[Paper Review] Lax pair, Darboux Transformations and solitonic solutions for a (2+1) dimensional NLSE

P.G. Estévez, G. A. Hernáez|ArXiv.org|Oct 14, 1999

Nonlinear Waves and Solitons2 references3 citations

TL;DR

This paper derives the Lax pair, Darboux transformations, and tau functions for a (2+1)-dimensional nonlinear Schrödinger equation (NLSE) using the Singular Manifold Method. By applying these integrability tools, the authors construct iteratively generated solitonic solutions, demonstrating the equation's complete integrability and enabling systematic generation of multi-soliton and other nonlinear wave structures in two spatial and one temporal dimension.

ABSTRACT

In this paper the Singular Manifold Method has allowed us to obtain the Lax pair, Darboux transformations and tau functions for a non-linear Schrödiger equation in 2+1 dimensions. In this way we can iteratively build different kind of solutions with solitonic behavior.

Motivation & Objective

• To establish the complete integrability of a (2+1)-dimensional nonlinear Schrödinger equation (NLSE) through the derivation of its Lax pair.
• To apply the Singular Manifold Method to systematically derive Darboux transformations for the (2+1)D NLSE.
• To construct explicit solitonic solutions using iterative Darboux transformations and tau functions.
• To demonstrate the feasibility of generating complex nonlinear wave patterns, including multi-soliton states, in higher-dimensional integrable systems.

Proposed method

• The Singular Manifold Method is employed to derive the Lax pair for the (2+1)D NLSE, identifying the linear spectral problem and its associated evolution equation.
• Darboux transformations are constructed based on the Lax pair, enabling the generation of new solutions from known seed solutions through a gauge-like transformation.
• Tau functions are derived from the Darboux transformation framework, providing a direct link to the bilinear form and soliton solutions.
• Iterative application of the Darboux transformation allows for the systematic construction of multi-soliton solutions with distinct spatial and temporal dynamics.
• The method relies on the integrability structure of the NLSE, ensuring consistency and closure under successive transformations.
• Theoretical derivations are supported by symbolic computation and visualized through nine figures illustrating solution profiles.

Experimental results

Research questions

• RQ1 Can the Lax pair for the (2+1)-dimensional NLSE be derived using the Singular Manifold Method?
• RQ2 How can Darboux transformations be systematically constructed for a (2+1)D integrable system like the NLSE?
• RQ3 What role do tau functions play in the solution hierarchy of the (2+1)D NLSE?
• RQ4 What types of solitonic solutions can be generated through iterative Darboux transformations in two spatial dimensions?
• RQ5 How does the integrability structure of the (2+1)D NLSE support the existence of multi-soliton and localized wave solutions?

Key findings

• The Lax pair for the (2+1)-dimensional NLSE is successfully derived using the Singular Manifold Method, confirming the equation's integrability.
• Darboux transformations are explicitly constructed, enabling the generation of new solutions from seed solutions through a systematic algebraic procedure.
• Tau functions are derived and shown to play a central role in expressing solitonic solutions in bilinear form.
• Iterative application of the Darboux transformation yields multi-soliton solutions with complex spatial profiles, including localized and propagating wave structures.
• The method allows for the systematic construction of solitonic solutions with adjustable parameters, demonstrating the model's rich solution space.
• The results are supported by nine figures illustrating the dynamics and profiles of the derived solitonic solutions in two spatial dimensions.