[Paper Review] Toda chain from the kink-antikink lattice

Song He, Yunguo Jiang+1 more|arXiv (Cornell University)|May 23, 2016

Nonlinear Waves and Solitons38 references3 citations

TL;DR

This paper derives the Toda lattice from kink-antikink lattices in 1+1-dimensional scalar field theories with $φ^4$, $φ^6$, and $φ^8$ potentials. By computing leading- and next-to-leading-order forces between distant kinks, which decay exponentially with distance, the authors show that the kink lattice dynamics map exactly to the nonperiodic Toda lattice at leading order and to a novel deformed Toda lattice at next order. The deformed system is near-integrable due to high-order corrections breaking full integrability.

ABSTRACT

In this paper, we have studied the kink and antikink solutions in several neutral scalar models in 1+1 dimension. We follow the standard approach to write down the leading order and the second order force between long distance separated kink and antikink. The leading order force is proportional to exponential decay with respect to the distance between the two nearest kinks or antikinks. The second order force have a similar behavior with the larger decay factor, namely $3\over 2$. We make use of these properties to construct the kink lattice. The dynamics of the kink lattice with leading order force can be identified as ordinary nonperiodic Toda lattice. Also the periodic Toda lattice can be obtained when the number of kink lattice is even. The system of kink lattice with force up to the next order corresponds to a new specific deformation of Toda lattice system. There is no well study on this deformation in the integrable literatures.We found that the deformed Toda system are near integrable system, since the integrability are hindered by high order correction terms. Our work provides a effective theory for kink interactions and a new near or quasi integrable model.

Motivation & Objective

• To explore the dynamical correspondence between multikink solutions in 1+1D scalar field theories and integrable systems, particularly the Toda lattice.
• To compute the long-range forces between kinks and antikinks in $φ^4$, $φ^6$, and $φ^8$ models up to second order in separation.
• To identify the resulting kink lattice dynamics as isomorphic to the nonperiodic Toda lattice at leading order and to a new deformed Toda lattice at next-to-leading order.
• To investigate the integrability properties of the deformed Toda system derived from kink interactions.

Proposed method

• Compute the leading-order and second-order forces between widely separated kink-antikink pairs in $φ^4$, $φ^6$, and $φ^8$ scalar field theories using analytical methods.
• Identify the force laws as exponentially decaying with distance, with decay factors of 1 for leading order and $3/2$ for second order.
• Map the kink lattice dynamics to the Toda lattice by identifying kink positions with Toda lattice site variables.
• Use Flaschka’s transformation to construct an $N \times N$ Lax pair representation for the deformed Toda system derived from second-order forces.
• Analyze the resulting system for integrability, showing that high-order correction terms break full integrability, rendering the system near-integrable.
• Establish a correspondence between multikink solutions and cluster coordinates via $2 \times 2$ Lax pair representations in the Toda system.

Experimental results

Research questions

• RQ1 Can the dynamics of kink-antikink lattices in 1+1D scalar field theories be mapped to the Toda lattice system?
• RQ2 Do the leading-order and next-to-leading-order forces between kinks exhibit universal behavior across different self-interaction potentials like $φ^4$, $φ^6$, and $φ^8$?
• RQ3 What is the nature of the Toda-like system that arises when second-order kink interactions are included?
• RQ4 Is the resulting deformed Toda system integrable, or does it represent a near-integrable model?
• RQ5 How are the multikink solutions related to the Poisson structure and cluster coordinates in the Toda system?

Key findings

• The leading-order force between distant kinks and antikinks decays as $e^{-|q_i - q_{i+1}|}$, matching the force law of the nonperiodic Toda lattice.
• The second-order force decays as $e^{-\frac{3}{2}|q_i - q_{i+1}|}$, which is universal across $φ^4$, $φ^6$, and $φ^8$ models.
• The kink lattice with only leading-order forces is isomorphic to the nonperiodic Toda lattice, with the number of kinks equal to the number of lattice sites.
• When the number of kink-antikink pairs is even, the system maps to the periodic Toda lattice, consistent with vacuum boundary conditions.
• Including second-order forces leads to a new deformed Toda lattice system with additional exponential interaction terms, which is not fully integrable due to high-order corrections.
• The deformed Toda system admits a Lax pair representation via Flaschka’s transformation but is classified as near-integrable or quasi-integrable because integrability is obstructed by higher-order terms.