Classifying Integrable Egoroff Hydrodynamic Chains

We introduce the notion of Egoroff hydrodynamic chains. We show how they are related to integrable (2+1)-dimensional equations of hydrodynamic type. We classify these equations in the simplest case. We find (2+1)-dimensional equations that are not just generalizations of the already known Khokhlov–Zabolotskaya and Boyer–Finley equations but are much more involved. These equations are parameterized by theta functions and by solutions of the Chazy equations. We obtain analogues of the dispersionless Hirota equations.     

The Calogero–Bogoyavlenskii–Schiff Equation in 2+1 Dimensions

We use the classical and nonclassical methods to obtain symmetry reductions and exact solutions of the (2+1)-dimensional integrable Calogero–Bogoyavlenskii–Schiff equation. Although this (2+1)-dimensional equation arises in a nonlocal form, it can be written as a system of differential equations and, in potential form, as a fourth-order partial differential equation. The classical and nonclassical methods yield some exact solutions of the (2+1)-dimensional equation that involve several arbitrary functions and hence exhibit a rich variety of qualitative behavior.     

Solutions of Jimbo-Miwa Equation and Konopelchenko-Dubrovsky Equations

The Jimbo-Miwa equation is the second equation in the well known KP hierarchy of integrable systems, which is used to describe certain interesting (3+1)-dimensional waves in physics but not pass any of the conventional integrability tests. The Konopelchenko-Dubrovsky equations arose in physics in connection with the nonlinear weaves with a weak dispersion. In this paper, we obtain two families of explicit exact solutions with multiple parameter functions for these equations by using Xu’s stable-range method and our logarithmic generalization of the stable-range method. These parameter functions make our solutions more applicable to related practical models and boundary value problems.     

Classical Symmetry Reductions of the Schwarz–Korteweg–de Vries Equation in 2+1 Dimensions

Classical reductions of a (2+1)-dimensional integrable Schwarz–Korteweg–de Vries equation are classified. These reductions to systems of partial differential equations in 1+1 dimensions admit symmetries that lead to further reductions, i.e., to systems of ordinary differential equations. All these systems have been reduced to second-order ordinary differential equations. We present some particular solutions involving two arbitrary functions.     

新(2+1)-维超可积系统的生成

本文利用二项式残数表示方法生成(2+1)-维超可积系统. 由这些系统得到了一个新的(2+1)-维超孤子族,它能约化为(2+1)-维超非线性Schrodinger方程. 特别地,我们得到两个具有重要物理应用的结果,一个是(2+1)-维超可积耦合方程,另一个是(2+1)-维的扩散方程. 最后借助超迹恒等式给出了新(2+1)-维超可积系统的Hamilton结构.     

Induced Surfaces and Their Integrable Dynamics

A method is considered to induce surfaces in three-dimensional (pseudo) Euclidean space via the solutions to two-dimensional linear problems (20 LPs) and their integrable dynamics (deformations) via the 2 + 1-dimensional nonlinear integrable equations associated with these 2D LPs. Coordinates X_i of the induced surfaces are defined as integrals over certain bilinear combinations of the wave functions ψ of these 20 LPs. General formulation as well as three concrete examples are considered. Some properties and features of such induction are discussed. Three-dimensional Riemann spaces associated with 2 + 1-dimensional nonlinear integrable equations are considered also.     

Remarks on the local version of the inverse scattering method

It is very likely that all local holomorphic solutions of integrable (1+1)-dimensional parabolic-type evolution equations can be obtained from the zero solution by formal gauge transformations that belong (as formal power series) to appropriate Gevrey classes. We describe in detail the construction of solutions by means of convergent gauge transformations and prove an assertion converse to the above conjecture; namely, we suggest a simple necessary condition for the existence of a local holomorphic solution to the Cauchy problem for the evolution equations under consideration in terms of scattering data of initial conditions.     

Relation among nonlinear evolution equations and their reductions

The LCZ soliton hierarchy is presented, and their generalized Hamiltonian structures are deduced. From the compatibility of soliton equations, it is shown that this soliton hierarchy is closely related to the Burger equation, the mKP equation and a new (2 + 1)-dimensional nonlinear evolution equation (NEE). Resorting to the nonlinearization of Lax pairs (NLP), all the resulting NEEs are reduced into integrable Hamiltonian systems of ordinary differential equations (ODEs). As a concrete application, the solutions for NEEs can be derived via solving the corresponding ODEs.     

The ( \frac{G'}{G}) -expansion method and its applications to some nonlinear evolution equations in the mathematical physics

In the present paper, we construct the traveling wave solutions involving parameters for some nonlinear evolution equations in the mathematical physics via the (2+1)-dimensional Painlevé integrable Burgers equations, the (2+1)-dimensional Nizhnik-Novikov-Vesselov equations, the (2+1)-dimensional Boiti-Leon-Pempinelli equations and the (2+1)-dimensional dispersive long wave equations by using a new approach, namely the ( $\frac{G'}{G})$ -expansion method, where G=G(ξ) satisfies a second order linear ordinary differential equation. When the parameters are taken special values, the solitary waves are derived from the traveling waves. The traveling wave solutions are expressed by hyperbolic, trigonometric and rational functions.     

Traveling-Wave Solutions of the Schwarz–Korteweg–de Vries Equation in 2+1 Dimensions and the Ablowitz–Kaup–Newell–Segur Equation Through Symmetry Reductions

One of the more interesting solutions of the (2+1)-dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation is the soliton solutions. We previously derived a complete group classification for the SKdV equation in 2+1 dimensions. Using classical Lie symmetries, we now consider traveling-wave reductions with a variable velocity depending on the form of an arbitrary function. The corresponding solutions of the (2+1)-dimensional equation involve up to three arbitrary smooth functions. Consequently, the solutions exhibit a rich variety of qualitative behaviors. In particular, we show the interaction of a Wadati soliton with a line soliton. Moreover, via a Miura transformation, the SKdV is closely related to the Ablowitz–Kaup–Newell–Segur (AKNS) equation in 2+1 dimensions. Using classical Lie symmetries, we consider traveling-wave reductions for the AKNS equation in 2+1 dimensions. It is interesting that neither of the (2+1)-dimensional integrable systems considered admit Virasoro-type subalgebras.     

Additional symmetries and Bäcklund transformations for the dispersionless Harry Dym hierarchy

We give a dispersionless Toda-like extension to the dispersionless Harry Dym (dDym) hierarchy. The extended dDym (EdDym) hierarchy has a dressing formulation, and its underlying solution structure can be investigated through the twistor construction. We show that additional symmetries of the solution space generate Backlund transformations of the EdDym hierarchy. We present some examples of constructing new solutions of the (2+1)-dimensional dDym and EdDym equations via Bäcklund transformations.     

New solutions of the Schwarzian Korteweg-de Vries equation in 2+1 dimensions based on weak symmetries

We consider the (2+1)-dimensional integrable Schwarzian Korteweg-de Vries equation. Using weak symmetries, we obtain a system of partial differential equations in 1+1 dimensions. Further reductions lead to second-order ordinary differential equations that provide new solutions expressible in terms of known functions. These solutions depend on two arbitrary functions and one arbitrary solution of the Riemann wave equation and cannot be obtained by classical or nonclassical symmetries. Some of the obtained solutions of the Schwarzian Korteweg-de Vries equation exhibit a wide variety of qualitative behaviors; traveling waves and soliton solutions are among the most interesting. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 380–390, June, 2007.     

Bäcklund-Schlesinger transformations for Davey-Stewartson equations

It is shown that integrable (1+2)-dimensional Davey-Stewartson (DS) and Boiti-Leon-Pempinelli (BLP) equations admit an explicit invertible Bäcklund auto-transformation. An algorithm is developed to construct exact solutions for flat- and horseshoe-type solitons of the DS system. Successive application of these transformations allows us to find solutions of (1+1)- and (0+2)-dimensional Toda lattice equations. We point out a similar auto-transformation for analogues of the DS system realized for an arbitrary associative algebra with a unity, in particular, for matrix DS equations. We also relate the (1+2)-dimensional models that we construct to (1+1)-dimensional J-S-systems.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 3, pp. 338–346, December, 1996.     

Symmetry Approach to the Integrability Problem

We review the results of the twenty-year development of the symmetry approach to classifying integrable models in mathematical physics. The generalized Toda chains and the related equations of the nonlinear Schrödinger type, discrete transformations, and hyperbolic systems are central in this approach. Moreover, we consider equations of the Painlevé type, master symmetries, and the problem of integrability criteria for (2+1)-dimensional models. We present the list of canonical forms for (1+1)-dimensional integrable systems. We elaborate the effective tests for integrability and the algorithms for reduction to the canonical form.     

Integrable elliptic pseudopotentials

We construct integrable pseudopotentials with an arbitrary number of fields in terms of an elliptic generalization of hypergeometric functions in several variables. These pseudopotentials are multiparameter deformations of ones constructed by Krichever in studying the Whitham-averaged solutions of the KP equation and yield new integrable (2+1)-dimensional systems of hydrodynamic type. Moreover, an interesting class of integrable (1+1)-dimensional systems described in terms of solutions of an elliptic generalization of the Gibbons-Tsarev system is related to these pseudopotentials.     

Search for high dimensional integrable models

The possible high dimensional integrable models are studied in three different aspects: (i) starting from a strong symmetry operator of a known (1+1) -dimensional integrable model, we can construct a type of (n+1)-dimensional integrable models, high dimensional breaking soliton equations; (ii) from every concrete realization of the generalized Virasoro algebra, we can get many high dimensional integrable models in the meaning that the models possess generalized Virasoro symmetry algebra; (iii) starting from the Schwartz equations which possess conformal invariance, we can also get various high dimensional integrable models in the meaning that they possess Painlevé property. Project supported by the National Natural Science Foundation of China and the Natural Science Foundation of Zhejiang Province.     

One variant of a (2 + 1)-dimensional Volterra system and its (1 + 1)-dimensional reduction

A new system is generated from a multi-linear form of a (2+1)-dimensional Volterra system. Though the system is only partially integrable and needs additional conditions to possess two-soliton solutions, its (1+1)-dimensional reduction gives an integrable equation which has been studied via reduction skills. Here, we give this (1+1)-dimensional reduction a simple bilinear form, from which a Bäcklund transformation is derived and the corresponding nonlinear superposition formula is built.     

Soliton interactions of integrable models

The solution of integrable (n+1)-dimensional KdV system in bilinear form yields a dromion solution that is localized in all directions. The interactions between two dromions are studied both in analytical and in numerical for three (n+1)-dimensional KdV-type equations (n=1, 2, 3). The same interactive properties between two dromions (solitons) are revealed for these models. The interactions between two dromions (solitons) may be elastic or inelastic for different form of solutions.     

Higher-Dimensional Integrable Newton Systems with Quadratic Integrals of Motion

Newton systems     , with integrals of motion quadratic in velocities, are considered. We show that if such a system admits two quadratic integrals of motion of the so-called cofactor type , then it has in fact n quadratic integrals of motion and can be embedded into a  (2 n + 1)  -dimensional bi-Hamiltonian system, which under some non-degeneracy assumptions is completely integrable. The majority of these cofactor pair Newton systems are new, but they also include conservative systems with elliptic and parabolic separable potentials, as well as many integrable Newton systems previously derived from soliton equations. We explain the connection between cofactor pair systems and solutions of a certain system of second-order linear PDEs (the fundamental equations ), and use this to recursively construct infinite families of cofactor pair systems.