[Paper Review] Covariant hodograph transformations between nonlocal short pulse models and AKNS$(-1)$ system

Kui Chen, Shimin Liu+1 more|arXiv (Cornell University)|Dec 22, 2017

Nonlinear Waves and Solitons21 references3 citations

TL;DR

This paper establishes covariant hodograph transformations between multi-component nonlocal short pulse (SP) models and the AKNS(−1) system, demonstrating that independent variables transform consistently under nonlocal reductions. It shows that nonlocal reductions on the SP and AKNS(−1) systems are covariant via these transformations, enabling solution transfer between the systems using the well-developed AKNS spectral problem framework.

ABSTRACT

The paper presents hodograph transformation between nonlocal short pulse models and the first member in the AKNS negative hierarchy (AKNS($-1$)). We consider real and complex multi-component cases. It is shown that the independent variables of the short pulse models and AKNS($-1$) that are connected via hodograph transformation are covariant in nonlocal reductions.

Motivation & Objective

• To establish a systematic link between nonlocal multi-component short pulse models and the AKNS(−1) system via hodograph transformations.
• To analyze how independent variables transform under nonlocal reductions, ensuring covariance between the SP and AKNS(−1) systems.
• To extend known connections between the AKNS(−1) system, the sine-Gordon equation, and the short pulse equation to nonlocal and multi-component settings.
• To provide a framework for generating solutions of nonlocal SP models from solutions of nonlocal AKNS(−1) systems using covariant transformations.

Proposed method

• Derives hodograph transformations linking the vector short pulse system and the AKNS(−1) system using coordinate changes involving integrated variables.
• Introduces a specific integration operator ∂⁻¹ₓ defined via asymmetric limits to preserve symmetry under nonlocal reductions.
• Applies nonlocal reductions (e.g., reverse-time, reverse-space, and combined) to both SP and AKNS(−1) systems and analyzes their effect on transformed coordinates (y,z).
• Uses the Lax pair structure of AKNS(−1) and the known bilinear forms to ensure integrability and solution compatibility.
• Constructs explicit mappings between solutions of nonlocal AKNS(−1) and nonlocal SP systems through the hodograph transformation and reduction covariance.
• Validates the transformation by verifying that nonlocal reductions on the SP side correspond to consistent, covariant reductions on the AKNS(−1) side in the transformed variables.

Experimental results

Research questions

• RQ1 How can hodograph transformations be extended from local to nonlocal multi-component short pulse models?
• RQ2 What is the behavior of independent variables (x,t) and transformed variables (y,z) under nonlocal reductions in the context of hodograph transformations?
• RQ3 Are nonlocal reductions on the short pulse model and the AKNS(−1) system covariant under hodograph transformations?
• RQ4 Can solutions of nonlocal AKNS(−1) systems be used to generate solutions of nonlocal short pulse models via these transformations?
• RQ5 What role does the choice of integration operator ∂⁻¹ₓ play in ensuring covariance under nonlocal reductions?

Key findings

• The hodograph transformation between nonlocal SP models and AKNS(−1) preserves covariance under nonlocal reductions, as shown by explicit transformation of independent variables (x,t) to (y,z).
• For nonlocal reductions such as (x,t) → (−x,−t) or (−x,t), the transformed variables satisfy y(−x,−t) = −y(x,t) and z(−x,−t) = −z(x,t), ensuring consistent symmetry.
• The transformation ensures that nonlocal reductions on the SP system correspond to equivalent reductions on the AKNS(−1) system in the (y,z) variables, establishing solution equivalence.
• The integration operator ∂⁻¹ₓ defined via asymmetric limits (10) is essential for maintaining the symmetry z(−x,−t) = −z(x,t) under nonlocal reductions.
• Covariant transformations allow solutions of nonlocal AKNS(−1) systems to be mapped to solutions of nonlocal SP models, leveraging the rich solution structure of the AKNS hierarchy.
• The AKNS(−1) system serves as a central integrable model, with its spectral problem enabling the study of nonlocal SP equations through transformation and reduction.