[Paper Review] Exact solutions of multicomponent nonlinear Schrödinger equations under general plane-wave boundary conditions

Takayuki Tsuchida|arXiv (Cornell University)|Aug 29, 2013

Nonlinear Waves and Solitons86 references10 citations

TL;DR

This paper presents a novel method to construct exact multicomponent soliton solutions of integrable nonlinear Schrödinger equations under general nonvanishing plane-wave boundary conditions, where different components approach plane waves with distinct wavenumbers and frequencies. By applying Bäcklund–Darboux transformations and introducing a re-parametrization of the non-diagonal matrix in the Lax pair, the authors derive explicit, closed-form expressions for matrix exponentials, enabling the construction of bright and dark solitons with internal degrees of freedom—offering a more explicit and tractable alternative to prior methods relying on Cardano’s formula.

ABSTRACT

We construct exact soliton solutions of integrable multicomponent nonlinear Schrödinger (NLS) equations under general nonvanishing boundary conditions. Different components of the vector (or matrix) dependent variable can approach plane waves with different wavenumbers and frequencies at spatial infinity. We apply Bäcklund-Darboux transformations to the cubic NLS equations with a self-focusing nonlinearity, a self-defocusing nonlinearity or a mixed focusing-defocusing nonlinearity. Both bright-soliton solutions and dark-soliton solutions are obtained, depending on the signs of the nonlinear terms and the type of Bäcklund-Darboux transformation. The multicomponent solitons generally possess internal degrees of freedom and provide highly nontrivial generalizations of the scalar NLS solitons. The main step in the construction of the multicomponent solitons is to compute the matrix exponential of a constant non-diagonal matrix arising from the Lax pair. With a suitable re-parametrization of the non-diagonal matrix, the matrix exponential can be computed explicitly in closed form for the most interesting cases such as the two-component vector NLS equation. In particular, we do not resort to Cardano's formula in diagonalizing a $3 imes 3$ matrix, so our expressions for the multicomponent solitons are in some sense more explicit and useful than those obtained in [Q-H. Park and H. J. Shin, Phys. Rev. E 61 (2000) 3093].

Motivation & Objective

• To construct exact soliton solutions of multicomponent nonlinear Schrödinger equations under general nonvanishing boundary conditions, where different components approach plane waves with distinct wavenumbers and frequencies.
• To overcome the limitations of the inverse scattering method in the non-degenerate case, where background plane waves differ across components.
• To develop a more explicit and computationally tractable method for computing matrix exponentials arising from the Lax pair, avoiding the use of Cardano’s formula for cubic roots.
• To derive both bright and dark soliton solutions with internal degrees of freedom, generalizing scalar NLS solitons in a nontrivial way.
• To provide a systematic framework for constructing one-soliton solutions using Bäcklund–Darboux transformations, with potential extension to multisoliton solutions.

Proposed method

• Application of elementary and binary Bäcklund–Darboux transformations to generate new solutions from seed solutions of the multicomponent NLS equation.
• Use of the Lax-pair representation and Miura maps to relate the nonlinear equation to a linear eigenvalue problem.
• Computation of the matrix exponential of a constant non-diagonal matrix derived from the Lax pair, which is central to the soliton construction.
• Introduction of a novel re-parametrization of the transformation parameters to express the matrix exponential in closed form without diagonalization or Cardano’s formula.
• Explicit derivation of soliton solutions for the two-component vector NLS equation by solving the resulting linear system via the re-parametrized matrix exponential.
• Utilization of the coalescence limit of the binary Bäcklund–Daroux transformation to generate vector dark solitons with internal degrees of freedom.

Experimental results

Research questions

• RQ1 How can exact soliton solutions be constructed for multicomponent NLS equations when different components approach plane waves with distinct wavenumbers and frequencies at spatial infinity?
• RQ2 What is the role of Bäcklund–Darboux transformations in generating soliton solutions under general nonvanishing boundary conditions, especially when the inverse scattering method fails?
• RQ3 Can the matrix exponential arising from the Lax pair be computed explicitly without diagonalizing the matrix or using Cardano’s formula for cubic roots?
• RQ4 What are the structural and dynamical properties of multicomponent solitons with internal degrees of freedom, and how do they generalize scalar NLS solitons?
• RQ5 How do the re-parametrization of transformation parameters affect the form and tractability of the resulting soliton solutions?

Key findings

• The paper derives explicit, closed-form expressions for matrix exponentials of constant non-diagonal matrices arising in the Lax pair, avoiding the use of Cardano’s formula and enabling more tractable soliton solutions.
• The method successfully constructs both bright and dark soliton solutions for multicomponent NLS equations under general plane-wave boundary conditions, including cases with different wavenumbers and frequencies in different components.
• Vector dark solitons with internal degrees of freedom are obtained through the coalescence limit of the binary Bäcklund–Darboux transformation, which is essential for capturing nontrivial soliton dynamics.
• The re-parametrization of transformation parameters leads to more explicit and useful expressions than previous approaches, particularly in the two-component case.
• The approach generalizes scalar NLS solitons to multicomponent systems with nontrivial internal degrees of freedom, including beating effects between soliton and background plane waves.
• The method is extendable to multisoliton solutions via iterative application of the Bäcklund–Darboux transformation, though explicit expressions become too complex for practical use.