Applications of a few Lie algebras

Two isomorphic groups R ² andM are firstly constructed. Then we extend them into the differential manifold R ²ⁿ and n products of the group M for which four kinds of Lie algebras are obtained. By using these Lie algebras and the Tu scheme, integrable hierarchies of evolution equations along with multi-component potential functions can be generated, whose Hamiltonian structures can be worked out by the variational identity. As application illustrations, two integrable Hamiltonian hierarchies with 4 component potential functions are obtained, respectively, some new reduced equations are followed to present. Specially remark that the integrable hierarchies obtained by taking use of the approach presented in the paper are not integrable couplings. Finally, we generalize an equation obtained in the paper to introduce a general nonlinear integrable equation with variable coefficients whose bilinear form, B¨acklund transformation, Lax pair and infinite conserved laws are worked out, respectively, by taking use of the Bell polynomials.     

Commutator identities on associative algebras and the integrability of nonlinear evolution equations

We show that commutator identities on associative algebras generate solutions of the linearized versions of integrable equations. In addition, we introduce a special dressing procedure in a class of integral operators that allows deriving both the nonlinear integrable equation itself and its Lax pair from such a commutator identity. The problem of constructing new integrable nonlinear evolution equations thus reduces to the problem of constructing commutator identities on associative algebras. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 477–491, March, 2008.     

Decomposition of a hierarchy of nonlinear evolution equations

The generalized Hamiltonian structures for a hierarchy of nonlinear evolution equations are established with the aid of the trace identity. Using the nonlinearization approach, the hierarchy of nonlinear evolution equations is decomposed into a class of new finite-dimensional Hamiltonian systems. The generating function of integrals and their generator are presented, based on which the finite-dimensional Hamiltonian systems are proved to be completely integrable in the Liouville sense. As an application, solutions for the hierarchy of nonlinear evolution equations are reduced to solving the compatible Hamiltonian systems of ordinary differential equations.     

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.     

Integrability conditions for a certain class of nonlinear evolution equations and Kähler geometry

Multicomponent evolution equations associated with linear connections on complex manifolds are considered. It is proved that under some general assumptions an equation from this class is integrable by inverse scattering method if the corresponding linear connection is the Levi-Civita connection of an indefinite Kählerian metric of constant holomorphic sectional curvature. This result is based on a certain characterization of the above-mentioned Levi-Civita connections. It is shown that the obtained integrable equations are generalized ferromagnetics, and recurrent formulas for their local conservation laws are given.     

Time-space integrable decompositions of nonlinear evolution equations

Separation of the time and space variables of evolution equations is analyzed, without using any structure associated with evolution equations. The resulting theory provides techniques for constructing time-space integrable decompositions of evolution equations, which transform an evolution equation into two compatible Liouville integrable ordinary differential equations in the time and space variables. The techniques are applied to the KdV, MKdV and diffusion equations, thereby yielding several new time-space integrable decompositions of these equations.     

A family of integrable differential-difference equations and its Bargmann symmetry constraint

A family of integrable differential-difference equations is constructed through discrete zero curvature equation. The Hamiltonian structures of the resulting differential-difference equations are established by the discrete trace identity. The Bargmann symmetry constraint of the resulting family is presented. Under this symmetry constraint, every differential-difference equation in the resulting family is factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense.     

利用李代数构造非线性可积及双可积哈密顿耦合

With the help of a Lie algebra,two kinds of Lie algebras with the forms of blocks are introduced for generating nonlinear integrable and bi-integrable couplings.For illustrating the application of the Lie algebras,an integrable Hamiltonian system is obtained,from which some reduced evolution equations are presented.Finally,Hamiltonian structures of nonlinear integrable and bi-integrable couplings of the integrable Hamiltonian system are furnished by applying the variational identity.The approach presented in the paper can also provide nonlinear integrable and bi-integrable couplings of other integrable system.     

一族新的Lax可积系及其Liouville可积性

该文讨论了一个新的等谱特征值问题．按屠规彰格式导出了相应的Lax可积的非线性发展方程族，利用迹恒等式给出了它的Hamilton结构并且证明它是Liouville可积的．     

The Liouville integrability of integrable couplings of Volterra lattice equation

A hierarchy of integrable couplings of Volterra lattice equations with three potentials is proposed, which is derived from a new discrete six-by-six matrix spectral problem. Moreover, by means of the discrete variational identity on semi-direct sums of Lie algebra, the two Hamiltonian forms are deduced for each lattice equation in the resulting hierarchy. A strong symmetry operator of the resulting hierarchy is given. Finally, we prove that the hierarchy of the resulting Hamiltonian equations are all Liouville integrable discrete Hamiltonian systems.     

The geometric series method for constructing exact solutions to nonlinear evolution equations

It is proved that, for the majority of integrable evolution equations, the perturbation series constructed based on the exponential solution of the linearized problem is geometric or becomes geometric as a result of changing the variable in the equation or after a transformation of the series. Using this property, a method for constructing exact solutions to a wide class of nonintegrable equations is proposed; this method is based on the requirement for the perturbation series to be geometric and on the imposition of constraints on the values of the coefficients and parameters of the equation under which the sum of the series is the solution to be found. The effectiveness of using the diagonal Padé approximants the minimal order of which is determined by the order of the pole of the solution to the equation is demonstrated.     

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.     

Painlevé analysis of nonlinear evolution equations—an algorithmic method

This paper refines existing techniques into an algorithmic method for deriving the generalization of a Lax Pair directly from a general integrable nonlinear evolution equation via the use of truncated Painlevé expansions. The resulting algorithm is also applicable to multicomponent integrable systems, and is thus expected to be of great value for complicated variants of such systems in various applications areas. Although a related method has existed for simple scalar integrable evolution equations for many years now, nevertheless no systematic procedure has been given that would work in general for scalar as well as for multicomponent systems. The method presented here largely systematizes the necessary operations in applying the Painlevé method to a general integrable evolution equation or system of equations. We demonstrate that by following the concept of enforcing integrability at each step (referred to here as the Principle of Integrability), one is led to an appropriate generalization of a Lax Pair, although perhaps in nonlinear form, called a “Lax Complex”. One new feature of this procedure is that it utilizes, as needed, a technique from the well-known Estabrook–Wahlquist method for determining necessary integrating factors. The end result of this procedure is to obtain a Lax Complex, whose integrability condition will contain the original evolution equation as a necessary condition. This in itself is sufficient to ensure that the Lax Complex may be used to construct Bäcklund solutions of the evolution equation, to obtain Darboux Transformations, and also to obtain Hirota’s tau functions, in a manner analogous to the procedure for single component systems. The additional problem of finding a general procedure for the linearization of any Lax Complex is not treated in this paper. However, we do demonstrate that a particular technique, which can be derived self-consistently from the Painlevé–Bäcklund equations, has proven to be sufficient so far. The Nonlinear Schrödinger equation is used to illustrate the method, and then the method is applied to obtain, for the first time via the Painlevé method, a Lax Complex for the vector Manakov system. Limitations in the algorithm remain, especially for cases with more than one principal branch, and these are briefly mentioned as directions for future work.     

The method of generalized Cole-Hopf substitutions and new examples of linearizable nonlinear evolution equations

We propose a new approach for constructing nonlinear evolution equations in matrix form that are integrable via substitutions similar to the Cole-Hopf substitution linearizing the Burgers equation. We use this new approach to find new integrable nonlinear evolution equations and their hierarchies. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 1, pp. 58–71, January, 2009     

Poisson brackets of mappings obtained as (<Emphasis Type="Italic">q</Emphasis>,−<Emphasis Type="Italic">p</Emphasis>) reductions of lattice equations

In this paper, we present Poisson brackets of certain classes of mappings obtained as general periodic reductions of integrable lattice equations. The Poisson brackets are derived from a Lagrangian, using the so-called Ostrogradsky transformation. The (q,?p) reductions are (p + q)-dimensional maps and explicit Poisson brackets for such reductions of the discrete KdV equation, the discrete Lotka–Volterra equation, and the discrete Liouville equation are included. Lax representations of these equations can be used to construct sufficiently many integrals for the reductions. As examples we show that the (3,?2) reductions of the integrable partial difference equations are Liouville integrable in their own right.     

Nonexistence of spatially localized free vibrations for a class of nonlinear wave equations

We prove the nonexistence of free vibrations of arbitrary period with polynomially decreasing profiles for a large class of nonlinear wave equations in one space dimension Our class of admissible models includes examples of non integrable wave equations with certain polynomial nonlinearities, as well as examples of completely integrable ones with exponential nonlinearities related to Mikhailov's equations. Our result thus proves a particular case of a conjecture first formulated by Eleonskii, Kulagin, Novozhilova and Silin, and dispels some confusion regarding the relationship between the existence of so-called breather-solutions and the complete integrability of the wave equation. Our class of admissible nonlinearities also contains a particular instance of the nonlinear scalar Higgs' equation, but does not contain the Sine-Gordon equation which is known to possess a 2π-periodic solution in time with exponential fall-off in the spatial direction. Our results may be considered as complementary to recent results by Coron and Weinstein. Our arguments are entirely global, and rest upon methods from the calculus of variations. Work supported in part by the Los Alamos National Laboratory under contract COL-2335, by a University of Texas summer grant and by the ETH-Forschungsinstitut für Mathematik.     

一个新的Liouville可积系统及其Lax表示，Bi—Hamilton结构

从一个特征值问题出发,首先推导一族非线性发展方程,其中包括著名MKdV方程做为特殊约化,进一步证明这族方程在Liounille意义下可积并具有Bi-Hamilton结构,而在位执函数和特征函数之间的一定约束下,特征值问题被非线性化为一完全可积的有限维Hamilton系统。     

The hierarchy of an integrable coupling and its Hamiltonian structure

A type of higher dimensional loop algebra is constructed from which an isospectral problem is established. It follows that an integrable coupling, actually an extended integrable model of the existed solitary hierarchy of equations, is obtained by taking use of the zero curvature equation, whose Hamiltonian structure is worked out by employing the constructed quadratic identity.     

一个新的非等谱可积族和一些相关对称

利用李群$M_nC$的一个子群我们引入一个线性非等谱问题,该问题的相容性条件可导出演化方程的一个非等谱可积族,该可积族可约化成一个广义非等谱可积族.这个广义非等谱可积族可进一步约化成在物理学中具有重要应用的标准非线性薛定谔方程和KdV方程.基于此,我们讨论在广义非等谱可积族等谱条件下的一个广义AKNS族$u_t=K_m(u)$的$K$对称和$\tau$对称.此外,我们还考虑非等谱AKNS族$u_t=\tau_{N+1}^l$的$K$对称和$\tau$对称.最后,我们得到这两个可积族的对称李代数,并给出这些对称和李代数的一些应用,即生成了一些变换李群和约化方程的无穷小算子.     

Formulas that Represent Cauchy Problem Solution for Momentum and Position Schrödinger Equation

In the paper we derive two formulas representing solutions of Cauchy problem for two Schrödinger equations: one-dimensional momentum space equation with polynomial potential, and multidimensional position space equation with locally square integrable potential. The first equation is a constant coefficients particular case of an evolution equation with derivatives of arbitrary high order and variable coefficients that do not change over time, this general equation is solved in the paper. We construct a family of translation operators in the space of square integrable functions and then use methods of functional analysis based on Chernoff product formula to prove that this family approximates the solution-giving semigroup. This leads us to some formulas that express the solution for Cauchy problem in terms of initial condition and coefficients of the equations studied.