Noether-Type Symmetries and Associated Conservation Laws of Some Systems of Nonlinear PDEs

The algorithm for constructing conservation laws of Euler Lagvange type equations via Noether-type symmetry operators associated with partial Lagrangian has been presented. As applications, many new conservation laws of some important systems of nonlinear partial differential equations have been obtained.     

On symmetries,conservation laws and exact solutions of the nonlinear Schrödinger–Hirota equation

In this paper, conservation laws and exact solution are found for nonlinear Schrödinger–Hirota equation. Conservation theorem is used for finding conservation laws. We get modified conservation laws for given equation. Modified simple equation method is used to obtain the exact solutions of the nonlinear Schrödinger–Hirota equation. It is shown that the suggested method provides a powerful mathematical instrument for solving nonlinear equations in mathematical physics and engineering.     

Conservation Laws of K（m,n） and mK（m,n） Equations

Based on the rank analysis method, algorithmization idea, and symbolic computation, in this paper we have presented a method to construct the conservation laws for nonlinear evolution equations. The polynomial conservation laws for K (n 2,n) equations and mK (m,n) equations are found by using of this approach and some new results have been obtained.     

Conservation Laws of K(m, n) and mK(m, n) Equations

Based on the rank analysis method, algorithmization idea, and symbolic computation, in this paper we have presented a method to construct the conservation laws for nonlinear evolution equations. The polynomial conservation laws for K (n 2, n) equations and mnK(m, n) equations are found by using of this approach and some new results have been obtained.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

Conservation Law Classification and Integrability of Generalized Nonlinear Second-Order Equation

This paper is concerned with the generalized nonlinear second-order equation. By the direct construction method, all of the first-order multipliers of the equation are obtained, and the corresponding complete conservation laws (CLs) of such equations are provided. Furthermore, the integrability of the equation is considered in terms of the conservation laws. In addition, the relationship of multipliers and symmetries of the equations is investigated.     

Infinitesimal symmetries and conservation laws of the DNLSE hierarchy and the Noether's theorem

The hierarchy of integrable nonlinear equations associated with the quadratic bundle is considered. The expressions for the solution of linearization of these equations and their conservation law in the terms of solutions of corresponding Lax pairs are found. It is shown for the first member of the hierarchy that the conservation law is connected with the solution of linearized equation due to the Noether's theorem. The local hierarchy and three nonlocal ones of the infinitesimal symmetries and conservation laws explicitly expressed through the variables of the nonlinear equations are derived.     

Infinitely-many conservation laws for two (2+1)-dimensional nonlinear evolution equations in fluids

In this paper, a method that can be used to construct the infinitely-many conservation laws with the Lax pair is generalized from the (1+1)-dimensional nonlinear evolution equations (NLEEs) to the (2+1)-dimensional ones. Besides, we apply that method to the Kadomtsev–Petviashvili (KP) and Davey–Stewartson equations in fluids, and respectively obtain their infinitely-many conservation laws with symbolic computation. Based on that method, we can also construct the infinitely-many conservation laws for other multidimensional NLEEs possessing the Lax pairs, including the cylindrical KP, modified KP and (2+1)-dimensional Gardner equations, in fluids, plasmas, optical fibres and Bose–Einstein condensates.     

On Differential Equations Describing 3-Dimensional Hyperbolic Spaces

In this paper, we introduce the notion of a (2+1)-dimensional differential equation describing three-dimensional hyperbolic spaces (3-h.s.). The (2+1)-dimensional coupled nonlinear Schrödinger equation and its sister equation, the (2+1)-dimensional coupled derivative nonlinear Schrödinger equation, are shown to describe 3-h.s. The (2+1)-dimensional generalized HF model: S_t={(1/2i)[S,S_y]+2iσS}_x, σ_x=-(1/4i)tr(SS_xS_y), in which S∈[GL_C(2)]/[GL_C(1)×GL_C(1)], provides another example of (2+1)-dimensional differential equations describing 3-h.s. As a direct consequence, the geometric construction of an infinite number of conservation laws of such equations is illustrated. Furthermore we display a new infinite number of conservation laws of the (2+1)-dimensional nonlinear Schrödinger equation and the (2+1)-dimensional derivative nonlinear Schrödinger equation by a geometric way.     

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.     

An Extension of Algorithm on Symbolic Computations of Conserved Densities for High-Dimensional Nonlinear Systems

An improved algorithm for symbolic computations of polynomial-type conservation laws (PCLaws) of a general polynomial nonlinear system is presented. The algorithm is implemented in Maple and can be successfully used for high-dimensional models. Furthermore, the algorithm discards the restriction to evolution equations. The program can also be used to determine the preferences for a given parameterized nonlinear systems. The code is tested on several known nonlinear equations from the soliton theory.     

Infinite sets of conservation laws for linear and nonlinear field equations

The relation between an infinite set of conservation laws of a linear field equation and the enveloping algebra of the space-time symmetry group is established. It is shown that each symmetric element of the enveloping algebra of the space-time symmetry group of a linear field equation generates a one-parameter group of symmetries of the field equation. The cases of the Maxwell and Dirac equations are studied in detail. Then it is shown that (at least in the sense of a power series in the coupling constant) the conservation laws of the linear case can be deformed to conservation laws of a nonlinear field equation which is obtained from the linear one by adding a nonlinear term invariant under the group of space-time symmetries. As an example, our method is applied to the Korteweg-de Vries equation and to the massless Thirring model.     

Conservation laws for classes of nonlinear evolution equations solvable by the spectral transform

The existence of conservation laws for novel classes of nonlinear evolution equations (with linearlyx-dependent coefficients) solvable by the spectral transform is investigated. A remarkably explicit representation is moreover obtained for the conserved quantities of the old classes of nonlinear evolution equations (withx-independent coefficients; including the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the nonlinear Schrödinger equation, etc.).     

Bäcklund transformations and new solutions of nonlinear wave equations in four-dimensional space-time

Bäcklund transformations for several nonlinear field equations in four-dimensional space-time relating two solutions of the same equation (symmetry), or two different equations (dynamical), are given. These transformations can be used to generate new families of solutions and infinitely many conservation laws for nonlinear equations.Bäcklund transformations and solutions of nonlinear equations have been studied extensively in one-space and one-time dimension. We give here a fairly general method for a class of equations in four-dimensional space-time which paves the way for many further generalizations.Supported in part by the U.S. National Academy of Sciences Foundation Grant No. INT 73-20002 A01 (formerly GF-41958).     

Conservation Laws and Darboux Transformation for Sharma-Tasso-Olver Equation

A hierarchy of new nonlinear evolution equations associated with a 2?2 matrix spectral problem is derived. One of the nontrivial equations in this hierarchy is the famous Sharma-Tasso-Olver equation. Then infinitely many conservation laws of this equation are deduced. Darboux transformation for the Sharma-Tasso-Olver equation is constructed with the aid of a gauge transformation.     

Reciprocal bäcklund transformations of conservation laws

Bäcklund transformations of a reciprocal-type are developed for a broad class of conservation laws. The basic result may be used to generate auto-Bäcklund transformations for reciprocally associated nonlinear evolution equations. A permutability diagram for the generation of solutions is presented.     

Conserved Gross–Pitaevskii equations with a parabolic potential

An integrable Gross–Pitaevskii equation with a parabolic potential is presented where particle density ∣u∣² is conserved. We also present an integrable vector Gross–Pitaevskii system with a parabolic potential, where the total particle density ${\sum }_{j=1}^{n}| {u}_{j}{| }^{2}$ is conserved. These equations are related to nonisospectral scalar and vector nonlinear Schrödinger equations. Infinitely many conservation laws are obtained. Gauge transformations between the standard isospectral nonlinear Schrödinger equations and the conserved Gross–Pitaevskii equations, both scalar and vector cases are derived. Solutions and dynamics are analyzed and illustrated. Some solutions exhibit features of localized-like waves.     

Conservation laws of the complex short pulse equation and coupled complex short pulse equations

In this paper, the complex short pulse equation and the coupled complex short pulse equations that can describe the ultra-short pulse propagation in optical fibers are investigated. The two complex nonlinear models are turned into multi-component real models by proper transformations. Lie symmetries are obtained via the classical Lie group method, and the results for the coupled complex short pulse equations contain the existing results as particular cases. Based on the linearizing operator and adjoint linearizing operator for the two real systems, adjoint symmetries can be obtained. Explicit conservation laws are constructed using the symmetry/adjoint symmetry pair (SA) method. Relationships between the nonlinear self-adjointness method and the SA method are investigated.     

On Integrable Properties for Two Variable-Coefficient Evolution Equations

With the help of the extended binary Bell polynomials, the new bilinear representations, Bcklund trans-formations, Lax pair and infinite conservation laws for two types of variable-coefficient nonlinear integrable equations are obtained, respectively, which are more straightforward than previous corresponding results obtained. Finally, we obtain new multi-soliton wave solutions of a reduced soliton equations with variable coefficients.     

Hamiltonian System and Infinite Conservation Laws Associated with a New Discrete Spectral Problem

Starting from a new discrete spectral problem, the corresponding hierarchy of nonlinear lattice equations is proposed. It is shown that the lattice soliton hierarchy possesses the bi-Hamiltonian structures and infinitely many common commuting conserved functions. Further, infinite conservation laws of the hierarchy are presented.