Large Time Asymptotics of Solutions to the Generalized Korteweg–de Vries Equation

✍ Scribed by Nakao Hayashi; Pavel I. Naumkin

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 389 KB
• Volume: 159
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3291

✦ Synopsis

We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Korteweg de Vries (gKdV) equation u t + ( |u| &1 u) x + 1 3 u xxx =0, where x, t # R when the initial data are small enough. If the power \ of the nonlinearity is greater than 3 then the solution of the Cauchy problem has a quasilinear asymptotic behavior for large time. More precisely, we show that the solution u(t) satisfies the decay estimate &u(t)& L ; C(1+t) &(1Â3)(1&1Â;) for ; # (4, ], &uu x (t)& L Ct &2Â3 (1+t) &1Â3 and using these estimates we prove the existence of the scattering state

for any small initial data belonging to the weighted Sobolev space H

x ) 1Â2 f& L 2 < ], where U(t) is the Airy free evolution group.