Conservation laws for a super G-J hierarchy with self-consistent sources

Based on a well known super Lie algebra, a super integrable system is presented. Then, the super G-J hierarchy with self-consistent sources are obtained. Furthermore, we establish the infinitely many conservation laws for the integrable super G-J hierarchy. The methods derived by us can be generalized to other nonlinear equations hierarchies with self-consistent sources.     

Constructing super D-Kaup-Newell hierarchy and its nonlinear integrable coupling with self-consistent sources

How to construct new super integrable equation hierarchy is an important problem. In this paper, a new Lax pair is proposed and the super D-Kaup-Newell hierarchy is generated, then a nonlinear integrable coupling of the super D-Kaup-Newell hierarchy is constructed. The super Hamiltonian structures of coupling equation hierarchy is derived with the aid of the super variational identity. Finally, the self-consistent sources of super integrable coupling hierarchy is established. It is indicated that this method is a straight- forward and efficient way to construct the super integrable equation hierarchy.     

Super Jaulent-Miodek hierarchy and its super Hamiltonian structure,conservation laws and its self-consistent sources

A super Jaulent-Miodek hierarchy and its super Hamiltonian structures are constructed by means of a kind of Lie super algebras and super trace identity. Moreover, the self-consistent sources of the super Jaulent-Miodek hierarchy is presented based on the theory of self-consistent sources. Further- more, the infinite conservation laws of the super Jaulent-Miodek hierarchy are also obtained. It is worth noting that as even variables are boson variables, odd variables are fermi variables in the spectral problem, the commutator is different from the ordinary one.     

A non-isospectral integrable couplings of Volterra lattice hierarchy with self-consistent sources

A kind of non-isospectral integrable couplings of discrete soliton equations hierarchy with self-consistent sources associated with [Y.F. Zhang, E.G. Fan, Characteristic Numbers of Matrix Lie Algebras, Commun. Theor. Phys (China) 49 (2008) 845] is presented. As an application example, the hierarchy of non-isospectral cubic Volterra lattice hierarchy with self-consistent sources is derived. Furthermore, we construct a non-isospectral integrable couplings of cubic Volterra lattice hierarchy with self-consistent sources by using the loop algebra .     

The finite-dimensional super integrable system of a super NLS-mKdV equation

Based on the constructed Lie superalgebra, the super Hamiltonian structure of a NLS-mKdV hierarchy is obtained by making use of super-trace identity. Moreover, an explicit super Bargmann symmetry constraint and its associated binary nonlinearization of Lax pairs are carried out for the super NLS-mKdV system.     

Conservation laws for a modified lubrication equation

In this work we study the conservation laws of a modified lubrication equation, which describes the dynamics of the interfacial motion in phase transition. We show that the equation is nonlinear self-adjoint and has an exact Lagrangian with an auxiliary function. As a result, by a general theorem on conservation laws proved by Nail Ibragimov recently and Noether’s theorem, some new conservation laws for the equation are obtained. Our results show that the non-locally defined conservation laws generated by Noether’s theorem are equivalent to the local ones given by Ibragimov’s theorem.     

The exact solutions to a Ragnisco-Tu hierarchy with self-consistent sources

The Ragnisco-Tu hierarchy with self-consistent sources is derived. The exact solutions of the hierarchy are obtained via the inverse scattering transform (IST). An explicit form for a solution of the Ragnisco-Tu equation is presented.     

非线性对流扩散方程的守恒律

利用直接方法研究了非线性对流扩散方程的守恒律,得到了关于非线性对流扩散方程的守恒律乘子性质的一个定理.利用这个定理,可以简化守恒律乘子的确定方程.随后通过对确定方程中的变量函数进行分析,发现在四种情况下乘子的确定方程是可解的.最后解出这些守恒律乘子,利用积分公式法分别得到了四种情况下对应于各个守恒律乘子的守恒律.     

An integrable lattice hierarchy, associated integrable coupling, Darboux transformation and conservation laws

A new integrable lattice hierarchy is constructed from a discrete matrix spectral problem, some related properties of the new hierarchy are discussed. The Hamiltonian structures and Liouville integrability of the new hierarchy are established by using the discrete trace identity. A kind of integrable coupling for the new hierarchy is constructed through enlarging spectral problems. A Darboux transformation (DT) with two variable parameters and the infinitely many conservation laws for a typical lattice equation in the new hierarchy are constructed based on its Lax representation, the explicit solutions are obtained via the DT, the structures for those solutions are graphically investigated. All these properties might be helpful to understanding some physical phenomena.     

The Nizhnik-Veselov-Novikov equation with self-consistent sources

We construct the coupled generalization of the Nizhnik-Veselov-Novikov (NVN) equation, i.e., the NVN equation with self-consistent sources (ESCS) and solve it using the so-called source generation procedure. We obtain BKP-type and DKP-type Pfaffian solutions of the NVN ESCS. In addition, we give a bilinear Bäcklund transformation and a Lax pair for the NVN ESCS. Furthermore, this Lax pair can be reduced to a Lax pair for the NVN equation.     

Noncommutative integrable coupling system of TC equation hierarchy

A noncommutative version of the TC soliton equation hierarchy is presented, which possesses the zero curvature representation. Then, we show that noncommutative (NC) TC equation can be derived from the noncommutative (anti-)self-dual Yang-Mills equation by reduction. Finally, an integrable coupling system of the NC TC equation hierarchy is constructed by using of the enlarged Lax pairs.     

Broer-Kaup-Kupershmidt族的非线性双可积耦合及其自相容源

基于新的非半单矩阵李代数,介绍了构造孤子族非线性双可积耦合的方法,由相应的变分恒等式给出了孤子族非线性双可积耦合的Hamilton结构.作为应用,给出了Broer-Kaup-Kupershmidt族的非线性双可积耦合及其Hamilton结构.最后指出了文献中的一些错误,利用源生成理论建立了新的公式,并导出了带自相容源Broer-Kaup-Kupershmidt族的非线性双可积耦合方程.     

On the pfaffianized-KP equation with self-consistent sources

A new procedure called ‘source generation’ is applied to the pfaffianized KP equation. As a result, the pfaffianized-KP equation with self-consistent sources (ESCS) is obtained. This coupled system cannot only be reduced to the pfaffianized KP equation, but also reduced to the KP equation with self-consistent sources (KPESCS). So the pfaffianized-KP ESCS can be viewed as a pfaffian version of the KPESCS, which indicates the commutativity of the ‘source generation’ procedure and pfaffianization.     

Conservation laws with singular nonlocal sources

Several physical phenomena are modeled by conservation laws with fluxes or sources that are singular in the origin. Here an integro-differential regularization of those problems is proposed. The existence of positive solutions with finite total variation is proved.     

Hierarchies,integrable decompositions and conservation laws of Geng equation

In this paper, two hierarchies of the Geng equations are presented, including positive non-isospectral hierarchy and negative non-isospectral hierarchy. Moreover, integrable couplings of the corresponding Geng hierarchies are also constructed by enlarging the associated matrix spectral problem. Three new integrable decompositions and conservation laws of the isospectral Geng equation are also obtained. The Gauge transformations are used to obtain the associated binary symmetry constraints of the Geng equation at the first time.     

一个新的离散的演化方程族及其Hamilton结构

A new discrete isospectral problem is introduced, from which a hierarchy of Laxintegrable lattice equation is deduced. By using the trace identity, the correspondingHamiltonian structure is given and its Liouville integrability is proved.     

A hierarchy of differential-difference equations,conservation laws and new integrable coupling system

Starting from a discrete spectral problem with two arbitrary parameters, a hierarchy of nonlinear differential-difference equations is derived. The new hierarchy not only includes the original hierarchy, but also the well-known Toda equation and relativistic Toda equation. Moreover, infinitely many conservation laws for a representative discrete equation are given. Further, a new integrable coupling system of the resulting hierarchy is constructed.     

Conservation laws for the Black–Scholes equation

In the search for solutions to the important partial differential equation due to Black, Scholes and Merton potential symmetries are very useful as new solutions of the equation can be obtained as a result. These potential symmetries require that the equation be written in conserved form, ie. we need to determine conservation laws for the equation. We calculate the conservation laws utilizing the point symmetries of the equation following the method of Kara and Mahomed [A.H. Kara, F.M. Mahomed, The relationship between symmetries and conservation laws, Int. J. Theor. Phys. 39 (2000) 23–40].