ASYMPTOTIC-BEHAVIOR OF NONSOLITON SOLUTIONS OF THE VARIABLE-COEFFICIENT AND NONISOSPECTRAL KORTEWEG-DEVRIES 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 Absolute value of x --> infinity. Using the method of inverse scattering we study the asymptotic behavior of the solution u(x, t) in the coordinate regions (1) t greater-than-or-equal-to t0, x greater-than-or-equal-to -mu + nut; (2) t greater-than-or-equal-to t(c), x greater-than-or-equal-to -mu - {nuT - 4[(3/2) L(0) F(K0, 3h, t) + F(K1, h, t)]} exp(-integral-t/0 h dt), where mu, nu, t0, t(c) are nonnegative constants; T = [3F(K0, 3h, t)]1/3, F(chi, kappa, t) = integral-t/0 [chi(s) exp(integral-s/0 kappa 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. (C) 1994 Academic Press, Inc.