New articles on Nonlinear Sciences


[1] 2405.17848

Generic quartic solitons in optical media

Our analysis suggests strongly that stationary pulses exist in nonlinear media with second-, third-, and fourth-order dispersion. A theory, based on the variational approach, is developed for finding approximate parameters of such solitons. It is obtained that the soliton velocity in the retarded reference frame can be different from the inverse of the group velocity of linear waves. It is shown that the interaction of the pulse spectrum with that of linear waves can affect the existence of stationary solitons. These theoretical results are supported by numerical simulations. Transformations between solitons of different systems are derived. A generalization for solitons in media with the highest even-order dispersion is suggested.


[2] 2405.17973

Symmetries of linear and nonlinear partial differential equations

We consider higher symmetries and operator symmetries of linear partial differential equations. The higher symmetries form a Lie algebra, and operator ones form an associative algebra. The relationship between these symmetries is established. We show that symmetries of linear equations sometimes generate symmetries of nonlinear ones. New symmetries of two-dimensional stationary equations of gas dynamics are found.


[3] 2405.18197

The dynamic of the positons for the reverse space-time nonlocal short pulse equation

In this paper, the Darboux transformation (DT) of the reverse space-time (RST) nonlocal short pulse equation is constructed by a hodograph transformation and the eigenfunctions of its Lax pair. The multi-soliton solutions of the RST nonlocal short pulse equation are produced through the DT, which can be expressed in terms of determinant representation. By taking different values of eigenvalues, bounded soliton solutions and unbounded soliton solutions can be obtained. In addition, based on the degenerate Darboux transformation, the $N$-positon solutions of the RST nonlocal short pulse equation are computed from the determinant expression of the multi-soliton solution. Furthermore, different kinds of mixed solutions are also presented, and the interaction properties between positons and solitons are investigated.


[4] 2405.17662

Nonlinear steepest descent on a torus: A case study of the Landau-Lifshitz equation

We obtain rigorous large time asymptotics for the Landau-Lifshitz equation in the soliton free case by extending the nonlinear steepest descent method to genus 1 surfaces. The methods presented in this paper pave the way to rigorous analysis of other integrable equations on the torus and enables asymptotic analysis on different regimes of the Landau-Lifshitz equation.


[5] 2405.17865

Quantum Integrable Systems on a Classical Integrable Background

In this paper, we develop the framework for quantum integrable systems on an integrable classical background. We call them hybrid quantum integrable systems (hybrid integrable systems), and we show that they occur naturally in the semiclassical limit of quantum integrable systems. We start with an outline of the concept of hybrid dynamical systems. Then we give several examples of hybrid integrable systems. The first series of examples is a class of hybrid integrable systems that appear in the semiclassical limit of quantum spin chains. Then we look at the semiclassical limit of the quantum spin Calogero--Moser system. The result is a hybrid integrable system driven by usual classical Calogero--Moser (CM) dynamics. This system at the fixed point of the multi-time classical dynamics CM system gives commuting spin Hamiltonians of Haldane--Shastry model.


[6] 2405.18006

Mathematical models of the Arabidopsis circadian oscillator

We review the construction and evolution of mathematical models of the Arabidopsis circadian clock, structuring the discussion into two distinct historical phases of modeling strategies: extension and reduction. The extension phase explores the bottom-up assembly of regulatory networks introducing as many components and interactions as possible in order to capture the oscillatory nature of the clock. The reduction phase deals with functional decomposition, distilling complex models to their essential dynamical repertoire. Current challenges in this field, including the integration of spatial considerations and environmental influences like light and temperature, are also discussed. The review emphasizes the ongoing need for models that balance molecular detail with practical simplicity.