Explicit solutions of superized toda lattices

Explicit solutions are constructed for superized Toda lattices associated with systems of simple roots of classical Lie superalgebras. The completeness of flows and the asymptotic behavior of systems are studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 123, pp. 98–111, 1983.     

On the Cauchy problem for the Kortewegde Vries equation with steplike finite-gap initial data II. Perturbations with finite moments

We solve the Cauchy problem for the Korteweg-de Vries equation with steplike quasi-periodic, finite-gap initial conditions under the assumption that the perturbations have a given number of finite derivatives with finite moments.     

Involutive solutions for toda and langmuir lattices through nonlinearization of lax pairs

Under the constraint determined by a relation (a _n ,b _n )^T={f(?)} _n between the reflectionless potentials and the eigenfunctions of the general discrete Schrödinger eigenvalue problem, the Lax pair of the Toda lattice is nonlinearized to be a finite-dimensional difference system and a nonlinear evolution equation, while the solution varietyN of the former is an invariant set of S-flows determined by the latter, and the constants of the motion for the algebraic system are presented.f maps the solution of the algebraic system into the solution of a certain stationary Toda equation. Similar results concerning the Langmuir lattice are given, and a relation between the two difference systems, which are the spatial parts of the nonlinearized Lax pairs of the Toda lattice and Langmuir lattice, is discussed.     

Kovalevskaya numbers of generalized toda chains

Translated from Matematicheskie Zametki, Vol. 46, No. 5, pp. 17–28, November, 1989.     

The KPI equation with unconstrained initial data

The solutionu(t, x, y) of the Kadomtsev-Petviashvili I (KPI) equation with given initial data u(0,x, y) belonging to the Schwartz space is considered. No additional special constraints, usually considered in the literature as dxu(0,x,y)=0 are required to be satisfied by the initial data. The spectral theory associated with KPI is studied in the space of the Fourier transform of the solutions. The variablesp={p ₁,p ₁} of the Fourier space are shown to be the most convenient spectral variables to use for spectral data. Spectral data are shown to decay rapidly at largep but to be discontinuous atp=0. Direct and inverse problems are solved with special attention to the behavior of all the quantities involved in the neighborhood oft=0 andp=0. It is shown in particular that the solutionu(t, x, y) has a time derivative discontinuous att = 0 and that at anyt 0 it does not belong to the Schwartz space no matter how small in norm and rapidly decaying at large distances the initial data are chosen.Work supported in part by Ministero delle Universitá e della Recerca Scientifica e Technologica, India.     

Degenerate parabolic equations with initial data measures

We address the problem of existence of solutions to degenerate (and nondegenerate) parabolic equations under optimal assumptions on the initial data, which are assumed to be measures. The requirements imposed on the initial data are connected both with the degeneracy of the principal part of the equation, and with the form of the nonlinear forcing term. The latter depends on the space gradient of a power of the solution. Applications to related problems are also outlined.

     

On a class of toda chains

The main result of this paper is a list of integrable generalizations of the classical and relativistic Toda chains. This list includes three new types in addition to the well-known chains. Each of them defines a Bäcklund transformation for an NSE-type system. The classification is carried out with the help of a simple symmetry test.     

Singular parabolic equations with measures as initial data

In this paper we study the Cauchy problem for the singular evolution p-Laplacian equations with gradient term and source on the assumption of measures as initial conditions. For the supercritical case q>p−1+p/N, we obtain that for every nonnegative solution there exists a nonnegative Radon measure μ as initial trace and μ has some local regularity.     

The equations of electrohydrodynamics with highly irregular initial data

The existence of a weak solution to the equations of electrohydrodynamicsis proved. The data are allowed to be highly irregular; specifically,the initial charge density may be a measure (with integrabledensity) concentrated on the boundary. Historical comments aremade concerning a relevant paper of B. Riemann.     

Stability analysis of time-fractional differential equations with initial data

In this work, we investigate the stability of solution for a class of time-fractional differential equation with initial data by the help of a fractional Duhamel principle; we use it and the superposition principle to obtain the solution of our problem. We present some theoretical results by some scientific theorems and lemmas.     

Large-deviation analysis of Burgers turbulence with white-noise initial data

The statistics of the solution to the inviscid Burgers equation are investigated when the initial velocity is a white-noise function. Explicit representations for the probability distribution functions (PDFs) of the velocity and the distance between two successive shocks are found in terms of the solution to a PDE system. By using these expressions and large-deviation theory, tight bounds are found for the tails of these distributions and for that of the shock-strength distribution. These bounds give the exact rates of decay for the tails. © 1998 John Wiley & Sons, Inc.     

Regularization of Steffensen's method with inexactly specified initial data

A minimization problem is considered for the case when the minimand function and the functions defining the set of equality and inequality constraints are known with an error. A regularized Steffensen's method for the solution of this problem is proposed. It is shown that, when the variation of the regularization and penalty parameters is compatible with the errors, the sequence generated by this method converges in norm to the minimum point with minimum norm.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 15–23, 1986.     

Order affine completeness of lattices with Boolean congruence lattices

This paper grew out from attempts to determine which modular lattices of finite height are locally order affine complete. A surprising discovery was that one can go quite far without assuming the modularity itself. The only thing which matters is that the congruence lattice is finite Boolean. The local order affine completeness problem of such lattices L easily reduces to the case when L is a subdirect product of two simple lattices L ₁ and L ₂. Our main result claims that such a lattice is locally order affine complete iff L ₁ and L ₂ are tolerance trivial and one of the following three cases occurs:

Explicit solutions of classical generalized toda models

The explicit solution of the generalized Toda models connected with the root systems of simple Lie algebras is constructed. This solution is obtained by a special projection of geodesic flows on some symmetric spaces on the phase space of the Toda models. In particular, for the usualN-body Toda model the solutions are expressed by minors of the exponent of the Jacobi matrix depending on initial data.This paper has been published early as a preprint ITEP-157 (1978)     

Certain methods for linear problems with inexact initial data

Linear problems with inexact initial data are examined. The stopping rules for certain iterative methods designed for solving linear equations and a linear elimination problem are proposed and analysed. In particular, these methods are applicable to ill-conditioned and ill-posed problems. Numerical results are presented that demonstrate the efficiency of these methods.