Note on nonlinear Schrödinger equation, KdV equation and 2D topological Yang–Mills–Higgs theory

Abstract

In this paper we discuss the relation between the (1+1)D nonlinear Schrödinger equation and the KdV equation. By applying the boson/vortex duality, we can map the classical nonlinear Schrödinger equation into the classical KdV equation in the small coupling limit, which corresponds to the UV regime of the theory. At quantum level, the two theories satisfy the Bethe Ansatz equations of the spin-1 2 XXX chain and the XXZ chain in the continuum limit respectively. Combining these relations with the dualities discussed previously in the literature, we propose a duality web in the UV regime among the nonlinear Schrödinger equation, the KdV equation and the 2D N = (2, 2) * topological Yang-Mills-Higgs theory.

Dark solitons, D-branes and noncommutative tachyon field theory

“…As mentioned in Section 3, the duality can be generalized to other dimensions. Since the (1+1)D Gross-Pitaevskii equation, also called the nonlinear Schrödinger equation, is an integrable model, we expect the integrability should be maintained in the dual theory [32]. Also, the (1+1)D nonlinear Schrödinger equation is dual to a 2D topological Yang-Mills-Higgs model at quantum level [33].…”

On Quasi-integrable Deformation Scheme of The KdV System

“…At the simplest level, such deformations lead to scaling of the KdV parameters and thus retaining integrability. In the perturbative domain, as the KdV and NLS systems are related through a weak-coupling map [8,9] between the solutions, we obtain a map between this quasi-KdV and the known quasi-NLS results [16]. We further infer about the connection of the quasi-KdV system to its non-holonomic (NH) deformation [18], the latter retaining integrability.…”

“…Moreover the usual Lax representation [7] of KdV system involves second and third order monic differential operators (L, A), whereas that of the NLS system is given by 2 × 2 matrices [2]. However, it is known that in a suitable weak-coupling limit, their respective solutions map into each-other [8,9], including their soliton solutions.…”

“…It was further found that the anomaly functions corresponding to the deformed curvature and that for the anomalous charge evolution have definite parity properties essential for the asymptotic integrability of the system. Owing to the closeness with the NLS system [8,9], the KdV system is also expected to display such quasi-integrability. However this third order equation in space eludes a direct dynamical interpretation in the usual classical sense and there are no 'potential' analogue here.…”