[Paper Review] Quasi-Integrability of The KdV System

Kumar Abhinav, Partha Guha|arXiv (Cornell University)|Dec 22, 2016

Nonlinear Waves and Solitons7 references3 citations

TL;DR

This paper proposes a quasi-integrable deformation of the Korteweg-de Vries (KdV) equation via off-shell Hamiltonian deformation within a loop algebra framework, yielding infinite anomalous conservation laws and a hierarchy of higher-derivative KdV systems with at least one conserved charge. The method establishes a direct link to quasi-integrable NLS systems through weak-coupling correspondence, enabling single-soliton solutions with scaling symmetry and extending the concept to complex coupled KdV systems.

ABSTRACT

The quasi-integrable KdV equation has been obtained from the corresponding deformation of the Hamiltonian for the usual KdV system. Following suitable gauge-fixing, it has been found that the quasi-conservation condition is satisfied and an infinite number of anomalous conservation laws are obtained, with some containing possible conserved charges. Judicious choice of deformation of the Hamiltonian clearly leads to a conserved charge, manifesting quasi-integrability, but also creates a hierarchy of higher-derivative equations with at least one conserved charge. A particular quasi-deformation parameterization of the Hamiltonian is found to complement the same of the NLS system, following an approximate equivalence of the two systems obtained earlier, in the weak coupling limit. Single-soliton solutions for all these cases are obtained, with manifest scaling due to the quasi-integrable deformation. Finally, quasi-conservation formulation for the complex coupled generalized KdV system is obtained.

Motivation & Objective

• To develop a quasi-integrable deformation of the KdV equation despite the absence of a standard Lax pair for such deformations.
• To address the challenge of deforming higher-order PDEs like KdV, which lack dynamic Lax pair evolution, by using off-shell Hamiltonian deformations.
• To establish a connection between quasi-integrable KdV and previously studied quasi-integrable NLS systems via weak-coupling correspondence.
• To derive a hierarchy of higher-derivative, quasi-integrable KdV extensions with at least one conserved charge.
• To generalize the quasi-conservation framework to the complex coupled generalized KdV system using loop algebra methods.

Proposed method

• Utilizes loop algebra representation of the KdV system to bypass limitations of traditional Lax pair deformation in higher-order systems.
• Applies off-shell Hamiltonian deformation by modifying the Hamiltonian density and amplitude power, avoiding reliance on equations of motion.
• Employs gauge-fixing to satisfy the quasi-conservation condition, leading to infinite anomalous conservation laws.
• Derives generalized Lax pairs and curvature components in terms of deformed Hamiltonian functionals, with explicit replacements mapping real to complex KdV systems.
• Uses a perturbative ansatz to connect the deformed KdV system to the quasi-integrable NLS system in the weak-coupling limit.
• Constructs explicit single-soliton solutions with scaling symmetry by solving the deformed equations under the derived quasi-conservation structure.

Experimental results

Research questions

• RQ1 Can the KdV equation be quasi-integrably deformed despite the absence of a dynamic Lax pair deformation mechanism?
• RQ2 How can a consistent quasi-conservation law structure be constructed for higher-derivative, third-order KdV-type systems?
• RQ3 What is the relationship between the quasi-integrable deformation of KdV and the previously studied quasi-integrable NLS system?
• RQ4 Can a hierarchy of higher-derivative, quasi-integrable KdV models be systematically derived from a single deformation parameterization?
• RQ5 How does the complex coupled generalized KdV system inherit the quasi-integrability properties from the real KdV case?

Key findings

• The quasi-integrable deformation of the KdV system is achieved via off-shell Hamiltonian deformation, yielding infinite anomalous conservation laws with at least one conserved charge.
• A particular deformation parameterization leads to a scaled version of the KdV equation, preserving single-soliton solutions with manifest scaling symmetry.
• The method produces a hierarchy of higher-derivative KdV extensions that are quasi-integrable, each with at least one conserved charge.
• A direct mapping to the quasi-integrable NLS system is established in the weak-coupling limit, confirming consistency with prior results on NLS quasi-integrability.
• The complex coupled generalized KdV system is shown to admit a quasi-conservation formulation, with anomaly terms expressible as total space-derivatives.
• Explicit single-soliton solutions are derived for all deformed KdV systems, confirming the presence of scaling invariance due to the quasi-integrable deformation.