Bilinear Representations of Integrable Equations

We present a method for deriving recursion operators and canonical Lax pairs directly from bilinear identities of the KP type. Examples include the KdV equation, the Boussinesq equation, and a real equivalent of the nonlinear Schrödinger equation.     

Construction of a hierarchy of negative‐order integrable Burgers equations of higher orders

In this work, we employ the recursion operator, the Burgers equation and its inverse operator, for constructing a hierarchy of negative‐order integrable Burgers equations of higher orders. The complete integrability of each established equation emerges by virtue of the correlation between integrability and recursion operators. We use the simplified Hirota's method to obtain multiple kink solutions for some of the derived equations, and in particular, for the generalized negative‐order Burgers equation.     

l-solutions and stability analysis of difference equations using the Kummer’s test

We address the p-summability and asymptotic stability properties in nonautonomous linear difference equations. We focus our discussion on two kind of difference equations. The first one is a first order system of linear nonautonomous difference equations, and our discussion involves the use of Kummer’s convergence test. The second one is a linear nonautonomous scalar higher order difference equation. In this case our discussion is based on a recently introduced transformation of a higher order system into a first-step recursion, where the companion matrices are well treatable from our point of view. We give insight on our ideas that are behind our methods, prove new results, and show applications.     

A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations

In this paper we propose a new modified recursion scheme for the resolution of multi-order and multi-point boundary value problems for nonlinear ordinary and partial differential equations by the Adomian decomposition method (ADM). Our new approach, including Duan’s convergence parameter, provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems, in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series. We utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components. Thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components, whereas several prior recursion schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots, which is required to complete calculation of the solution by several prior modified recursion schemes using the ADM.     

Riemann–Hilbert Problems,Matrix Orthogonal Polynomials and Discrete Matrix Equations with Singularity Confinement

In this paper, matrix orthogonal polynomials in the real line are described in terms of a Riemann–Hilbert problem. This approach provides an easy derivation of discrete equations for the corresponding matrix recursion coefficients. The discrete equation is explicitly derived in the matrix Freud case, associated with matrix quartic potentials. It is shown that, when the initial condition and the measure are simultaneously triangularizable, this matrix discrete equation possesses the singularity confinement property, independently if the solution under consideration is given by the recursion coefficients to quartic Freud matrix orthogonal polynomials or not.     

Nonlocal symmetries and recursion operators: Partial differential and differential-difference equations

A systematic method to derive the nonlocal symmetries for partial differential and differential-difference equations with two independent variables is presented and shown that the Korteweg-de Vries (KdV) and Burger's equations, Volterra and relativistic Toda (RT) lattice equations admit a sequence of nonlocal symmetries. An algorithm, exploiting the obtained nonlocal symmetries, is proposed to derive recursion operators involving nonlocal variables and illustrated it for the KdV and Burger's equations, Volterra and RT lattice equations and shown that the former three equations admit factorisable recursion operators while the RT lattice equation possesses (2×2) matrix factorisable recursion operator. The existence of nonlocal symmetries and the corresponding recursion operator of partial differential and differential-difference equations does not always determine their mathematical structures, for example, bi-Hamiltonian representation.     

Nonlocal Symmetries of the Camassa-Holm Type Equations

A class of nonlocal symmetries of the Camassa-Holm type equations with bi-Hamiltonian structures, including the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation and Degasperis-Procesi equation, is studied. The nonlocal symmetries are derived by looking for the kernels of the recursion operators and their inverse operators of these equations. To find the kernels of the recursion operators, the authors adapt the known factorization results for the recursion operators of the KdV, modified KdV, Sawada-Kotera and Kaup-Kupershmidt hierarchies, and the explicit Liouville correspondences between the KdV and Camassa-Holm hierarchies, the modified KdV and modified Camassa-Holm hierarchies, the Novikov and Sawada-Kotera hierarchies, as well as the Degasperis-Procesi and Kaup-Kupershmidt hierarchies.     

Darboux transformations and recursion operators for differential-difference equations

We review two concepts directly related to the Lax representations of integrable systems: Darboux transformations and recursion operators. We present an extensive list of integrable differential-difference equations with their Hamiltonian structures, recursion operators, nontrivial generalized symmetries, and Darboux-Lax representations. The new results include multi-Hamiltonian structures and recursion operators for integrable Volterra-type equations and integrable discretizations of derivative nonlinear Schrödinger equations such as the Kaup-Newell, Chen-Lee-Liu, and Ablowitz-Ramani-Segur (Gerdjikov-Ivanov) lattices. We also compute the weakly nonlocal inverse recursion operators.     

Stability with respect to the input data and monotonicity of an implicit finite-difference scheme for a quasilinear parabolic equation

We analyze the stability and monotonicity of a conservative difference scheme approximating an initial-boundary value problem for a quasilinear parabolic equation under specific conditions imposed solely on the problem input data. We prove some kinds of the maximum principle for the nonlinear equations that are used in the derivation of a priori estimates for the solution; we also prove estimates for some kinds of recursion inequalities that are used in the derivation of a priori estimates for higher-order derivatives, these estimates being necessary for proving the continuous dependence of the solution on small perturbations of the input data and for analyzing monotonicity in the nonlinear case. We show that, depending on the properties of the input data, higher derivatives can become infinite in finite critical time. We obtain conditions on the input data guaranteeing the stability of the difference scheme on the entire time interval.     

Lax pair, cyclic basis, and a new integrable system

A direct approach to zero-curvature representation, as introduced by Marvan is applied to generate a new class of integrable equations by starting with the generic Lax operator (x-part) for a coupled set of nonlinear equations. The class of equation so obtained is more general than those usually obtained with the help of standard recursion operator. Finally some particular type of equations are identified by the special choice of the arbitrary functions occurring in the final solution of the time part of the Lax equation. The methodology is specially useful when the x-part of the Lax operator does not contain any spectral parameter.     

Two Algorithms for Solving a Kind of Heat Conduction Equations

In this paper, a strategy is suggested for numerical solution of a kind of parabolic partial differential equations with nonlinear boundary conditions and discontinuous coefficients, which arise from practical engineering problems. First, a difference equation at the discontinuous point is established in which both the stability and the truncation error are consistent with the total difference equations. Then, on account of the fact that the coefficient matrix of the difference equations is tridiagonal and nonlinearity appears only in the first and the last equations, two algorithms are suggested: a mixed method combining the modified Gaussian elimination method with the successive recursion method, and a variant of the modified Gaussian elimination method. These algorithms are shown to be effective.     

A Direct Algorithm for Constructing Recursion Operators and Lax Pairs for Integrable Models

We suggest an algorithm for seeking recursion operators for nonlinear integrable equations. We find that the recursion operator R can be represented as a ratio of the form R = L₁^?1 L₂, where the linear differential operators L₁ and L₂ are chosen such that the ordinary differential equation (L₂ ?λL₁)U = 0 is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter λ ∈ C. To construct the operator L1, we use the concept of an invariant manifold, which is a generalization of a symmetry. To seek L₂, we then take an auxiliary linear equation related to the linearized equation by a Darboux transformation. It is remarkable that the equation L₁\(\tilde U\) = L2U defines a B¨acklund transformation mapping a solution U of the linearized equation to another solution \(\tilde U\) of the same equation. We discuss the connection of the invariant manifold with the Lax pairs and the Dubrovin equations.     

Nonlinear equations for the recurrence coefficients of discrete orthogonal polynomials

We study polynomials orthogonal on a uniform grid. We show that each weight function gives two potentials and each potential leads to a structure relation (lowering operator). These results are applied to derive second order difference equations satisfied by the orthogonal polynomials and nonlinear difference equations satisfied by the recursion coefficients in the three-term recurrence relations.     

Commutator identities on associative algebras,the non-Abelian Hirota difference equation and its reductions

We show that the non-Abelian Hirota difference equation is directly related to a commutator identity on an associative algebra. Evolutions generated by similarity transformations of elements of this algebra lead to a linear difference equation. We develop a special dressing procedure that results in an integrable non-Abelian Hirota difference equation and propose two regular reduction procedures that lead to a set of known equations, Abelian or non-Abelian, and also to some new integrable equations.     

Higher-order conservation laws for the nonlinear Poisson equation via characteristic cohomology

We study higher-order conservation laws of the nonlinearizable elliptic Poisson equation as elements of the characteristic cohomology of the associated exterior differential system. The theory of characteristic cohomology determines a normal form for differentiated conservation laws by realizing them as elements of the kernel of a linear differential operator. We show that the \mathbbS¹{\mathbb{S}^1} -symmetry of the PDE leads to a normal form for the undifferentiated conservation laws. Zhiber and Shabat (in Sov Phys Dokl Akad 24(8):607–609, 1979) determine which potentials of nonlinearizable Poisson equations admit nontrivial Lie–B?cklund transformations. In the case that such transformations exist, they introduce a pseudo-differential operator that can be used to generate infinitely many such transformations. We obtain similar results using the theory of characteristic cohomology: we show that for higher-order conservation laws to exist, it is necessary that the potential satisfies a linear second-order ODE. In this case, at most two new conservation laws in normal form appear at each even prolongation. By using a recursion motivated by Killing fields, we show that, for the simplest class of potentials, this upper bound is attained. The recursion circumvents the use of pseudo-differential operators. We relate higher-order conservation laws to generalized symmetries of the exterior differential system by identifying their generating functions. This Noether correspondence provides the connection between conservation laws and the canonical Jacobi fields of Pinkall and Sterling.     

压电热弹性体的变分原理及正则方程和齐次方程

结合对偶变量理论,为压电热弹性体混合层合板问题推导了齐次的控制方程和Hamilton等参元列式．首先根据广义的Hamilton变分原理推导了压电热弹性体非齐次的Hamilton正则方程．然后进一步考虑了热平衡方程与导热方程中变量的对偶关系,通过增加正则方程的维数,成功地将非齐次的正则方程转化为能独立求解压电热弹性体耦合问题的齐次控制方程．为了推导四节点Hamilton等参元列式的方便,可将温度梯度关系类比成本构关系并构建新的变分原理．齐次方程大大简化了人们在分析压电热弹性体耦合问题时,通常要求解非齐次方程和关于平衡方程和导热方程的二阶微分方程的繁琐方法,同时也减少了数值计算工作量．     

On the nature of the de Branges Hamiltonian

We prove the theorem announced by the author in 1995 in the paper “A criterion for the discreteness of the spectrum of a singular canonical system” (Funkts. Anal. Prilozhen., 29, No. 3).In developing the theory of Hilbert spaces of entire functions (we call them Krein-de Branges spaces), de Branges arrived at a certain class of canonical equations of phase dimension 2. He showed that, for any given Krein-de Branges space, there exists a canonical equation of the class indicated that restores a chain of Krein-de Branges spaces imbedded one into another. The Hamiltonians of such canonical equations are called de Branges Hamiltonians. The following question arises: Under what conditions will the Hamiltonian of a certain canonical equation be a de Branges Hamiltonian? The main theorem of the present work, together with Theorem 1 of the paper cited above, gives an answer to this question.     

Rotationally invariant hyperbolic waves

We use weakly nonlinear asymptotics to derive a canonical asymptotic equation for rotationally invariant hyperbolic waves. The equation can include weak dissipative, dispersive, or diffractive effects. We give applications to equations from magnetohydrodynamics, elasticity, and viscoelasticity.