Computer-Assisted Proofs of Gap Solitons in Bose-Einstein Condensates

TL;DR

The paper presents a computer-assisted framework to obtain rigorous existence proofs for gap solitons in the one-dimensional Gross–Pitaevskii equation with periodic potentials by reformulating the problem as finding a homoclinic orbit to a periodic orbit in a four-dimensional autonomous system ($u'' + a u - V(x)u - c u^3 = 0$ from the standing-wave ansatz $\psi(t,x)=e^{-iat}u(x)$). It constructs a local stable manifold using the Parameterization Method and certifies nearby true solutions through Newton–Kantorovich-type interval arithmetic bounds, yielding explicit $C^0$-error bounds between the exact soliton and a numerical approximation. The first computer-assisted proofs of gap solitons in the GP equation are achieved in non-perturbative parameter regimes, and the method is adaptable to broad classes of periodic potentials. Open-source Julia code implementing interval arithmetic bounds makes the approach accessible for verifying solitons under various parameter sets.

Abstract

We provide a framework for turning a numerical simulation of a gap soliton in the one-dimensional Gross-Pitaevskii equation into a rigorous mathematical proof of its existence. These nonlinear localized solutions play a central role in the study of Bose-Einstein condensates (BECs). We reformulate the problem of proving their existence as the search for homoclinic orbits in a dynamical system. We then apply computer-assisted proof techniques to obtain verifiable conditions under which a numerically approximated trajectory corresponds to a true homoclinic orbit. This work also presents the first examples of computer-assisted proofs of gap solitons in the Gross-Pitaevskii equation on non-perturbative parameter regimes.

Figures (2)

• Figure 1: The figure shows a validated soliton solution of the Gross-Pitaevskii (GP) equation with parameters $a = 1.1025$, $b = 0.55125$, and $c = -0.826875$. It also depicts the main elements of our approach: the solution to the boundary-value problem (blue), the stable manifold (orange), and the even extension of the soliton (black).
• Figure 2: Computer-assisted proofs for numerically approximated solitons originally presented in Alfimov_2002. The parameters of the equation are shown at the top of the figure. The illustration also shows the main components of our approach: the solution to the boundary-value problem (blue), the stable manifold (orange), and the even extension of the soliton (black).

Theorems & Definitions (34)

• Theorem 1
• Theorem 2: Newton-Kantorovich Theorem
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof