KdV Equation With Nontrivial Boundary Conditions at x → ±∞

Abstract

Let us consider KdV equation

$${{u}_{t}} - 6u{{u}_{x}} + {{u}_{{xxx}}} = 0$$

(1)

with initial condition u(x, 0), which is a smooth function of x, rapidly (for simplicity - in the sense of Shwartz class) approaching two different limits as x → ±∞:

$$u\left( {x,0} \right) \to v\pm \left( {x,0|\Gamma \pm ,D\pm } \right),x \to \pm \infty$$

(2)