Integrable nonlinear evolution equations and dynamical systems in multidimensions

The investigation of nonlinear evolution equations and dynamical systems integrable in multidimensions constitutes at present our main research interest. Here we survey findings obtained recently as well as over time: solvable equations (both PDEs and ODEs) are reported, philosophical motivations and methodological approaches are outlined. For more detailed treatments, including the display and analysis of solutions, the interested reader is referred to the original papers.On leave while serving as Secretary General, Pugwash Conferences on Science and World Affairs, Geneva, London, Rome.     

Hamilton equations for infinite-dimensional systems, and their variational equations

For systems with infinitely many degrees of freedom, we establish a relationship between the solutions and first integrals of noncanonical Hamilton equations, their variational equations, and the adjoint variational equations.     

Integrable seepage equations

Well-known results of the classification of parabolic-type differential equations, possessing an infinite Lie–Bäcklund algebra, are used to describe seepage models, which, using (in general) differential substitutions, can be reduced to the heat conduction equation. Relations between the functional parameters, characterizing the properties of the liquid and gas phases and the porous medium, are obtained that ensure the existence of such substitutions.     

Generalized conditional symmetries and exact solutions of non-integrable equations

We introduce the concept of a generalized conditional symmetry. This concept provides an algorithm for constructing physically important exact solutions of non-integrable equations. Examples include 2-shock and 2-soliton solutions. The existence of such exact solutions for non-integrable equations can be traced back to the relation of these equations with integrable ones. In this sense these exact solutions are remnants of integrability.Department of Mathematics and Computer Science and Institute for Nonlinear Studies, Clarkson University, Potsdam, New York 13699-5815. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 2, pp. 263–277, May, 1994.     

Integrable coupled dispersionless equations

A new type of coupled dispersionless equations, which has a real dark type of soliton and a complex bright type of envelope soliton, is proposed and solved by the inverse scattering method. The N-soliton solution is given and one and two solitons are explicitly shown.     

Analytic non-integrable Hamiltonian systems and irregular singularity

We study a C ^∞-Liouville-integrable and analytic non-integrable Hamiltonian system. We will show that an irregular singular character plays a crucial role in the analytic non-integrability of the system. Partially supported by Grant-in-Aid for Scientific Research (No. 11640183), Ministry of Education, Science and Culture, Japan.     

Integrable Lotka-Volterra systems

Infinite- and finite-dimensional lattices of Lotka-Volterra type are derived that possess Lax representations and have large families of first integrals. The obtained systems are Hamiltonian and contain perturbations of Volterra lattice. Examples of Liouville-integrable 4-dimensional Hamiltonian Lotka-Volterra systems are presented. Several 5-dimensional Lotka- Volterra systems are found that have Lax representations and are Liouville-integrable on constant levels of Casimir functions.      

Integrable superconductivity and Richardson equations

For the integrable generalized model of superconductivity, the solution of the Richardson equations is studied for a model spectrum. For the case of a narrow band, the solution is presented in terms of generalized Laguerre or Jacobi polynomials. In the asymptotic limit, when the Richardson equations are transformed into a singular integral equation, the properties of the integration contour are discussed and the spectral density is calculated. The conditions of appearance of gaps in the spectrum are investigated. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 3, pp. 314–326, March, 2007.     

On the Lagrangian form of the variational equations of Lagrangian dynamical systems

An extremal principle for obtaining the variational equations of a Lagrangian system is reviewed and formalized. Formalization is accomplished by relating the new Lagrangian function γ needed in such scheme to a prolongation of the original Lagrangian L. This formalization may be regarded as a necessary step before using the approach for stablishing nonintegrability of dynamical systems, or before applying it to analyse chaos-producing perturbations of integrable Lagrangian systems. The configuration manifold in which γ is defined is the double tangent bundle T(TQ) of the original configuration manifold Q modulo a flip mapping in such manifold. Our main result establishes that both the Euler–Lagrange equations and the corresponding variational equations of the original system can be viewed as the Lagrangian vector field associated with the composition of the first prolongation of the original Lagrangian with a flip mapping. Some applications of the approach to chaos and integrability issues are discussed.     

Integrability of Hamiltonian systems and differential Galois groups of higher variational equations

Given a complex analytical Hamiltonian system, we prove that a necessary condition for its meromorphic complete integrability is the commutativity of the identity component of the Galois group of each variational equation of arbitrary order along any integral curve. This was conjectured by the first author based on a suggestion by the third author. The first-order non-integrability criterion, obtained by the first and second authors using only first variational equations, is extended to higher orders by the present criterion. Using this result (at order two, three or higher) it is possible to solve important open problems of integrability which escaped the first order criterion.     

Reducible variational equations and the perturbation of invariant tori of Hamiltonian systems

Translated from Matematicheskie Zametki, Vol. 45, No. 5, pp. 38–49, May, 1989.     

Integrable nonlinear equations on a half-axis

A study is made of the nonlinear Schrödinger equation and the modified Kortewegde Vries equation on the half-axis 0x<. Classes of solutions are identified when there is sufficient initial data for unique solvability of a corresponding problem. The class of solutions investigated is characterized in terms of the Weyl-Titchmarsh function of an auxiliary linear problem. A solution is constructed in explicit form when the corresponding Weyl-Titchmarsh function v(t, z) is a rational function of z.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 11, pp. 1578–1584, November, 1991.     

Integrable systems of chiral type

We present new integrable systems close to the WZNW systems (Wess-Zumino-Novikov-Witten) and to the nonabelian affine Toda systems. One of the systems is a new integrable generalization of the sine-Gordon equation. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 12, No. 7, pp. 5–21, 2006.     

Integrable boundary conditions for many-component burgers equations

Infinite series of boundary conditions that are consistent with even-order higher symmetries and ensure the integrability of a Burgers type equation are constructed. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 888–901, December, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-011-16008 and by the International Science Foundation under grant Nos. MLY00 and RK2000.     

Wiener-hopf equations and variational inequalities

In this paper, we show that the general variational inequality problem is equivalent to solving the Wiener-Hopf equations. We use this equivalence to suggest and analyze a number of iterative algorithms for solving general variational inequalities. We also discuss the convergence criteria for these algorithms.