Geodesic
Initial Boundary Value Problem for the KdV Equation on a~Semiaxis with Homogeneous Boundary Conditions
Teoretičeskaâ i matematičeskaâ fizika, Tome 130 (2002) no. 1, pp. 31-53

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We consider the Korteweg–de Vries equation on the semiaxis with zero boundary conditions at $x=0$ and arbitrary smooth decreasing initial data. We show that the problem can be effectively integrated by the inverse scattering transform method if the associated linear equation has no discrete spectrum. Under these assumptions, we prove the global solvability of the problem. 
@article{TMF_2002_130_1_a2,
     author = {I. T. Habibullin},
     title = {Initial {Boundary} {Value} {Problem} for the {KdV} {Equation} on {a~Semiaxis} with {Homogeneous} {Boundary} {Conditions}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {31--53},
     publisher = {mathdoc},
     volume = {130},
     number = {1},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2002_130_1_a2/}
}
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I. T. Habibullin. Initial Boundary Value Problem for the KdV Equation on a~Semiaxis with Homogeneous Boundary Conditions. Teoretičeskaâ i matematičeskaâ fizika, Tome 130 (2002) no. 1, pp. 31-53. http://geodesic.mathdoc.fr/item/TMF_2002_130_1_a2/