On the Korteweg-de Vries Long-Wave Approximation of the Gross-Pitaevskii Equation I

Abstract

The fact that the Korteweg-de-Vries equation offers a good approximation of long-wave solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation was derived several years ago in the physical literature (see e.g. [17]). In this paper, we provide a rigorous proof of this fact, and compute a precise estimate for the error term. Our proof relies on the integrability of both the equations. In particular, we give a relation between the invariants of the two equations, which, we hope, is of independent interest.

Nongauge bright soliton of the nonlinear Schrödinger (NLS) equation and a family of generalized NLS equations

“…That is the case of the KdV [9] and the related BBM equation [10], which are well-established mathematical models of waves on shallow water surfaces and known to be connected to the standard NLS equation only by approximations [11][12][13][14]. Here, we show that there exists an exact relationship between these equations and the equation of parameter p = 2 from the family of generalized NLS equations…”

Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity: some results and open problems

“…It has been conjectured that, after a suitable anisotropic rescaling and up to a subsequence, the functions r 2 0 − |ψ cn | 2 and the phases of ψ cn should tend to solitary waves of the Korteweg-de Vries equation in one dimension, respectively to the solitary waves of the Kadomtsev-Petviashvili I equation in dimensions two and three. In the case of the Gross-Pitaevskii nonlinearity, this has been proved in [BGSS09] in dimension one, respectively in [BGS08] in dimension two for those solutions that minimize the energy at fixed momentum. All other cases XIV-4 are still unknown (a major difficulty in proving this convergence seems to be the obtention of good estimates on the energy of traveling waves).…”

Error bounds for the (KdV)/(KP-I) and (gKdV)/(gKP-I) asymptotic regime for nonlinear Schrödinger type equations

“…The first rigorous justifications of this long wave asymptotic regime for (NLS) are given in the papers [13] and [14], which work on the Gross-Pitaevskii equation in dimension d = 1. The point is that this equation is integrable, and these results rely on the higher order conservation laws of (GP).…”

“…The point is that this equation is integrable, and these results rely on the higher order conservation laws of (GP). For (GP) in dimension d = 1, the Cauchy problem is known (see [13]) to be globally well-posed (see also [56], [27], [28]) in the Zhidkov space Z σ (R) ≡ {v ∈ L ∞ (R), ∂ x v ∈ H σ−1 (R)}, where σ is a positive integer. We recall the main results of [13] and [14].…”

“…This is in particular due to the fact that the use, in [13], of the first three pairs of nontrivial conservation laws for (GP) yields…”