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

✍ Scribed by W.L. Chan; K.S. Li

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 662 KB
• Volume: 121
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1994.1055

✦ Synopsis

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| \rightarrow \infty). Using the method of inverse scattering we study the asymptotic behavior of the solution (u(x, t)) in the coordinate regions (1) (t \geqslant t_{0}, x \geqslant-\mu+v t ;(2) t \geqslant t_{c}, x \geqslant-\mu-\left{v \mathscr{T}-4\left[(3 / 2) L(0) F\left(K_{0}, 3 h, t\right)\right.\right.) (\left.\left.+F\left(K_{1}, h, t\right)\right]\right} \exp \left(-\int_{0}^{t} h d t\right)), where (\mu, v, t_{0}, t_{c}) are nonnegative constants; (\mathscr{T}=) (\left[3 F\left(K_{0}, 3 h, t\right)\right]^{1 / 3}, F(\chi, \kappa, t)=\int_{0}^{t}\left[\chi(s) \exp \left(\int_{0}^{s} \kappa d t\right)\right] d s). 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.