Group Analysis,Fractional Explicit Solutions and Conservation Laws of Time Fractional Generalized Burgers Equation

The generalized fractional Burgers equation is studied in this paper. Using the classical Lie symmetry method, all of the vector fields and symmetry reduction of the equation with nonlinearity are constructed. In particular,an exact solution is provided by using the ansatz method. In addition, other types of exact solution are obtained via the invariant subspace method. Finally, conservation laws for this equation are derived.     

Conservation Laws and Soliton Solutions for Generalized Seventh Order KdV Equation

With the assistance of the symbolic computation system Maple, rich higher order polynomial-type conservation laws and a sixth order t/x-dependent conservation law are constructed for a generalized seventh order nonlinear evolution equation by using a direct algebraic method. From the compatibility conditions that guaranteeing the existence of conserved densities, an integrable unnamed seventh order KdV-type equation is found. By introducing some nonlinear transformations, the one-, two-, and three-solition solutions as well as the solitary wave solutions are obtained.     

On New Invariant Solutions of Generalized Fokker-Planck Equation

The generalized one-dimensional Fokker-Planck equation is analyzed via potential symmetry method and the invariant solutions under potential symmetries are obtained. Among those solutions, some are new and first reported.     

Infinite Sequence of Conservation Laws and Analytic Solutions for a Generalized Variable-Coefficient Fifth-Order Korteweg-de Vries Equation in Fluids

In this paper, an infinite sequence of conservation laws for a generalized variable-coefficient fifth-order Korteweg-de Vries equation in fluids are constructed based on the Bäcklund transformation. Hirota bilinear form and symbolic computation are applied to obtain three kinds of solutions. Variablecoefficients can affect the conserved density, associated flux, andappearance of the characteristic lines. Effects of the wave number on the soliton structures are also discussed and types of soliton structures, e.g., the double-periodic soliton, parallel soliton and soliton complexes, are presented.     

Potential Symmetries and Associated Conservation Laws to Fokker-Planck and Burgers Equation

Kara and Mahomed have derived an identity, which does not rely on use of a Lagrangian as needed to obtain conservation laws by Noethers theorem. By using the identity and symbolic computation, conservation laws arising from nonlocal symmetries are obtained for Fokker-Planck equation and burgers equation.     

Double Reduction and Exact Solutions of Zakharov–Kuznetsov Modified Equal width Equation with Power Law Nonlinearity via Conservation Laws

The conservation laws for the （1＋2）-dimensional Zakharov-Kuznetsov modified equal width （ZK-MEW） equation with power law nonlinearity are constructed by using Noether＇s approach through an interesting method of increasing the order of this equation. With the aid of an obtained conservation law, the generalized double reduction theorem is applied to this equation. It can be shown that the reduced equation is a second order nonlinear ODE. FinaJ1y, some exact solutions for a particular case of this equation are obtained after solving the reduced equation.     

Conservation Laws and Analytic Soliton Solutions for Coupled Integrable Dispersionless Equations with Symbolic Computation

Under investigation in this paper are two coupled integrable dispersionless （CID） equations modeling the dynamics of the current-fed string within an external magnetic field. Through a set of the dependent variable transformations, the bilinear forms for the CID equations are derived. Based on the Hirota method and symbolic computation, the analytic N-soliton solutions are presented. Infinitely many conservation laws for the CID equations are given through the known spectral problem. Propagation characteristics and interaction behaviors of the solitons are analyzed graphically.     

Double Reduction and Exact Solutions of Zakharov-Kuznetsov Modified Equal width Equation with Power Law Nonlinearity via Conservation Laws

The conservation laws for the (1+2)-dimensional Zakharov-Kuznetsov modified equal width (ZK-MEW) equation with power law nonlinearity are constructed by using Noether's approach through an interesting method of increasing the order of this equation. With the aid of an obtained conservation law, the generalized double reduction theorem is applied to this equation. It can be shown that the reduced equation is a second order nonlinear ODE. Finally, some exact solutions for a particular case of this equation are obtained after solving the reduced equation.     

Conservation Laws and Analytic Soliton Solutions for Coupled Integrable Dispersionless Equations with Symbolic Computation

Under investigation in this paper are two coupled integrable dispersionless (CID) equations modeling the dynamics of the current-fed string within an external magnetic field. Through a set of the dependent variable transformations, the bilinear forms for the CID equations are derived. Based on the Hirota method and symbolic computation, the analytic N-soliton solutions are presented. Infinitely many conservation laws for the CID equations are given through the known spectral problem. Propagationcharacteristics and interaction behaviors of the solitons are analyzed graphically.     

Exact Analytical Solutions of the Generalized Calogero-Bogoyavlenskii-Schiff Equation Using Symbolic Computation

By means of generalized Riccati equation expansion method and symbolic computation, some exact analytical solutions, which contain soliton-like solutions and periodic-like solutions to the generalized Calogero-Bogoyavlenskii-Schiff (GCBS) equation, are obtained. From our results, the solitary-wave solutions and previously known soliton-like solutions of the special cases of GCBS equation can be recovered.     

Conservation Laws and Symmetries of the Levi Equation

The conservation laws of the Levi equation are presented. Two types of symmetry of the Levi equation hierarchy are deduced, Further it is proved that these symmetries construct an infinite-dimensional Lie algebra.     

Conservation Laws and Symmetries of the Levi Equation

The conservation laws of the Levi equation are presented. Two types of symmetry of the Levi equation hierarchy are deduced. Further it is proved that these symmetries construct an infinite-dimensional Lie algebra.     

Nonlinear Self-Adjointness and Conservation Laws for the Hyperbolic Geometric Flow Equation

We study the nonlinear self-adjointness of a class of quasilinear 2D second order evolution equations by applying the method of Ibragimov. Which enables one to establish the conservation laws for any differential equation. We first obtain conditions determining the self-adjointness for a sub-class in the general case. Then, we establish the conservation laws for hyperbolic geometric flow equation on Riemman surfaces.     

Integrable Hierarchy Covering the Lattice Burgers Equation in Fluid Mechanics:N-fold Darboux Transformation and Conservation Laws

Burgers-type equations can describe some phenomena in fluids,plasmas,gas dynamics,traffic,etc.In this paper,an integrable hierarchy covering the lattice Burgers equation is derived from a discrete spectral problem.N-fold Darboux transformation(DT) and conservation laws for the lattice Burgers equation are constructed based on its Lax pair.N-soliton solutions in the form of Vandermonde-like determinant are derived via the resulting DT with symbolic computation,structures of which are shown graphically.Coexistence of the elastic-inelastic interaction among the three solitons is firstly reported for the lattice Burgers equation,even if the similar phenomenon for certern continuous systems is known.Results in this paper might be helpful for understanding some ecological problems describing the evolution of competing species and the propagation of nonlinear waves in fluids.     

Integrable Hierarchy Covering the Lattice Burgers Equation in Fluid Mechanics: N-fold Darboux Transformation and Conservation Laws

Burgers-type equations can describe some phenomena in fluids, plasmas, gas dynamics, traffic, etc. In this paper, an integrable hierarchy covering the lattice Burgers equation is derived from a discrete spectral problem. N-fold Darboux transformation (DT) and conservation laws for the lattice Burgers equation are constructed based on its Lax pair. N-soliton solutions in the form of Vandermonde-like determinant are derived via the resulting DT with symbolic computation, structures of which are shown graphically. Coexistence of the elastic-inelastic interaction among the three solitons is firstly reported for the lattice Burgers equation, even if the similar phenomenon for certern continuous systems is known. Results in this paper might be helpful for understanding some ecological problems describing the evolution of competing species and the propagation of nonlinear waves in fluids.     

CTE Solvability,Exact Solutions and Nonlocal Symmetries of the Sharma-Tasso-Olver Equation

In this letter, we prove that the STO equation is CTE solvable and obtain the exact solutions of solitons fission and fusion. We also provide the nonlocal symmetries of the STO equation related to CTE. The nonlocal symmetries are localized by prolonging the related enlarged system.     

Conditional Stability of Solitary-Wave Solutions for Generalized Compound KdV Equation and Generalized Compound KdV-Burgers Equation

In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the generalized compound KdV equation and the generalized compound KdV-Burgers equations. Linear stability of the exact solitary-wave solutions is proved for the above two types of equations when the small disturbance of travellingwave form satisfies some special conditions.     

Conditional Stability of Solitary-Wave Solutions for Generalized Compound KdV Equation and Generalized Compound KdV-Burgers Equation

In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the generalized compound KdV equation and the generalized compound KdV-Burgers equations. Linear stability of the exact solitary-wave solutions is proved for the above two types of equations when the small disturbance of travelling wave form satisfies some special conditions.     

Symmetry Reductions, Exact Solutions and Conservation Laws of Asymmetric Nizhnik-Novikov-Veselov Equation

By applying a direct symmetry method, we get the symmetry of the asymmetric Nizhnik-Novikov-Veselov equation （ANNV）. Taking the special case, we have a finite-dimensional symmetry. By using the equivalent vector of the symmetry, we construct an eight-dimensional symmetry algebra and get the optimal system of group-invariant solutions. To every case of the optimal system, we reduce the ANNV equation and obtain some solutions to the reduced equations. Furthermore, we find some new explicit solutions of the ANNV equation. At last, we give the conservation laws of the ANNV equation.     

Symmetry Analysis and Conservation Laws to the(2+1)-Dimensional Coupled Nonlinear Extension of the Reaction-Diffusion Equation

In this paper, a detailed Lie symmetry analysis of the(2+1)-dimensional coupled nonlinear extension of the reaction-diffusion equation is presented. The general finite transformation group is derived via a simple direct method,which is equivalent to Lie point symmetry group actually. Similarity reduction and some exact solutions of the original equation are obtained based on the optimal system of one-dimensional subalgebras. In addition, conservation laws are constructed by employing the new conservation theorem.