Connection between the kinetic equations and the equations of hydrodynamics

One investigates the Cauchy problem for the nonlinear Boltzmann equation

Solitary waves of the generalized Kadomtsev-Petviashvili equations

In this paper, we consider the existence of solitary waves of the generalized Kadomtsev-Petviashvili equations by using variational methods.     

Connection between the Fokker-Planck-Kolmogorov and nonlinear Langevin equations

We recall the general proof of the statement that the behavior of every holonomic nonrelativistic system can be described in terms of the Langevin equation in Euclidean (imaginary) time such that for certain initial conditions, the different stochastic correlators (after averaging over the stochastic force) coincide with the quantum mechanical correlators. The Fokker-Planck-Kolmogorov (FPK) equation that follows from this Langevin equation is equivalent to the Schrödinger equation in Euclidean time if the Hamiltonian is Hermitian, the dynamics are described by potential forces, the vacuum state is normalizable, and there is an energy gap between the vacuum state and the first excited state. These conditions are necessary for proving the limit and ergodic theorems. For three solvable models with nonlinear Langevin equations, we prove that the corresponding Schrödinger equations satisfy all the above conditions and lead to local linear FPK equations with the derivative order not exceeding two. We also briefly discuss several subtle mathematical questions of stochastic calculus.     

Recent results on the generalized Kadomtsev-Petviashvili equations

We survey some recent results concerning the generalized Kadomtsev-Petviashvili equations, which are natural extensions of KdV type equations to higher dimensions. We will focus on rigorous results of the Cauchy problem and on the existence and properties of localized solitary waves.     

Supersymmetric structure of Kadomtsev-Petviashvili equation: Connection between supersymmetry and Bäcklund transformations

Factorization of the Sato integrodifferential operator that gives the hierarchy of the Kadomtsev-Petviashvili equation is considered. The supersymmetric structure of this hierarchy is investigated by means of the factorization. A connection between the Miura transformations, Bäcklund transformations, and supersymmetry is established.P. N. Lebedev Physics Institute, Rossiskoi Akademii Nauk. Translated from Teorecheskaya i Matematicheskaya Fizika, Vol. 91, No. 3, pp. 426–432, June, 1992.     

Darboux transformations for higher-rank Kadomtsev-Petviashvili and Krichever-Novikov equations

It is shown that the action of a special rank 2 or rank 3 Darboux transformation, calledtransference, upon a pair of commuting ordinary differential operators of orders 4 and 6 implements the Abelian sum on their elliptic joint spectrum. A consequence of this is that, under the deformation of these commuting operators by the KP flow, every rank 2 KP solution corresponds to a solution of the Krichever-Novikov (KN) equation, and vice versa, with the transference process providing the correspondence between (2+1) and (1+1) dimensions. For a singular joint spectrum, it is further shown that transference at the singular point produces a correspondence between solutions of the singular KN equation and those of the KdV equation. These correspondences are illustrated by considering examples of a nondecaying rational KdV or Boussinesq solutions and the corresponding rational, singular-KN and rational KP solutions which the transference process generates.     

A finite difference method for the Korteweg-de Vries and the Kadomtsev-Petviashvili equations

A linearized implicit finite difference method for the Korteweg-de Vries equation is proposed and straightforwardly extended to the Kadomtsev-Petviashvili equation. We investigate the order of accuracy of the method and prove the method to be unconditionally linearly stable. The numerical experiments for the Korteweg-de Vries and the Kadomtsev-Petviashvili equations are carried out with various conditions. Numerical results for the collision of two lump type solitary wave solutions to the Kadomtsev-Petviashvili equation are also reported.     

Uniform asymptotic formulas for curved solitons of the Kadomtsev-Petviashvili equations

A special class of solutions to the Kadomtsev-Petviashvili equations is investigated in the limit t . It is proved that these solutions split into an infinite series of curved solitons in the neighborhood of the wave front. Parameters of these solitons depend on the variable Y=y/t. Asymptotic formulas uniform in Y are obtained.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 2, pp. 205–211, August, 1996.     

Meromorphic exact solutions of the generalized (3+1)-dimensional Kadomtsev-Petviashvili equations

In this work, the generalized (3+1)-dimensional Kadomtsev-Petviashvili equation and its new form have been systematically investigated by using the complex method. The method is based on complex analysis and complex differential equations. And we get plentiful meromorphic exact solutions of these equations, which include rational solutions, exponential function solutions, and elliptic function solutions. The dynamic behaviors of these solutions are also shown by some graphs.     

Hierarchy of the Kadomtsev-Petviashvili equations under nonlocal constraints: Many-dimensional generalizations and exact solutions of reduced system

We present a spatially two-dimensional generalization of the hierarchy of Kadomtsev-Petviashvili equations under nonlocal constraints (the so-called 2-dimensionalk-constrained KP-hierarchy, briefly called the 2d k-c-hierarchy). As examples of (2+1)-dimensional nonlinear models belonging to the 2d k-c KP-hierarchy, both generalizations of already known systems and new nonlinear systems are presented. A method for the construction of exact solutions of equations belonging to the 2d k-c KP-hierarchy is proposed. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 78–97, January, 1999.     

On the Kadomtsev-Petviashvili Equation and Associated Constraints

The initial-boundary-value problem for the Kadomtsev-Petviashvili equation in infinite space is considered. When formulated as an evolution equation, found that a symmetric integral is the appropriate choice in the nonlocal term; namely, . If one simply chooses , then an infinite number of constraints on the initial data in physical space are required, the first being . The conserved quantities are calculated, and it is shown that they must be suitably regularized from those that have been used when the constraints are imposed.     

Complete integrability of the Kadomtsev-Petviashvili equation

The total hierarchy of the Kadomtsev-Petviashvili (KP) equation is transformed to the system of linear partial differential equations with constant coefficients. The complete integrability of the KP equation is proved by using this linear system. The existence and uniqueness theorem of the Cauchy problem of the KP hierarchy is obtained.     

Connection between commutative algebra and topology

The main aim of this paper is to show how commutative algebra is connected to topology. We give underlying topological idea of some results on completable unimodular rows.