Geodesic
Initial boundary value problems for the nonlinear Schrödinger equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 122 (2000) no. 1, pp. 128-143 Cet article a éte moissonné depuis la source Math-Net.Ru

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A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrцdinger equation and to its linearized version in the domain $\{x\geq l(t),t\geq0\}$. We show that there exist two cases: a) if $l''(t)<0$, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann–Hilbert problem, which is defined on a time-dependent contour; b) if $l''(t)>0$, then the Riemann–Hilbert problem is replaced by a respective scalar or matrix $\overline\partial$ problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form. 
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     author = {B. Pelloni},
     title = {Initial boundary value problems for the nonlinear {Schr\"odinger} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {128--143},
     year = {2000},
     volume = {122},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2000_122_1_a10/}
}
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B. Pelloni. Initial boundary value problems for the nonlinear Schrödinger equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 122 (2000) no. 1, pp. 128-143. http://geodesic.mathdoc.fr/item/TMF_2000_122_1_a10/

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