Integrability and solutions to multi-component degenerate CH-type equations

Zhen Xiaoyan

（Department of Mathematics，Ningbo University，Ningbo315211，China）

Integrability and solutions to multi-component degenerate CH-type equations

Zhen Xiaoyan

（Department of Mathematics，Ningbo University，Ningbo315211，China）

In this paper，we propose a multi-component degenerate CH-type system with cubic nonlinearity.This system is shown to be integrable with admitting Lax pair，bi-Hamiltonian structure and recursion operator.In particular，the two-component degenerate Novikov equation is mainly concerned and its exact singular solutions with a finite number of corners are obtained.

bi-Hamiltonian structure，short-wave limit，exact singular solution，multi-component Camassa-Holm type equation

1 Introduction

The Hunter-Saxton（HS）equation

was derived by Hunter and Saxton as a model for describing propagation of orientation waves in a massive nematic liquid crystal director field［1］.It can be derived as the high-frequency limit of the Camassa-Holm（CH）equation［2-3］，so it can be regarded as a degenerate CH equation. Similar to the CH equation，the HS equation is also integrable［4］，which admits bi-Hamiltonian structure，Lax-pair and rich symmetries［5-6］.Interestingly，the HS equation can be linearizedby a reciprocal transformation［5］.In the similar manner，the short-pulse equation［7］

They further showed that the Novikov equation（1）is related to a negative flow in the Sawada-Kotera hierarchy.Its infinitely many conserved quantities and a bi-Hamiltonian structure are also presented.

It is of interest to find multi-component generalizations of these CH-type equations［14-19］.Some of them have physical applications［20-21］and nice geometric formulations［14］. For instance，Geng and Xue［15］introduced the two-component Novikov equation：

It is also a completely integrable system，possessing Lax representation and bi-Hamiltonian structure.Moreover，they studied the special reductions of their general spectral problem.In this sense，almost all known 3×3 spectral problem for the CH-type equations are contained in this case.Recently，Popowicz［24］introduced the matrix version of the Lax representation of equation（3）in which mi=ui-uixx，ni=vi-vixx，i=1，2 are N-dimensional matrices.

In this paper，we consider the case where mi=-uixx，ni=-vixx，i=1，2 are N-dimensional vector function.Since the HS equation can be derived from high-frequency limitof the celebrated CH equation.A natural question is to extend such a study to the multicomponent systems.Analogous to the derivation of the HS equation，we construct multicomponent degenerate CH-type equation.

The structure of the paper is as follows.In Section 2，we will show that the multicomponent degenerate CH-type system is completely integrable with a Lax pair and bi-Hamiltonian structure.In Section 3，an infinite sequence of symmetries is constructed by its recursion operator.In Section 4，we consider the special reductions of our general spectral problem.

2 Construction of multi-component degenerate CH-type equation

3 An infinite sequence of symmetries

4 Reductions

Reference

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多分量退化的CH型方程的可积性及其解

甄肖燕

（宁波大学数学系，浙江，宁波315211）

主要研究多分量退化的含有立方项的CH型方程，并证明了其可积性：Lax表示，双哈密顿结构，以及递推算子.特别地，得到了一个退化的两分量的Novikov方程，并给出了其有限个拐点的奇性解.

双哈密顿结构，多分量CH型方程，极限约束，奇性解

O175.2

2016-01-10.

国家自然科学基金（11471174）.

甄肖燕（1990-），硕士生，研究方向：非线性偏微分方程的研究.

10.3969/j.issn.1008-5513.2016.02.008

2010 MSC:35A01Document Code:AArticle ID:1008-5513（2016）02-0169-13