[Paper Review] Construction of multi-soliton solutions for the L2-supercritical gKdV and NLS equations

Raphaël Côte, Yvan Martel+1 more|arXiv (Cornell University)|May 4, 2009

Advanced Mathematical Physics Problems22 references4 citations

TL;DR

This paper constructs multi-soliton solutions for the $L^2$-supercritical generalized Korteweg-de Vries (gKdV) and nonlinear Schrödinger (NLS) equations using a topological argument to control unstable directions in the linearized operator. The authors prove the existence of solutions that asymptotically approach the sum of $N$ solitons with distinct speeds as $t \to +\infty$, with exponential convergence in $H^1$ norm, extending previous results from the subcritical and critical regimes to the supercritical case where solitons are inherently unstable.

ABSTRACT

Multi-soliton solutions, i.e. solutions behaving as the sum of N given solitons as $t o +\infty$, were constructed in previous works for the L2 critical and subcritical (NLS) and (gKdV) equations. In this paper, we extend the construction of multi-soliton solutions to the L2 supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability.

Motivation & Objective

• To extend the construction of multi-soliton solutions from the $L^2$-subcritical and critical regimes to the $L^2$-supercritical case for gKdV and NLS equations.
• To address the challenge of soliton instability in the $L^2$-supercritical regime ($p > 5$ for gKdV, $p > 1 + 4/d$ for NLS), where standard methods fail due to unstable directions in the linearized operator.
• To develop a topological argument that controls the unstable directions associated with the $L^2$-supercritical setting, enabling the existence proof of multi-solitons.

Proposed method

• A compactness argument is employed to construct solutions that asymptotically resemble the sum of $N$ solitons as $t \to +\infty$, building on prior work in the subcritical case.
• A topological argument is introduced to control the unstable directions of the linearized operator around each soliton, which are present in the $L^2$-supercritical regime.
• The construction relies on the existence of $L^2$ eigenfunctions $Y^\pm$ for the linearized operator, constructed via ODE techniques by Pego and Weinstein, which are linked to the negative derivative of the $L^2$ norm of the soliton profile.
• The proof uses a functional $\Psi$ defined on a space of parameters, whose invertibility (via the implicit function theorem) ensures the existence of a solution satisfying the desired asymptotic behavior.
• The method involves a careful analysis of the interaction between solitons, using uniform estimates and exponential decay in time to control error terms.
• The solution is shown to converge to the sum of solitons in $H^1$ norm with exponential rate $Ce^{-\sigma_0^{3/2}t}$, and the result extends to higher regularity spaces $H^s$ for $s \geq 1$.

Experimental results

Research questions

• RQ1 Can multi-soliton solutions be constructed for the $L^2$-supercritical gKdV equation when solitons are known to be unstable?
• RQ2 How can the unstable directions in the linearized operator around each soliton be controlled to ensure the existence of multi-solitons in the supercritical regime?
• RQ3 Is it possible to extend the existence result for multi-solitons beyond the $L^2$-subcritical and critical cases to the $L^2$-supercritical setting using a topological approach?
• RQ4 What role do the $L^2$ eigenfunctions $Y^\pm$ of the linearized operator play in stabilizing the multi-soliton structure in the supercritical case?
• RQ5 Does the asymptotic behavior of the solution in the supercritical case still exhibit exponential convergence to the sum of solitons, as in the subcritical regime?

Key findings

• The paper establishes the existence of multi-soliton solutions for the $L^2$-supercritical gKdV equation with $p > 5$, for any $N \geq 1$ and any choice of distinct speeds $0 < c_1 < \cdots < c_N$.
• The constructed solution $u(t)$ satisfies $\left\|u(t) - \sum_{j=1}^N R_{c_j,x_j}(t)\right\|_{H^1} \leq C e^{-\sigma_0^{3/2}t}$ for all $t \geq T_0$, proving exponential convergence to the sum of solitons.
• The topological argument is essential in the proof, as it controls the unstable directions of the linearized operator, which are present due to $\frac{d}{dc}\int Q_c^2\big|_{c=c_0} < 0$ in the supercritical case.
• The solution belongs to $H^s$ for all $s \geq 1$, and the convergence holds in $H^s$ norm, not just $H^1$, as shown by Proposition 5 in [16].
• The method relies on the existence of $L^2$ eigenfunctions $Y^\pm$ for the linearized operator, constructed by Pego and Weinstein, which are shown to be equivalent to the instability condition.
• The implicit function theorem is applied to a map $\Psi$ whose invertibility ensures the existence of a unique solution parameterized by initial data, with the Jacobian $d\Psi$ shown to be invertible via the Gramm matrix of the functions $Z_j^\pm$.