Abstract

The C0-semigroup method is used to find exact solutions of the Korteweg-de Vries (KdV) equation ut = uxxx + 6uux. Various new kinds of solutions such as multisoliton solutions with unbounded phase numbers, "null" solutions and solutions of "wavelet-type" are found. The realization program consists of two steps. Firstly, the scalar KdV equation will be translated into an operator equation Ut = Uxxx +3(UU)x on the Banach algebra of a given Banach space. The solution formula for the operator equation has been constructed (Proposition 2.1). Secondly, once all of the operators U(x, t) are of rank one, solutions of the scalar KdV equation will be extracted by using the trace functional. The use of fractional powers of a closed unbounded linear operator in a Banach space also plays an essential role in the present work.

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