Quasiclassical integral equations and the asymptotic behavior of solutions to the Korteweg-de Vries equation for large times. (English. Russian original) Zbl 0956.35114

Budylin, A. M.; Buslaev, V. S.

Dokl. Math. 53, No. 3, 441-444 (1996); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 348, No. 4, 455-458 (1996).

The authors study the asymptotic behavior of the solution \(u(x,t)\) for \(t\to\infty\) to the Cauchy problem to the Korteweg-de Vries equation \[ u_t= u_{xxx}+ 6uu_x,\quad u|_{t= 0}= u_0,\quad u\in\mathbb{R},\quad t\geq 0 \] using its well-known connection to the spectral problem for the Schrödinger equation \[ \psi_{xx}+ v\psi+ k^2\psi= 0 \] (inverse problem method).

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
37K15	Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems

Keywords:

inverse problem method; asymptotic behavior; Cauchy problem; Korteweg-de Vries equation; spectral problem; Schrödinger equation