Changzheng Qu

Well posedness and blow-up solution for a new coupled Camassa–Holm equations with peakons

A new Camassa–Holm system with two-component admitting peakon solitons is proposed. The local well posedness for the system is established. A criterion and a condition on the initial data guaranteeing the development of singularities in finite time for strong solutions are obtained, and an existence result for a class of local weak solution is also given.

A new integrable two-component system with cubic nonlinearity

In this paper, a new integrable two-component system, mt=[m(uxvx−uv+uvx−uxv)]x,nt=[n(uxvx−uv+uvx−uxv)]x, where m=u−uxx and n=v−vxx, is proposed. Our system is a generalized version of the integrable system mt=[m(ux2−u2)]x, which was shown having cusped solution (cuspon) and W/M-shape soliton solutions by Qiao [J. Math. Phys. 47, 112701 (2006). The new system is proven integrable not only in the sense of Lax-pair but also in the sense of geometry, namely, it describes pseudospherical surfaces. Accordingly, infinitely many conservation laws are derived through recursion relations. Furthermore, exact solutions such as cuspons and W/M-shape solitons are also obtained.

Approximate potential symmetries for partial differential equations

The method of approximate potential symmetries for partial differential equations with a small parameter is introduced. By writing a given perturbed partial differential equation R in a conserved form, an associated system S with potential variables as additional variables is obtained. Approximate Lie point symmetries admitted by S induce approximate potential symmetries of R. As applications of the theory, approximate potential symmetries for a perturbed wave equation with variable wave speed and a nonlinear diffusion equation with perturbed convection terms are obtained. The corresponding approximate group-invariant solutions are also derived.

New variable separation approach: application to nonlinear diffusion equations

The concept of the derivative-dependent functional separable solution (DDFSS), as a generalization to the functional separable solution, is proposed. As an application, it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach. As a consequence, a complete list of canonical forms for such equations which admit the DDFSS is obtained and some exact solutions to the resulting equations are described.

Conditional Lie Bäcklund symmetries and solutions to (n+1)-dimensional nonlinear diffusion equations

This paper discusses a class of (n+1)-dimensional nonlinear diffusion equations with source term which arises in nonlinear shear flows of non-Newtonian fluids. It is shown that some radially symmetric equations admit certain types of conditional Lie Backlund symmetries. As a result, exact solutions to the resulting equations are obtained. Those solutions extend the known ones such as instantaneous source solutions of the porous medium equation with absorption term. The phenomena of extinction and blow up and behavior to many of the solutions are described.

Symmetry groups and separation of variables of a class of nonlinear diffusion-convection equations

In this paper, a class of nonlinear diffusion-convection equations, ut = (D(u)uxn)x+P(u)ux , which has quite a large number of physical applications, is analysed by using symmetry group methods which include the classical method, the potential symmetry method and the generalized conditional symmetry method. A complete classification of the functional forms of the diffusion and convection coefficients is presented when the equation admits Lies point symmetry groups and potential symmetry groups. The separation of variables for the equation is investigated using the generalized conditional symmetry approach. For some interesting cases, exact solutions using the method of separation of variables are discussed in detail.

Maximal dimension of invariant subspaces admitted by nonlinear vector differential operators

In this paper, the dimension of invariant subspaces admitted by m-component nonlinear vector differential operators is estimated. It is shown that if the m-component nonlinear vector differential operators of order k preserves the invariant subspace Wn11×⋯×Wnmm, where Wnqq is the space generated by solutions of linear ordinary differential equations of order nq (q = 1, …, m), then max {n1, …, nm} ⩽ 2mk + 1. To illustrate the approach, examples of nonlinear vector differential operators admitting invariant subspace Wn11×Wn22 with max {n1, n2} = 9 and three-component nonlinear vector differential operators admitting invariant subspace Wn11×Wn22×Wn33 with max {n1, n2, n3} = 13 are presented.

Contact Symmetry Algebras of Scalar Ordinary Differential Equations

Using Lies classification of irreducible contact transformations in thecomplex plane, we show thata third-order scalar ordinary differential equation (ODE)admits an irreducible contact symmetry algebra if and only if it is transformableto q(3)=0 via a local contact transformation. This result coupled with the classification of third-order ODEs with respect to point symmetriesprovide an explanation of symmetry breaking for third-order ODEs. Indeed, ingeneral, the point symmetry algebra of a third-order ODE is not asubalgebra of the seven-dimensional point symmetry algebra of q(3)=0.However, the contact symmetry algebra of any third-order ODE, except forthird-order linear ODEs with four- and five-dimensional pointsymmetry algebras, is shown to be a subalgebra of the ten-dimensional contact symmetryalgebra of q(3)==0. We also show that a fourth-orderscalar ODE cannot admit an irreducible contact symmetry algebra. Furthermore, weclassify completely scalar nth-order (n≥5) ODEs which admitnontrivial contact symmetry algebras.

Preliminary group classification of a class of fourth-order evolution equations

We perform preliminary group classification of a class of fourth-order evolution equations in one spatial variable. Following the approach developed by Basarab-Horwath et al. [Acta Appl. Math. 69, 43 (2001)], we construct all inequivalent partial differential equations belonging to the class in question which admit semisimple Lie groups. In addition, we describe all fourth-order evolution equations from the class under consideration which are invariant under solvable Lie groups of dimension n<=4. We have constructed all Galilei-invariant equations belonging to the class of evolution differential equations under study. The list of so obtained invariant equations contains both the well-known fourth-order evolution equations and a variety of new ones possessing rich symmetry and as such may be used to model nonlinear processes in physics, chemistry, and biology.

Group-theoretical framework for potential symmetries of evolution equations

We develop algebraic approach to the problem of classification of potential symmetries of nonlinear evolution equations. It is essentially based on the recently discovered fact [R. Zhdanov, J. Math. Phys. 50, 053522 (2009)], that any such symmetry is mapped into a contact symmetry. The approach enables using the classical results on classification of contact symmetries of nonlinear evolution equations by Sokolov and Magadeev to classify evolution equations admitting potential symmetries. We construct several examples of new nonlinear fourth-order evolution equations admitting potential symmetries. Since the symmetries obtained depend on nonlocal variables, they cannot be derived by the infinitesimal Lie approach.