Integrable structures for a generalized Monge-Ampère equation

We consider a third-order generalized Monge-Ampère equation u_yyy ? u _xxy ² + u_xxxu_xyy = 0, which is closely related to the associativity equation in two-dimensional topological field theory. We describe all integrable structures related to it: Hamiltonian, symplectic, and also recursion operators. We construct infinite hierarchies of symmetries and conservation laws.     

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.     

On Integrability of the Camassa–Holm Equation and Its Invariants

Using geometrical approach exposed in (Kersten et al. in J. Geom. Phys. 50:273–302, [2004] and Acta Appl. Math. 90:143–178, [2005]), we explore the Camassa–Holm equation (both in its initial scalar form, and in the form of 2×2-system). We describe Hamiltonian and symplectic structures, recursion operators and infinite series of symmetries and conservation laws (local and nonlocal). This work was supported in part by the NWO–RFBR grant 047.017.015 and RFBR–Consortium E.I.N.S.T.E.I.N. grant 06-01-92060.     

离散晶格的正族、负族及其Hamilton结构

构造了一个新的等谱问题，利用相容性条件，推导出离散晶格方程的正族和负族。再利用迹恒等式，建立其Hamilton 结构。获得的离散方程族的达布变换、双线性化、对称、守恒率及其精确解也值得进一步研究。     

Discrete equation on a square lattice with a nonstandard structure of generalized symmetries

We clarify the integrability nature of a recently found discrete equation on the square lattice with a nonstandard symmetry structure. We find its L-A pair and show that it is also nonstandard. For this discrete equation, we construct the hierarchies of both generalized symmetries and conservation laws. This equation yields two integrable systems of hyperbolic type. The hierarchies of generalized symmetries and conservation laws are also nonstandard compared with known equations in this class.     

Recursion operators,conservation laws,and integrability conditions for difference equations

We attempt to propose an algebraic approach to the theory of integrable difference equations. We define the concept of a recursion operator for difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. As in the case of partial differential equations, these canonical densities can serve as integrability conditions for difference equations. We obtain the recursion operators for the Viallet equation and all the Adler-Bobenko-Suris equations.     

A basis of hierarchy of generalized symmetries and their conservation laws for the (3+1)-dimensional diffusion equation

We determine, by hierarchy, dependencies between higher order linear symmetries which occur when generating them using recursion operators. Thus, we deduce a formula which gives the number of independent generalized symmetries (basis) of several orders. We construct a basis for conservation laws (with respect to the group admitted by the system of differential equations) and hence generate infinitely many conservation laws in each equivalence class.     

Hamiltonian spatial structure for three-dimensional water waves in a moving frame of reference

Summary The governing equations for three-dimensional time-dependent water waves in a moving frame of reference are reformulated in terms of the energy and momentum flux. The novelty of this approach is that time-independent motions of the system—that is, motions that are steady in a moving frame of reference—satisfy a partial differential equation, which is shown to be Hamiltonian. The theory of Hamiltonian evolution equations (canonical variables, Poisson brackets, symplectic form, conservation laws) is applied to the spatial Hamiltonian system derived for pure gravity waves. The addition of surface tension changes the spatial Hamiltonian structure in such a way that the symplectic operator becomes degenerate, and the properties of this generalized Hamiltonian system are also studied. Hamiltonian bifurcation theory is applied to the linear spatial Hamiltonian system for capillary-gravity waves, showing how new waves can be found in this framework.     

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.     

Backward error analysis for multi-symplectic integration methods

Summary. A useful method for understanding discretization error in the numerical solution of ODEs is to compare the system of ODEs with the modified equations obtained through backward error analysis, and using symplectic integration for Hamiltonian ODEs provides more incite into the modified equations. In this paper, the ideas of symplectic integration are extended to Hamiltonian PDEs, and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations. In particular, local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize, but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution. These results are also applied to the nonlinear wave equation. Mathematics Subject Classification (1991):65M06, 65P10, 37K05     

Symmetries for the Ablowitz–Ladik Hierarchy: Part I. Four‐Potential Case

In the paper, we first investigate symmetries of isospectral and non‐isospectral four‐potential Ablowitz–Ladik hierarchies. We express these hierarchies in the form of u_n,t= L^mH⁽⁰⁾ , where m is an arbitrary integer (instead of a nature number) and L is the recursion operator. Then by means of the zero‐curvature representations of the isospectral and non‐isospectral flows, we construct symmetries for the isospectral equation hierarchy as well as non‐isospectral equation hierarchy, respectively. The symmetries, respectively, form two centerless Kac‐Moody‐Virasoro algebras. The recursion operator L is proved to be hereditary and a strong symmetry for this isospectral equation hierarchy. Besides, we make clear for the relation between four‐potential and two‐potential Ablowitz–Ladik hierarchies. The even order members in the four‐potential Ablowitz–Ladik hierarchies together with their symmetries and algebraic structures can be reduced to two‐potential case. The reduction keeps invariant for the algebraic structures and the recursion operator for two potential case becomes L² .     

Three series of invariant manifolds of the Sawada-Kotera equation

We find a new infinite sequence of invariant manifolds for the Sawada-Kotera equation, in addition to the known two sequences of its symmetries and conservation laws. The elements of these three sequences are related cyclically by recursion relations similar to the Lenard formula for the KdV equation. For any n > 0, there are two invariant manifolds of order 2n, which allows one to construct two n-soliton solutions of the Sawada-Kotera equation.     

On the Lie algebra structures connected with Hamiltonian dynamical systems

We construct the hierarchies of master symmetries constituting Virasoro-type algebras for the Hamiltonian vector fields preserving a recursion operator. Similarly, repeatedly contracting a Hamiltonian vector field with the corresponding recursion operator, we define an Abelian Lie algebra of the thus obtained hierarchy of vector fields. The approach is shown to be applicable for the Volterra and Toda lattices. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 699–705, May, 1997.     

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.     

Symmetries and Their Lie Algebra of a Variable Coefficient Korteweg-de Vries Hierarchy

Isospectral and non-isospectral hierarchies related to a variable coefficient Painlev′e integrable Korteweg-de Vries(Kd V for short) equation are derived. The hierarchies share a formal recursion operator which is not a rigorous recursion operator and contains t explicitly. By the hereditary strong symmetry property of the formal recursion operator, the authors construct two sets of symmetries and their Lie algebra for the isospectral variable coefficient Korteweg-de Vries(vc Kd V for short) hierarchy.     

Symmetries and conservation laws for a sixth-order Boussinesq equation

This paper considers a generalization depending on an arbitrary function f(u) of a sixth-order Boussinesq equation which arises in shallow water waves theory. Interestingly, this equation admits a Hamiltonian formulation when written as a system. A classification of point symmetries and conservation laws in terms of the function f(u) is presented for both, the generalized Boussinesq equation and the equivalent Hamiltonian system.     

Invertible Coupled KdV and Coupled Harry Dym Hierarchies

In this paper, we discuss the conditions under which the coupled KdV and coupled Harry Dym hierarchies possess inverse (negative) parts. We further investigate the structure of nonlocal parts of tensor invariants of these hierarchies, in particular, the nonlocal terms of vector fields, conserved one‐forms, recursion operators, Poisson and symplectic operators. We show that the invertible coupled KdV hierarchies possess Poisson structures that are at most weakly nonlocal while coupled Harry Dym hierarchies have Poisson structures with nonlocalities of the third order.     

A Unified Approach to Computation of Integrable Structures

We expose a unified computational approach to integrable structures (including recursion, Hamiltonian, and symplectic operators) based on geometrical theory of partial differential equations.     

Equations of Pseudo-Spherical Type (After S.S. Chern and K. Tenenblat)

This paper surveys some recent developments around the notion of a scalar partial differential equation describing pseudo-spherical surfaces due to Chern and Tenenblat. It is shown how conservation laws, pseudo-potentials, and linear problems arise naturally from geometric considerations, and it is also explained how Darboux and B?cklund transformations can be constructed starting from geometric data. Classification results for equations in this class are stated, and hierarchies of equations of pseudo-spherical type are introduced, providing a connection between differential geometry and the study of hierarchies of equations which are the integrability condition of sl(2, R)-valued linear problems. Furthermore, the existence of correspondences between any two solutions to equations of pseudo-spherical type is reviewed, and a correspondence theorem for hierarchies is also mentioned. As applications, an elementary immersion result for pseudo-spherical metrics arising from the Chern?CTenenblat construction is proven, and non-local symmetries of the Kaup?CKupershmidt, Sawada?CKotera, fifth order Korteweg?Cde Vries and Camassa?CHolm (CH) equation with non-zero critical wave speed are considered. It is shown that the existence of a non-local symmetry of a particular type is enough to single the first three equations out from a whole family of equations describing pseudo-spherical surfaces while, in the CH case, it is shown that it admits an infinite-dimensional Lie algebra of non-local symmetries which includes the Virasoro algebra.