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Zhijun Qiao

Publications and source records attributed to Zhijun Qiao.

At least 37 records · Page 2Linked to original sources
arXiv2015

Blow-up phenomena for an integrable two-component Camassa-Holm system with cubic nonlinearity and peakon solutions

This paper is devoted to an integrable two-component Camassa-Holm system with cubic nonlinearity, which includes the cubic Camassa-Holm equation (also called the Fokas-Olver-Rosenau-Qiao equation) as a special case. The one peaked solitons (peakons) and two peakon solutions are described in an explicit formula. Then, the local well-posedness for the Cauchy problem of the system is studied. Moreover, we target at the precise blow-up scenario for strong solutions to the system, and establish a new blow-up result with respect to the initial data.

math.AP↗
arXiv2015

A synthetical two-component model with peakon solutions

A generalized two-component model with peakon solutions is proposed in this paper. It allows an arbitrary function to be involved in as well as including some existing integrable peakon equations as special reductions. The generalized two-component system is shown to possess Lax pair and infinitely many conservation laws. Bi-Hamiltonian structures and peakon interactions are discussed in detail for typical representative equations of the generalized system. In particular, a new type of $N$-peakon solution, which is not in the traveling wave type, is obtained from the generalized system.

nlin.SI↗
arXiv2015

Integrable system with peakon, weak kink, and kink-peakon interactional solutions

In this paper, we study an integrable system with both quadratic and cubic nonlinearity: $m_t=bu_x+1/2k_1[m(u^2-u^2_x)]_x+1/2k_2(2m u_x+m_xu)$, $m=u-u_{xx}$, where $b$, $k_1$ and $k_2$ are arbitrary constants. This model is kind of a cubic generalization of the Camassa-Holm (CH) equation: $m_t+m_xu+2mu_x=0$. The equation is shown integrable with its Lax pair, bi-Hamiltonian structure, and infinitely many conservation laws. In the case of $b=0$, the peaked soliton (peakon) and multi-peakon solutions are studied. In particular, the two-peakon dynamical system is explicitly presented and their collisions are investigated in details. In the case of $b\neq0$ and $k_2=0$, the weak kink and kink-peakon interactional solutions are found. Significant difference from the CH equation is analyzed through a comparison. In the paper, we also study all possible smooth one-soliton solutions for the system.

nlin.SI↗
arXiv2015

Multi-component generalization of Camassa-Holm equation

In this paper, we propose a multi-component system of Camassa-Holm equation, denoted by CH($N$,$H$) with 2N components and an arbitrary smooth function $H$. This system is shown to admit Lax pair and infinitely many conservation laws. We particularly study the case of N=2 and derive the bi-Hamiltonian structures and peaked soliton (peakon) solutions for some examples.

nlin.SI↗
arXiv2015

A new two-component integrable system with peakon solutions

A new two-component system with cubic nonlinearity and linear dispersion: \begin{eqnarray*} \left\{\begin{array}{l} m_t=bu_{x}+\frac{1}{2}[m(uv-u_xv_x)]_x-\frac{1}{2}m(uv_x-u_xv), \\ n_t=bv_{x}+\frac{1}{2}[ n(uv-u_xv_x)]_x+\frac{1}{2} n(uv_x-u_xv), \\m=u-u_{xx},~~ n=v-v_{xx}, \end{array}\right. \end{eqnarray*} where $b$ is an arbitrary real constant, is proposed in this paper. This system is shown integrable with its Lax pair, bi-Hamiltonian structure, and infinitely many conservation laws. Geometrically, this system describes a nontrivial one-parameter family of pseudo-spherical surfaces. In the case $b=0$, the peaked soliton (peakon) and multi-peakon solutions to this two-component system are derived. In particular, the two-peakon dynamical system is explicitly solved and their interactions are investigated in details. Moreover, a new integrable cubic nonlinear equation with linear dispersion \begin{eqnarray*} m_t=bu_{x}+\frac{1}{2}[m(|u|^2-|u_x|^2)]_x-\frac{1}{2}m(uu^\ast_x-u_xu^\ast), \quad m=u-u_{xx}, \end{eqnarray*} is obtained by imposing the complex conjugate reduction $v=u^\ast$ to the two-component system. The complex valued $N$-peakon solution and kink wave solution to this complex equation are also derived.

nlin.SI↗
arXiv2013

A three-component Camassa-Holm system with cubic nonlinearity and peakons

In this paper, we propose a three-component Camassa-Holm (3CH) system with cubic nonlinearity and peakons. The 3CH model is proven integrable in the sense of Lax pair, Hamiltonian structure, and conservation laws. We show that this system admits peaked soliton (peakon) and multi-peakon solutions. Additionally, reductions of the 3CH system are investigated so that a new integrable perturbed CH equation with cubic nonlinearity is generated to possess peakon solutions.

nlin.SI↗
arXiv2013

The semi-discrete AKNS system: Conservation laws, reductions and continuum limits

In this paper, the semi-discrete Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy is shown in spirit composed by the Ablowitz-Ladik flows under certain combinations. Furthermore, we derive its explicit Lax pairs and infinitely many conservation laws, which are non-trivial in light of continuum limit. Reductions of the semi-discrete AKNS hierarchy are investigated to include the semi-discrete Korteweg-de Vries (KdV), the semi-discrete modified KdV, and the semi-discrete nonlinear Schrödinger hierarchies as its special cases. Finally, under the uniform continuum limit we introduce in the paper, the above results of the semi-discrete AKNS hierarchy, including Lax pairs, infinitely many conservation laws and reductions, recover their counterparts of the continuous AKNS hierarchy.

nlin.SI↗
arXiv2013

On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity

In this paper, we study the Cauchy problem for a generalized integrable Camassa-Holm equation with both quadratic and cubic nonlinearity. By overcoming the difficulties caused by the complicated mixed nonlinear structure, we firstly establish the local well-posedness result in Besov spaces, and then present a precise blow-up scenario for strong solutions. Furthermore, we show the existence of single peakon by the method of analysis.

math.AP↗
arXiv2012

The algebro-geometric solutions for Degasperis-Procesi hierarchy

Though completely integrable Camassa-Holm (CH) equation and Degasperis-Procesi (DP) equation are cast in the same peakon family, they possess the second- and third-order Lax operators, respectively. From the viewpoint of algebro-geometrical study, this difference lies in hyper-elliptic and non-hyper-elliptic curves. The non-hyper-elliptic curves lead to great difficulty in the construction of algebro-geometric solutions of the DP equation. In this paper, we derive the DP hierarchy with the help of Lenard recursion operators. Based on the characteristic polynomial of a Lax matrix for the DP hierarchy, we introduce a third order algebraic curve $\mathcal{K}_{r-2}$ with genus $r-2$, from which the associated Baker-Akhiezer functions, meromorphic function and Dubrovin-type equations are established. Furthermore, the theory of algebraic curve is applied to derive explicit representations of the theta function for the Baker-Akhiezer functions and the meromorphic function. In particular, the algebro-geometric solutions are obtained for all equations in the whole DP hierarchy.

nlin.SI↗
arXiv2012

The algebro-geometric solutions for the modified Camassa-Holm hierarchy

This paper is dedicated to provide theta function representation of algebro-geometric solutions and related crucial quantities for the modified Camassa-Holm (MCH) hierarchy through %and studying a algebro-geometric initial value problem. Our main tools include the polynomial recursive formalism to derive the MCH hierarchy, the hyperelliptic curve with finite number of genus, the Baker-Akhiezer functions, the meromorphic function, the Dubrovin-type equations for auxiliary divisors, and the associated trace formulas. With the help of these tools, the explicit representations of the Baker-Ahhiezer functions, the meromorphic function, and the algebro-geometric solutions are obtained for the entire MCH hierarchy.

nlin.SI↗
arXiv2011

On Negative Order KdV Equations

In this paper, based on the regular KdV system, we study negative order KdV (NKdV) equations about their Hamiltonian structures, Lax pairs, infinitely many conservation laws, and explicit multi-soliton and multi-kink wave solutions thorough bilinear Bäcklund transformations. The NKdV equations studied in our paper are differential and actually derived from the first member in the negative order KdV hierarchy. The NKdV equations are not only gauge-equivalent to the Camassa-Holm equation through some hodograph transformations, but also closely related to the Ermakov-Pinney systems, and the Kupershmidt deformation. The bi-Hamiltonian structures and a Darboux transformation of the NKdV equations are constructed with the aid of trace identity and their Lax pairs, respectively. The single and double kink wave and bell soliton solutions are given in an explicit formula through the Darboux transformation. The 1-kink wave solution is expressed in the form of $tanh$ while the 1-bell soliton is in the form of $sech$, and both forms are very standard. The collisions of 2-kink-wave and 2-bell-soliton solutions, are analyzed in details, and this singular interaction is a big difference from the regular KdV equation. Multi-dimensional binary Bell polynomials are employed to find bilinear formulation and Bäcklund transformations, which produce $N$-soliton solutions. A direct and unifying scheme is proposed for explicitly building up quasi-periodic wave solutions of the NKdV equations. Furthermore, the relations between quasi-periodic wave solutions and soliton solutions are clearly described. Finally, we show the quasi-periodic wave solution convergent to the soliton solution under some limit conditions.

nlin.SI↗
arXiv2011

Negative order KdV equation with both solitons and kink wave solutions

In this paper, we report an interesting integrable equation that has both solitons and kink solutions. The integrable equation we study is $(\frac{-u_{xx}}{u})_{t}=2uu_{x}$, which actually comes from the negative KdV hierarchy and could be transformed to the Camassa-Holm equation through a gauge transform. The Lax pair of the equation is derived to guarantee its integrability, and furthermore the equation is shown to have classical solitons, periodic soliton and kink solutions.

math-ph↗
arXiv2010

Stability of solitary waves of a generalized two-component Camassa-Holm system

We study here the existence of solitary wave solutions of a generalized two-component Camassa-Holm system. In addition to those smooth solitary-wave solutions, we show that there are solitary waves with singularities: peaked and cusped solitary waves. We also demonstrate that all smooth solitary waves are orbitally stable in the energy space. We finally give a sufficient condition for global strong solutions to the equation without certain parameters.

math.AP↗
arXiv2002

Integrable hierarchy, $3\times 3$ constrained systems, and parametric and peaked stationary solutions

This paper gives a new integrable hierarchy of nonlinear evolution equations. The DP equation: $m_t+um_x+3mu_x=0, m=u-u_{xx}$, proposed recently by Desgaperis and Procesi \cite{DP[1999]}, is the first one in the negative hierarchy while the first one in the positive hierarchy is:\ $m_t=4(m^{-{2/3}})_x-5(m^{-{2/3}})_{xxx}+ (m^{-{2/3}})_{xxxxx}$. The whole hierarchy is shown Lax-integrable through solving a key matrix equation. To obtain the parametric solutions for the whole hierarchy, we separatedly discuss the negative and the positive hierarchies. For the negative hierarchy, its $3\times3$ Lax pairs and corresponding adjoint representations are nonlinearized to be Liouville-integrable Hamiltonian canonical systems under the so-called Dirac-Poisson bracket defined on a symplectic submanifold of $\R^{6N} $. Based on the integrability of those finite-dimensional canonical Hamiltonian systems we give the parametric solutions of the all equations in the negative hierarchy. In particular, we obtain the parametric solution of the DP equation. Moreover, for the positive hierarchy, we consider the different constraint and use a similar procudure to the negative case to obtain the parametric solutions of the positive hierarchy. In particular, we give the parametric solution of the 5th-order PDE $m_t=4(m^{-{2/3}})_x-5(m^{-{2/3}})_{xxx}+ (m^{-{2/3}})_{xxxxx}$. Finally, we discuss the stationary solutions of the 5th-order PDE, and particularly give its four peaked stationary solutions. The stationary solutions may be included in the parametric solution, but the peaked stationary solutions not. The 5th-order PDE does not have the cusp soliton although it looks like a higher order Harry-Dym equation.

nlin.SI↗
arXiv2002

An integrable hierarchy, parametric solution and traveling wave solution

This paper gives an integrable hierarchy of nonlinear evolution equations. In this hierarchy there are the following representative equations: \beqq & & u_t=\pa^5_x u^{-{2/3}}, & & u_t=\pa^5_x\frac{(u^{-{1/3}})_{xx} -2(u^{-{1/6}})_{x}^2}{u}; & & u_{xxt}+3u_{xx}u_x+u_{xxx}u=0. \eeqq The first two are in the positive order hierarchy while the 3rd one is in the negative order hierarchy. The whole hierarchy is shown integrable through solving a key $3\times 3 $ matrix equation. The $3\times3$ Lax pairs and their adjoint representations are nonlinearized to be two Liouville-integrable canonical Hamiltonian systems. Based on the integrability of $6N$-dimensional systems we give the parametric solution of the positive hierarchy. In particular, we obtain the parametric solution of the equation $u_t=\pa^5_x u^{-{2/3}}$. Moreover, we give the traveling wave solution (TWS) of the above three equations. The TWSs of the first two equations have singularity and look like cusp (cusp-like), but the TWS of the 3rd one is continuous. For the 5th-order equation, its parametric solution can not include its singular TWS. We also analyse the Gaussian initial solutions for the equations $u_t=\pa^5_x u^{-{2/3}}$, and $u_{xxt}+3u_{xx}u_x+u_{xxx}u=0.$ One is stable, the other not. Finally, we extend the equation $u_t=\pa^5_x u^{-{2/3}}$ to a large class of equations $ u_t=\partial_x^l u^{-m/n}, l\ge1, n\not=0, m,n \in \Z, $ which still have the singular cusp-like traveling wave solutions.

nlin.SI↗
arXiv2002

Generalized r-matrix structure and algebro-geometric solution for integrable systems

The purpose of this paper is to construct a generalized r-matrix structure of finite dimensional systems and an approach to obtain the algebro-geometric solutions of integrable nonlinear evolution equations (NLEEs). Our starting point is a generalized Lax matrix instead of usual Lax pair. The generalized r-matrix structure and Hamiltonian functions are presented on the basis of fundamental Poisson bracket. It can be clearly seen that various nonlinear constrained (c-) and restricted (r-) systems, such as the c-AKNS, c-MKdV, c-Toda, r-Toda, c-Levi, etc, are derived from the reduction of this structure. All these nonlinear systems have {\it r}-matrices, and are completely integrable in Liouville's sense. Furthermore, our generalized structure is developed to become an approach to obtain the algebro-geometric solutions of integrable NLEEs. Finally, the two typical examples are considered to illustrate this approach: the infinite or periodic Toda lattice equation and the AKNS equation with the condition of decay at infinity or periodic boundary.

nlin.SI↗