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By Bernd (d-fsu-mi) Huang, M. J. Ablowitz, P. A. Clarkson, Nonlinear Evolution Equations, R. Bhata and P. Rosenthal


On realizations of solutions of the KdV equation by the C0-semigroup method. (English summary) Amer. J. Math. 122 (2000), no. 2, 403–438. This paper uses an operator approach to construct solutions of the KdV equation. The KdV equation is first replaced by a related equation for a function taking values in the space of (unbounded) linear operators on some Banach space. Solutions of this operator-valued equation are constructed. If these solutions are of rank 1, the authors show that the trace of this operator-valued solution solves the integrated form of the original scalar KdV equation. Many technical difficulties connected to the unboundedness of the operators must be overcome in the construction. Several new types of solutions are constructed by this procedure, including some solutions of the KdV equation in the quarter plane {(x, t) | x> 0, t> 0} which decay exponentially in space and disappear after a finite time. Reviewed by C. Eugene Wayn

Year: 2013
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