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Some Problems for Cubic Nonlinear Equations on a Half-Line
Authors:Pham Loi Vu
Abstract:The paper deals with a problem of developing an inverse-scattering based formalism for solving problems for the cubic nonlinear (or the modified Korteweg–de Vries (KdV)) equations: q t +q xxx +6q 2 q x =0, 0lex<infin, –infin<t<infin,q t +q xxx –6q 2 q x =0, with the given initial and boundary conditions: q(x,0)=q(x),q(0,t)=p(t), p(t)isinL 1(–infin,infin). The relation between the solution of the initial-boundary value problem (1), (3), (4) and that of the KdV equation on the half-line is shown. The Cauchy problem for the cubic nonlinear equation: q t +q xxx –6|q|2 q x =0, 0lex<infin, –infin<t<infin, with the given initial condition (3) is considered also. Here we solve the above problems on the half-line 0lex<infin but with –infin<t<infin.
Keywords:initial-boundary value problems for the modified Korteweg–  de Vries equations  the presence of unknown boundary values of solutions at the origin x=0
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