Two-component Hunter Saxton System Research Articles

Generalized Nonlocal Symmetries of Two-Component Camassa–Holm and Hunter–Saxton Systems

The two-component Camassa–Holm system and two-component Hunter–Saxton system are completely integrable models. In this paper, it is shown that these systems admit nonlocal symmetries by their geometric integrability. As an application, we obtain the recursion operator and conservation laws by using this kind of nonlocal symmetries.

Hamiltonian regularisation of the unidimensional barotropic Euler equations

Recently, a Hamiltonian regularised shallow water (Saint-Venant) system has been introduced by Clamond and Dutykh (2018). This system is Galilean invariant, linearly non-dispersive and conserves formally an H1-like energy. In this paper, we extend this regularisation in two directions. First, we consider the more general barotropic Euler system, the shallow water equations being formally a very special case. Second, we introduce a class regularisations, showing thus that this Hamiltonian regularisation of Clamond and Dutykh (2018) is not unique. Considering the high-frequency approximation of this regularisation, we obtain a new two-component Hunter–Saxton system. We prove that both systems – the regularised barotropic Euler system and the two-component Hunter–Saxton system – are locally (in time) well-posed, and, if singularities appear in finite time, they are necessary in the first derivatives.

Lie symmetry reductions and exact solutions to a generalized two-component Hunter-Saxton system

Based on the classical Lie group method, a generalized two-component Hunter-Saxton system is studied in this paper. All of the its geometric vector fields, infinitesimal generators and the commutation relations of Lie algebra are derived. Furthermore, the similarity variables and symmetry reductions of this new generalized two-component Hunter-Saxton system are derived. Under these Lie symmetry reductions, some exact solutions are obtained by using the symbolic computation. Moreover, a conservation law of this system is presented by using the multiplier approach.

Existence and Lipschitz stability for α-dissipative solutions of the two-component Hunter–Saxton system

We establish the concept of [Formula: see text]-dissipative solutions for the two-component Hunter–Saxton system under the assumption that either [Formula: see text] or [Formula: see text] for all [Formula: see text]. Furthermore, we investigate the Lipschitz stability of solutions with respect to time by introducing a suitable parametrized family of metrics in Lagrangian coordinates. This is necessary due to the fact that the solution space is not invariant with respect to time.

Group Analysis of the Generalized Hunter-Saxton System

We find the Lie point symmetries of the generalized two-component Hunter-Saxton system. Then we show that it is nonlinearly self-adjoint and establish the corresponding conservation laws using a recent theorem of Nail Ibragimov which enables one to determine conservation laws for problems without variational structure. Finally we obtain some invariant solutions.

A Lipschitz metric for conservative solutions of the two-component Hunter–Saxton system

We establish the existence of conservative solutions of the initial value problem of the two-component Hunter--Saxton system on the line. Furthermore we investigate the stability of these solutions by constructing a Lipschitz metric.

On a variation of the two-component Hunter–Saxton system

We discuss a novel two-component Hunter–Saxton system in the periodic setting. This paper is mainly concerned with the blow-up phenomena. The precise blow-up scenario and blow-up criteria under proper assumptions on the initial data are established. As a by-product, the blow-up rate is obtained.

Single peak solitary wave and compacton solutions of the generalized two-component Hunter–Saxton system

Dynamical system theory is applied to the generalized two-component Hunter–Saxton system. Two singular straight lines are found in the associated topological vector field. The influence of parameters as well as the singular lines on the smoothness property of the traveling wave solutions is explored in detail. We obtain the single peak solitary wave and compacton solutions for the generalized two-component Hunter–Saxton system. Asymptotic analysis and numerical simulations are provided for smooth solitary wave, peakon, cuspon and compacton solutions of the generalized two-component Hunter–Saxton system.

Notes on “Solitary wave solutions of the generalized two-component Hunter–Saxton system”

The goals of this paper are to employ the approach of dynamical system to obtain some new results of the following generalized two-component Hunter–Saxton system: {utxx+2σuxuxx+σuuxxx−ρρx+Aux=0,ρt+(ρu)x=0,σ∈R,A≥0, which are not obtained in Moon (2013).

Bifurcations and exact travelling wave solutions of the generalized two-component Hunter-Saxton system

In this paper, we apply the method of dynamical systems to a generalized two-component Hunter-Saxton system. Through qualitative analysis, we obtain bifurcations of phase portraits of the traveling system. Under different parameter conditions, exact explicit smooth solitary wave solutions, solitary cusp wave solutions, as well as periodic wave solutions are obtained. To guarantee the existence of these solutions, rigorous parametric conditions are given.

Bifurcation Analysis and Different Kinds of Exact Travelling Wave Solutions of a Generalized Two-Component Hunter-Saxton System

This paper focuses on a generalized two-component Hunter-Saxton system. From a dynamic point of view, the existence of different kinds of periodic wave, solitary wave, and blow-up wave is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, some exact parametric representations of the travelling waves are presented.

Global estimates and blow-up criteria for the generalized Hunter-Saxton system

The generalized, two-component Hunter-Saxton system comprises several well-known models of fluid dynamics and serves as a tool for the study of one-dimensional fluid convection and stretching. In this article a general representation formula for periodic solutions to the system, which is valid for arbitrary values of parameters $(\lambda,\kappa) \in \mathbb{R} \times \mathbb{R}$, is derived. This allows us to examine in great detail qualitative properties of blow-up as well as the asymptotic behaviour of solutions, including convergence to steady states in finite or infinite time.

Solitary wave solutions of the generalized two-component Hunter–Saxton system

The existence of solitary wave solutions of the generalized two-component Hunter–Saxton system is determined. It is also shown that there are peaked and cusped solitary waves with singularities among those smooth solitary wave solutions.

The Hunter-Saxton System and the Geodesics on a Pseudosphere

We show that the two-component Hunter-Saxton system with negative coupling constant describes the geodesic flow on an infinite-dimensional pseudosphere. This approach yields explicit solution formulae for the Hunter-Saxton system. Using this geometric intuition, we conclude by constructing global weak solutions. The main novelty compared with similar previous studies is that the metric is indefinite.

Wave-Breaking Criterion for the Generalized Weakly Dissipative Periodic Two-Component Hunter-Saxton System

This paper studies the wave-breaking criterion for the generalized weakly dissipative two-component Hunter-Saxton system in the periodic setting. We get local well-posedness for the generalized weakly dissipative two-component Hunter-Saxton system. We study a wave-breaking criterion for solutions and results of wave-breaking solutions with certain initial profiles.

Wave breaking and global existence for the generalized periodic two-component Hunter–Saxton system

In this paper, we study the wave-breaking phenomena and global existence for the generalized two-component Hunter–Saxton system in the periodic setting. We first establish local well-posedness for the generalized two-component Hunter–Saxton system. We obtain a wave-breaking criterion for solutions and results of wave-breaking solutions with certain initial profiles. We also determine the exact blow-up rate of strong solutions. Finally, we give a sufficient condition for global solutions.

Global weak solutions for a periodic two-component Hunter-Saxton system

This paper is concerned with global existence of weak solutions for a periodic two-component Hunter-Saxton system. We first derive global existence for strong solutions to the system with smooth approximate initial data. Then we show that the limit of approximate solutions is a global weak solution of the two-component Hunter-Saxton system.

Global weak solutions and smooth solutions for a two-component Hunter-Saxton system

In this paper, we mainly study the initial value problem of a two-component Hunter-Saxton system. By the method of characteristics, we first show that the system has global smooth solutions and blowing up smooth solutions. By using the obtained a priori estimates on smooth solutions, we then prove the existence of global weak solutions to the system.

Analyticity of the Cauchy problem for two-component Hunter–Saxton systems

This paper is mainly concerned with the periodic Cauchy problem for a generalized two-component μ -Hunter–Saxton system with analytic initial data. The analyticity of its solutions is proved in both variables, globally in space and locally in time. The obtained result can be also applied to its special cases—the classical integrable two-component Hunter–Saxton system, the generalized μ -Hunter–Saxton equation and the classical Hunter–Saxton equation.

On initial boundary value problems for variants of the Hunter–Saxton equation

The Hunter-Saxton equation serves as a mathematical model for orientation waves in a nematic liquid crystal. The present paper discusses a modified variant of this equation, coming up in the study of critical points for the speed of orientation waves, as well as a two-component extension. We establish well-posedness and blow-up results for some initial boundary value problems for the modified Hunter-Saxton equation and the two-component Hunter-Saxton system.