A class of integrable evolutionary vector equations

We present the results of classifying integrable evolutionary N-component vector equations and construct Bäcklund transformations for each equation as proof of the exact integrability.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 13–20, January, 2005.     

Bilinear Bäcklund transformation and Lax pair for a coupled Ramani equation

In this paper, a coupled Ramani equation is proposed. The bilinear Bäcklund transformation and Lax pair for this equation are derived starting from its bilinear form. Multisoliton solutions to the system can also be obtained.     

The Kinematics of the Planar Motion of Ideal Fiber-Reinforced Fluids: An Integrable Reduction and Bäcklund Transformation

We establish that the kinematic constraints on the steady planar motion of an ideal fiber-reinforced fluid can be consolidated in a single third-order nonlinear equation. Remarkably, this equation admits a solitonic reduction related to the classical sine-Gordon equation. The kinematic conditions in this case admit a novel duality property and a Bäcklund transformation.     

Classification of Integrable Divergent N-Component Evolution Systems

We use a symmetry approach to solve the classification problem for integrable N-component evolution systems having the form of conservation laws. We obtain complete lists of both isotropic and anisotropic systems of this type and find auto-Bäcklund transformations with a spectral parameter for all systems.     

A finite difference method for singularly perturbed differential-difference equations with layer and oscillatory behavior

In this paper, we present a finite difference method for singularly perturbed linear second order differential-difference equations of convection–diffusion type with a small shift, i.e., where the second order derivative is multiplied by a small parameter and the shift depends on the small parameter. Similar boundary value problems are associated with expected first-exit times of the membrane potential in models of neurons. Here, the study focuses on the effect of shift on the boundary layer behavior or oscillatory behavior of the solution via finite difference approach. An extensive amount of computational work has been carried out to demonstrate the proposed method and to show the effect of shift parameter on the boundary layer behavior and oscillatory behavior of the solution of the problem.     

The decay mode solutions for the cylindrical KP equation

The decay mode solutions for the cylindrical Kadomtsev-Petviashvili equation can be obtained by the Bäcklund transformation and Hirota method.     

Integrable Boundary Value Problem for the Boussinesq Equation

We discuss a method for seeking integrable boundary conditions for nonlinear equations. For the Boussinesq equation, we find a new boundary condition that is compatible with the Lax pair and has an infinite set of higher symmetries and a Bäcklund transformation. We construct a class of explicit partial solutions of an equation satisfying this boundary condition.     

A study on the bilinear Caudrey-Dodd-Gibbon equation

In this paper we consider the Hirota transformation of the Caudrey-Dodd-Gibbon equation (CDGE) from another point of view. As a result, the local equivalence between the CDGE and its bilinear equation is established, and a new type of Bäcklund transformation, which is defined by a second-order ODE along with the appropriate initial values, is presented to construct new solutions for the bilinear CDGE from the seed solutions of original CDGE.     

On some aspects of Bäcklund transformations

The Painlevé differential equations (P₂-P₆) possess Bäcklund transformations which relate one solution to another solution either of the same equation, with different values of the parameters, or another such equation. We review a method for deriving difference equations, the discrete Painlevé equations in particular, from Bäcklund transformations of the continuous Painlevé equations. Then, we prove the existence of an algebraic formula relating three inconsecutive solutions of the same Bäcklund hierarchy for P₃ and P₄.     

Bäcklund transformations and soliton solutions for the KdV6 equation

Using Hirota technique, a Bäcklund transformation in bilinear form is obtained for the KdV6 equation. Furthermore, we present a modified Bäcklund transformation by a dependent variable transformation, it is shown that a new representation of N-soliton solution and some novel solutions to the KdV6 equation are derived by performing an appropriate limiting procedure on the known soliton solutions.     

Nonclassical symmetries of evolutionary partial differential equations and compatibility

The determining equations for the nonclassical reductions of a general nth order evolutionary partial differential equations is considered. It is shown that requiring compatibility with a first order quasilinear partial differential equation, the determining equations are obtained. Burgers' equation and the KdV equation and generalizations serve as examples illustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries.     

Soliton solutions and Bäcklund transformation for the complex Ginzburg-Landau equation

Symbolically investigated in this paper is the complex Ginzburg-Landau (CGL) equation. With the Hirota method, both bright and dark soliton solutions for the CGL equation are obtained simultaneously. New Bäcklund transformation in the bilinear form is derived. Relevant properties and features are discussed. Solitons can be compressed (amplified) when the nonlinear (linear) dispersion effect is enhanced. Meanwhile, central frequency of the soliton can be affected by the nonlinear and linear dispersion effects. Besides, directions of the movement for the soliton central frequency can be adjusted. Results of this paper would be of certain value to the studies on the soliton compression and amplification.     

Some Local Properties of Bäcklund Transformations

For Bäcklund transformations, treated as relations in the categoryof diffieties, local conditions of effectivity and normality are introduced,having implications for the solution generating properties. We check themfor the pKdV, the sine-Gordon, and the Tzitzéica equation.     

Rank 2 distributions of Monge equations: Symmetries, equivalences, extensions

By developing the Tanaka theory for rank 2 distributions, we completely classify classical Monge equations having maximal finite-dimensional symmetry algebras with fixed (albeit arbitrary) orders. Investigation of the corresponding Tanaka algebras leads to a new Lie–Bäcklund theorem. We prove that all flat Monge equations are successive integrable extensions of the Hilbert–Cartan equation. Many new examples are provided.     

一类微分-差分方程组的内禀对称和等价群变换

研究一类微分-差分方程组的对称和等价群变换.采取内禀的无穷小算子方法,给出了方程组的内禀对称和等价群变换.为结合抽象Lie代数结构,给方程完全分类提供了理论基础.     

Algebraic and Differential Nonlinear Superposition Formulas

Knowledge of the Lax pair and the Darboux transformation for a completely integrable system provides an iterative approach for generating exact solutions. This approach involves solving for the eigenfunction of the Lax pair at each step. But this process can be considerably simplified using the Bäcklund transformation and Bianchi's permutability theorem. This allows constructing the so-called nonlinear superposition formula, which provides a new solution of the system in terms of three previous solutions. The advantage of this approach is that the differential order of the nonlinear superposition formulas is lower than that of the Lax pairs, and in some cases, these formulas reduce to algebraic equations. We consider the construction of new nonlinear superposition formulas in the form of both differential equations and algebraic equations.     

Conservation laws and Bäcklund transformations associated with the Born-Infeld equation

For the Born-Infeld equation in the hyperbolic domain of its solutions, we obtain first-order conservation laws depending on two arbitrary functions. It is shown that each conservation law is related to some Bäcklund transformation that transforms the Born-Infeld equation into some related equation.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 551–565.Original Russian Text Copyright © 2005 by O. F. Menshikh.This revised version was published online in April 2005 with a corrected issue number.     

On RF-pairs, Bäcklund transformations and linearization of nonlinear equations

Some classes of nonlinear equations of mathematical physics are described that admit order reduction through the use of a hydrodynamic-type transformation, where the unknown function is taken as a new independent variable and an appropriate partial derivative is taken as the new dependent variable. RF-pairs and associated Bäcklund transformations are constructed for evolution equations of general form. The results obtained are used for order reduction of hydrodynamic equations (Navier-Stokes and boundary layer) and constructing exact solutions to these equations. A generalized Calogero equation and a number of other new linearizable nonlinear differential equations of the second, third and forth orders are considered. Some integro-differential equations are analyzed.     

Darboux and Bäcklund transformations of the bidirectional Sawada-Kotera equation

Darboux and Bäcklund transformations of the bidirectional Sawada-Kotera equation are derived with the help of the resulting Riccati equation. As an application, some explicit solutions of the bidirectional Sawada-Kotera equation are obtained, including rational solutions, periodic solutions, and soliton solutions.