[Paper Review] Fast-moving finite and infinite trains of solitons for nonlinear Schr\\"odinger equations

Stefan Le Coz, Dong Li+1 more|arXiv (Cornell University)|Apr 10, 2013

Advanced Mathematical Physics Problems24 references21 citations

TL;DR

This paper establishes the existence and uniqueness of infinite soliton trains—solutions to the energy-subcritical nonlinear Schrödinger equation that asymptotically behave as a sum of infinitely many solitary waves—under conditions of high relative speeds and integrability of soliton parameters. The construction relies on a contraction argument in a tailored function space, leveraging exponential separation of fast-moving solitons to treat the nonlinear interaction as a perturbation.

ABSTRACT

We study *infinite soliton trains* solutions of nonlinear Schr\\"odinger equations (NLS), i.e. solutions behaving at large time as the sum of infinitely many solitary waves. Assuming the composing solitons have sufficiently large relative speeds, we prove the existence and uniqueness of such a soliton train. We also give a new construction of multi-solitons (i.e. finite trains) and prove uniqueness in an exponentially small neighborhood, and we consider the case of solutions composed of several solitons and kinks (i.e. solutions with a non-zero background at infinity).

Motivation & Objective

• To construct and prove the existence of infinite soliton train solutions for the nonlinear Schrödinger equation (NLS) in the energy-subcritical regime.
• To extend the theory of multi-solitons beyond finite trains by establishing rigorous existence for infinite sequences of solitons with high relative velocities.
• To provide a new framework for constructing multi-solitons and multi-kinks in non-integrable NLS settings using perturbation and contraction techniques.
• To analyze the asymptotic behavior of solutions that decompose into infinitely many solitons, addressing a key case in the Soliton Resolution Conjecture.
• To prove uniqueness in an exponentially small neighborhood of the soliton train profile, ensuring robustness of the solution structure.

Proposed method

• Construct the infinite soliton train $ R_\infty = \sum_{j=1}^\infty \tilde{R}_j $, where each $ \tilde{R}_j $ is a solitary wave with frequency $ \omega_j $, velocity $ v_j $, and phase $ \gamma_j $, using the standard soliton ansatz.
• Define the perturbation $ \eta = u - R_\infty $, leading to a Duhamel integral formulation for $ \eta $ that captures the nonlinear dynamics around the train profile.
• Apply a contraction argument in a Strichartz-type function space $ X([t,\infty)\times\mathbb{R}^d) $, using the exponential decay of soliton tails at high relative speeds to control nonlinear terms.
• Employ Hölder and Strichartz estimates to bound the nonlinear terms $ f(R_\infty + \eta) - f(R_\infty) $, relying on the integrability condition $ \sum_j \omega_j^{\frac{1}{\alpha} - \frac{d}{2r_1}} < \infty $ and high-speed separation $ \sqrt{\min\{\omega_j, \omega_k\}} |v_k - v_j| \geq v_* > 0 $.
• Use the fact that fast-moving solitons have exponentially decaying overlap, allowing the nonlinear interaction to be treated as a small perturbation in the contraction setting.
• Establish uniqueness by proving that the solution $ u $ is the unique fixed point of the perturbation equation in a sufficiently small, exponentially decaying neighborhood of $ R_\infty $.

Experimental results

Research questions

• RQ1 Can infinite trains of solitons exist as solutions to the energy-subcritical nonlinear Schrödinger equation?
• RQ2 Under what conditions on the soliton parameters (frequency, velocity, phase) does such an infinite train solution exist and remain unique?
• RQ3 How does the high relative speed between solitons enable the construction of such solutions despite the nonlinearity?
• RQ4 Can the method be extended to construct multi-kink solutions with non-zero background at infinity?
• RQ5 What are the necessary and sufficient conditions on the nonlinearity exponent $ \alpha $ and space dimension $ d $ for the existence of such solutions?

Key findings

• An infinite soliton train solution exists for the NLS equation when the solitons have sufficiently large relative speeds and the parameter sequence $ \omega_j $ satisfies the integrability condition $ \sum_j \omega_j^{\frac{1}{\alpha} - \frac{d}{2r_1}} < \infty $ for some $ r_1 \in (\frac{d\alpha}{2}, \alpha + 2) $.
• The solution $ u $ asymptotically approaches the infinite train $ R_\infty $ in the sense $ \lim_{t \to \infty} \|u - R_\infty\|_{X([t,\infty)\times\mathbb{R}^d)} = 0 $, with $ X $ being a suitable Strichartz-type norm.
• Uniqueness of the solution is proven in an exponentially small neighborhood of $ R_\infty $, ensuring the solution is robust under small perturbations.
• The construction is flexible and extends to multi-kink solutions, where the background at infinity is non-zero, by modifying the profile and perturbation equation accordingly.
• For the case of power nonlinearity $ f(u) = |u|^\alpha u $, the method works for all $ \alpha \in (0, \alpha_{\max}) $, with $ \alpha_{\max} = \infty $ for $ d = 1,2 $ and $ \alpha_{\max} = \frac{4}{d-2} $ for $ d \geq 3 $.
• The existence region for the exponents $ \beta_1, \beta_2 $ in the nonlinear term estimates is characterized by the curves $ \Gamma(r_1) $ and $ \Sigma(r_2) $, with the intersection point determining admissible parameters for the Strichartz estimates.