Exploring the depths of symmetry in the mKdV equation: physical interpretations and multi-wave solutions

Xiazhi Hao; S. Y. Lou

Home/Archive/arXiv

arXiv · 2501.02912

Exploring the depths of symmetry in the mKdV equation: physical interpretations and multi-wave solutions

Xiazhi Hao·S. Y. Lou

View original source ↗Read source document ↗Export metadata ↓

Abstract

This manuscript embarks on an in-depth exploration of the modified Korteweg-de Vries (mKdV) equation, with a particular emphasis on unraveling the intricate structure of its infinite symmetries and their physical interpretations. Central to this investigation are the $K$-symmetries and $\tau$-symmetries, which are delineated by a recursive relationship and constitute an infinite ensemble that underpins the conservation laws. We engage with an existing symmetry conjecture, which posits that the currently identified symmetries represent a subset of a more expansive, yet to be unearthed, set. This conjecture is substantiated through an analysis of the soliton solutions associated with the mKdV equation, demonstrating that these symmetries can be decomposed into linear combinations of center and wave number translation symmetries. Further, by imposing an infinite sequence of symmetry constraints, it becomes feasible to derive exact multi-wave solutions. This methodology, predicated on the proposed symmetry conjecture, facilitates the extraction of exact solutions, encompassing complexiton, breather, multi-soliton solutions, among others.

Explore related subjects

nlin.SI

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Xiazhi Hao, S. Y. Lou. 2025-01-06. Exploring the depths of symmetry in the mKdV equation: physical interpretations and multi-wave solutions. https://arxiv.org/abs/2501.02912

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rogue-like waves from collision of mKdV solitons

Interactions of two solitary waves with an up and a down orientation in the modified Korteweg-de Vries equation are shown to produce rogue-like waves. For waves that asymptotically vanish, the maximum ratio between the height of the interaction profile and the height of the tallest incoming wave is 2.41 when the waves have approximately equal speeds, and this ratio decreases to 2 when the speed ratio is 1.5. For waves that approach a non-zero constant at infinity, the same ratio reaches a maximum of 2.65 when the speed ratio of the waves is 6.32.

nlin.SI↗

arXiv2026

On B\"acklund transformations preserving the Darboux integrability of hyperbolic equations

This paper deals with two kinds of B\"acklund transformations for scalar hyperbolic partial differential equations. We prove that both these types of transformations map solutions of a Darboux integrable equation into solutions of, generally speaking, another but also Darboux integrable equation. The latter fact can be used to roughly check the completeness of a list of Darboux integrable equations. To illustrate this, we apply the above transformations to several equations from a well-known list of Darboux integrable equations and, as a result, obtain a Darboux integrable equation which is absent in this list, but is already known at present. As a generalization of the last equation, we construct a family of Darboux integrable equations that is parametrized by three arbitrary functions, each of which depends on two arguments. This family is probably new.

nlin.SI↗

arXiv2026

Complex singularities for Burgers' equation with piecewise-continuous initial conditions

There is a body of research devoted to understanding how complex singularities of solutions of nonlinear partial differential equations (pdes) spontaneously emerge at $t=0^+$ and propagate for $t>0$, and how their behaviour affects the solution on the real axis. Despite the importance of the small-time limit in these studies, there is still a lack of understanding of how complex singularities are born at $t=0^+$, including for initial conditions that are not analytic functions of the spatial variable. In this paper, we use Burgers' equation as a prototype nonlinear pde and study the complex-plane singularities for initial conditions that are piecewise smooth. Using matched asymptotic expansions, we show how infinitely many singularities emerge from points of discontinuity in a pattern that can be described using branches of the Lambert-$W$ function. For various initial conditions, we observe how these singularities rearrange themselves to align with the appropriate exactly-described long-time behaviour, including sigmoid-shaped travelling waves, constant-area (triangular wave) similarity solutions and $N$-wave solutions. In terms of Burgers' equation, our small-time asymptotic analysis of the singularity propagation for piecewise-continuous initial conditions illustrates the types of generic behaviours that arise for inner regions when diffusion dominates advection. More generally, this work is a step towards understanding complex-plane behaviour of solutions of nonlinear partial differential equations with non-analytic initial conditions.

nlin.SI↗