VL - 14 IS - 1 SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.2007.14.1.4 DO - https://doi.org/10.2991/jnmp.2007.14.1.4 ID - Vu2007 ER -