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Integrable nonlinear evolution equations on the half-line. (English) Zbl 1010.35089
Summary: A rigorous methodology for the analysis of initial-boundary value problems on the half-line, $$0<x<\infty$$, $$t>0$$, is applied to the nonlinear Schrödinger (NLS), to the sine-Gordon (sG) in laboratory coordinates, and to the Korteweg-deVries (KdV) equation with dominant surface tension. Decaying initial conditions as well as a smooth subset of the boundary values $$\{\partial_x^l q(0,t) = g_l(t)\}^{n-1}_0$$ are given, where $$n = 2$$ for the NLS and the sG and $$n = 3$$ for the KdV. For the NLS and the KdV equations, the initial condition $$q(x,0) = q_{0}(x)$$ as well as one and two boundary conditions are given respectively; for the sG equation the initial conditions $$q(x,0) = q_{0}(x), q_{t}(x,0) = q_{1}(x)$$, as well as one boundary condition are given. The construction of the solution $$q(x,t)$$ of any of these problems involves two separate steps:
(a) Given decaying initial conditions define the spectral (scattering) functions $${a(k),b(k)}$$. Associated with the smooth functions $$\{ g_l(t) \}^{n-1}_0$$, define the spectral functions $${A(k),B(k)}$$. Define the function $$q(x,t)$$ in terms of the solution of a matrix Riemann-Hilbert problem formulated in the complex $$k$$-plane and uniquely defined in terms of the spectral functions $$\{a(k),b(k),A(k),B(k)\}$$. Under the assumption that there exist functions $$\{g_l(t)\}^{n-1}_0$$ such that the spectral functions satisfy a certain global algebraic relation, prove that the function $$q(x,t)$$ is defined for all $$0<x<\infty$$, $$t>0$$, it satisfies the given nonlinear PDE, and furthermore that $$q(x,0) = q_0(x),\{\partial^l_xq(0,t) = g_l(t)\}^{n-1}_0$$.
(b) Given a subset of the functions $$\{g_l(t)\}^{n-1}_0$$ as boundary conditions, prove that the above algebraic relation characterizes the unknown part of this set. In general this involves the solution of a nonlinear Volterra integral equation which is shown to have a global solution. For a particular class of boundary conditions, called linearizable, this nonlinear equation can be bypassed and $$\{A(k),B(k)\}$$ can be constructed using only the algebraic manipulation of the global relation. For the NLS, the sG, and the KdV, the following particular linearizable cases are solved: $$q_x(0,t)- \chi q(0,t) = 0$$, $$q(0,t) = \chi$$, $$\{ q(0,t) = \chi$$, $$q_{xx}(0,t) = \chi+3\chi^2\}$$, respectively, where $$\chi$$ is a real constant.

##### MSC:
 35Q53 KdV equations (Korteweg-de Vries equations) 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q15 Riemann-Hilbert problems in context of PDEs 37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
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