Provable bounds for the Korteweg–de Vries reduction in multi-component nonlinear Schrödinger equation

Abstract

We study the dynamics of multi-component Bose gas described by the Vector Nonlinear Schrödinger Equation (VNLS), aka the Vector Gross-Pitaevskii Equation (VGPE) . Through a Madelung transformation, the VNLS can be reduced to coupled hydrodynamic equations in terms of multiple density and velocity fields. Using a multi-scaling and a perturbation method along with the Fredholm alternative, we reduce the problem to a Korteweg de-Vries (KdV) system. This is of great importance to study more transparently, the obscure features hidden in VNLS. This ensures that hydrodynamic effects such as dispersion and nonlinearity are captured at an equal footing. Importantly, before studying the KdV connection, we provide a rigorous analysis of the linear problem. We write down a set of theorems along with proofs and associated corollaries that shine light on the conditions of existence and nature of eigenvalues and eigenvectors of the linear problem. This rigorous analysis is paramount for understanding the nonlinear problem and the KdV connection. We provide strong evidence of agreement between VNLS systems and KdV equations by using soliton solutions as a platform for comparison. Our results are expected to be relevant not only for cold atomic gases, but also for nonlinear optics and other branches where VNLS equations play a defining role.I.

Soliton-like behaviour in non-integrable systems

“…1 into suitable hydrodynamic variables (density and velocity fields). We then recap relevant results from an earlier work [48] and present new results on the spectral analysis. In Sec.…”

“…• The MNLS-to-KdV reduction is based on a detailed spectral analysis of the linear version of MNLS and in the current work we address issues that were left open from an earlier work [48]. For example, we establish explicit expressions for eigenvectors which gives an analytical handle on deriving the coefficients of the effective chiral field theory.…”

“…10 is an explicit representation for the simple eigenvector associated with eigenvalue λ 2 for the matrix ρα. From our previous results [48], we have explicit formulae for elements of the eigenvector. Indeed…”

“…Note ρ is an N × N diagonal matrix with strictly positive elements ρ 0k (the background densities) and δρ , δv are the N × 1 column vectors for the perturbations in density and velocity. The matrix α is given by [48] αij = gi , i = j, h, i = j. where gi = g i + 2G i ρ 0i…”

“…The spectral analysis of the 2N × 2N matrix A (Eq. 3) can be accomplished by a spectral analysis of the N × N matrix αρ [48]. Indeed, under the assumption that α is symmetric positive definite (for α as defined above this is true when h < min gi [48]), all eigenvalues of αρ are real and positive which we denote by λ 2 .…”