Integrable Nonlinear Evolution Equations on the HalfLine
Abstract
A rigorous methodology for the analysis of initialboundary value problems on the halfline, is applied to the nonlinear §(NLS), to the sineGordon (sG) in laboratory coordinates, and to the KortewegdeVries (KdV) with dominant surface tension. Decaying initial conditions as well as a smooth subset of the boundary values 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 , define the spectral functions {A(k),B(k)}. Define the function q(x,t) in terms of the solution of a matrix RiemannHilbert problem formulated in the complex kplane and uniquely defined in terms of the spectral functions {a(k),b(k),A(k),B(k)}. Under the assumption that there exist functions such that the spectral functions satisfy a certain global algebraic relation, prove that the function q(x,t) is defined for all , it satisfies the given nonlinear PDE, and furthermore that . (b) Given a subset of the functions 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: , respectively, where χ is a real constant.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 2002
 DOI:
 10.1007/s0022000206818
 Bibcode:
 2002CMaPh.230....1F