Asymptotic Behavior of Nonsoliton Solutions of the Variable Coefficient and Nonisospectral Korteweg-de Vries Equation

We consider the initial value problem of the variable coefficient and nonisospectral Korteweg-de Vries equation with variable boundary condition and smooth initial data decaying rapidly to zero as |x| → ∞. Using the method of inverse scattering we study the asymptotic behavior of the solution u(x, t) in the coordinate regions (1) t ≥ t0, x ≥ −μ + νt; (2) t ≥ tc, x≥−μ − {νT − 4[(3/2) L(0) F(K0, 3h, t) + F(K1, h, t)]} exp(−∫t0h dt), where μ, ν, t0, tc are nonnegative constants; T = [3F(K0, 3h, t)]1/3, F(χ, κ, t) = ∫t0 [χ(s) exp(∫s0 κ dt)] ds. It is shown that the bounds for the nonsoliton parts of the solutions depend on x and t. They decay to zero in the above regions as t becomes large.