Construction and characterization of solutions converging to solitons for supercritical gKdV equations. (English) Zbl 1240.35433

Combet, Vianney

Differ. Integral Equ. 23, No. 5-6, 513-568 (2010).

Summary: We consider the generalized Korteweg-de Vries equation \(\partial _{t}u +\partial _{x}^3u + \partial _{x}(u^{p})=0,\; (t,x)\in \mathbb {R}^2,\) in the supercritical case \(p>5\), and we are interested in solutions which converge to a soliton in large time in \(H^l\). In the subcritical case (\(p<5\)), such solutions are forced to be exactly solitons by variational characterization ([J. L. Bona, P. E. Souganidis and W. A. Strauss, Proc. R. Soc. Lond., Ser. A 411, 395–412 (1987; Zbl 0648.76005)], [M. I. Weinstein, Commun. Pure Appl. Math. 39, 51–67 (1986; Zbl 0594.35005)]), but no such result exists in the supercritical case. In this paper, we first construct a “special solution” in this case by a compactness argument, i.e., a solution which converges to a soliton without being a soliton. Secondly, using a description of the spectrum of the linearized operator around a soliton [R. L. Pego and M. I. Weinstein, Philos. Trans. R. Soc. Lond., Ser. A 340, No. 1656, 47–94 (1992; Zbl 0776.35065], we construct a one-parameter family of special solutions which characterizes all such special solutions. In the case of the nonlinear Schrödinger equation, a similar result was proved in [T. Duyckaerts and F. Merle, Geom. Funct. Anal. 18, No. 6, 1787–1840 (2008; Zbl 1232.35150)] and [T. Duyckaerts and S. Roudenko, Rev. Mat. Iberoam. 26, No. 1, 1–56 (2010; Zbl 1195.35276)].

MSC:

35Q51	Soliton equations
35Q53	KdV equations (Korteweg-de Vries equations)
35B40	Asymptotic behavior of solutions to PDEs
35C08	Soliton solutions
37K40	Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems

Keywords:

generalized Korteweg-de Vries equation; soliton; nonlinear Schrödinger equation

Citations:

Zbl 0648.76005; Zbl 0594.35005; Zbl 0776.35065; Zbl 1195.35276; Zbl 1232.35150