Construction of multi-soliton solutions for the \(L^2\)-supercritical gKdV and NLS equations. (English) Zbl 1273.35234

The paper proves the existence of multi-soliton solutions of the generalized Korteweg-de Vries (gKdV) and the nonlinear Schrödinger (NLS) equations in the \(L^2\)-supercritical case. The gKdV problem reads \[ u_t +(u_{xx}+u^p)_x=0, \quad (x,t)\in \mathbb{R}^2, p>5, \] and the NLS problem is \[ i u_t +\Delta u +|u|^{p-1}u=0, \quad (x,t)\in \mathbb{R}^{d+1}, p\in \left(1+\tfrac{4}{d},\tfrac{d+2}{d-2}\right), \] where \(\tfrac{d+2}{d-2}\) is replaced by \(\infty\) for \(d=1,2\).
In these regimes soliton solutions of both problems are known to be unstable. Multi-soliton solutions are solutions which at large times behave like the sum of \(N\) given solitons. In more detail, the paper proves that given \(N\) values of speed and spatial shift, a multi-soliton exists which converges to the sum of \(N\) solitons with these speeds and shifts as \(t\to\infty\).
The idea of the proof for the gKdV is the following. In the first step solutions \(u_n\) of gKdV are proved for a sequence of time intervals \(I_n=[T_0,S_n]\) with \(S_n\to\infty\) such that at \(t=S_n\) the solution is a sum of \(N\) solitons plus a correction term given by eigenfunctions of \(L\partial_x\), where \(L\) is the energy Hessian around a soliton. A crucial uniform (in \(n\)) \(H^1\)-estimate of the difference of \(u_n\) and the sum of \(N\) solitons is obtained next. The difference converges to zero exponentially fast as \(t\to\infty\). Finally by a compactness argument \(u_n(t)\) is shown to converge strongly to a gKdV solution \(u^*(t)\in H^{1/2}(\mathbb{R})\) and weakly in \(H^1(\mathbb{R})\). By ruling out the possibility of blowup the solution \(u^*\) exists on \([T_0,\infty)\). The above estimate then shows that \(u^*\) converges to the sum of \(N\) solitons exponentially fast as \(t\to\infty\). The proof for the NLS is analogous.
This work is an extension of [F. Merle, Commun. Math. Phys. 129, No. 2, 223–240 (1990; Zbl 0707.35021)], [Y. Martel, Am. J. Math. 127, No. 5, 1103–1140 (2005; Zbl 1090.35158)] and [Y. Martel and F. Merle, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23, No. 6, 849–864 (2006; Zbl 1133.35093)].


35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text: DOI arXiv Euclid


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