Abstract

Soliton solutions and the tau function of the KdV equation are studied within the stochastic analytic framework. A key role is played by the It^ o formula and the Cameron-Martin transformation. x Introduction In this paper, we investigate the Korteweg-de Vries (KdV) equation within the framework of stochastic analysis. We shall study soliton solutions with the help of the It^ o formula, whose original form was achieved in 1942 ([9]). The Cameron-Martin transformation, which was established in the early 1940’s ([2, 3]), also plays a key role. Let x > 0 andW n be the space of R n -valued continuous functions on [0;x] starting at the origin, and let P be the Wiener measure onW n . Following the idea of Cameron-Martin [3], we can show that

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