Integrable pseudopotentials related to generalized hypergeometric functions

We construct integrable pseudopotentials with an arbitrary number of fields in terms of generalized hypergeometric functions. These pseudopotentials yield some integrable (2 + 1)-dimensional hydrodynamic type systems. In two particular cases these systems are equivalent to integrable scalar 3-dimensional equations of second order. An interesting class of integrable (1 + 1)-dimensional hydrodynamic type systems is also generated by our pseudopotentials.     

Integrable (2+1)-dimensional systems of hydrodynamic type

We describe the results that have so far been obtained in the classification problem for integrable (2+1)-dimensional systems of hydrodynamic type. The Gibbons-Tsarev (GT) systems are most fundamental here. A whole class of integrable (2+1)-dimensional models is related to each such system. We present the known GT systems related to algebraic curves of genus g = 0 and g = 1 and also a new GT system corresponding to algebraic curves of genus g = 2. We construct a wide class of integrable models generated by the simplest GT system, which was not considered previously because it is “trivial.”     

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.     

(2+1)-dimensional Broer–Kaup system of shallow water waves and similarity solutions with symbolic computation

The Broer–Kaup system is among the important integrable models for the shallow water waves. For a (2+1)-dimensional Broer–Kaup system and with symbolic computation, we present some similarity solutions, which are expressible in terms of the Jacobian elliptic functions and second Painlevé transcendent. Our results are in agreement with the Painlevé conjecture.     

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.     

A family of (2+1)-dimensional hydrodynamic type systems possessing a pseudopotential

We construct a family of integrable hydrodynamic type systems with three independent and n ≥ 2 dependent variables in terms of solutions of a linear system of PDEs with rational coefficients. We choose the existence of a pseudopotential as a criterion of integrability. In the case n = 2 this family is a general solution of the classification problem for such systems. We also give an elliptic analog of this family in the case n > 2.     

(2+1)-dimensional Broer–Kaup system of shallow water waves and similarity solutions with symbolic computation

The Broer–Kaup system is among the important integrable models for the shallow water waves. For a (2+1)-dimensional Broer–Kaup system and with symbolic computation, we present some similarity solutions, which are expressible in terms of the Jacobian elliptic functions and second Painlevé transcendent. Our results are in agreement with the Painlevé conjecture.Received: February 26, 2003; revised: August 11, 2003     

The Application of Neumann Type Systems for Solving Integrable Nonlinear Evolution Equations

An algorithm to obtain finite‐gap solutions of integrable nonlinear evolution equations (INLEEs) is provided by using the Neumann type systems in the framework of algebraic geometry. From the nonlinearization of Lax pairs, some INLEEs in 1+1 and 2+1 dimensions are reduced into a class of new Neumann type systems separating the spatial and temporal variables of INLEEs over a symplectic submanifold (M, ω²) . Based on the Lax representations of INLEEs, we deduce the Lax–Moser matrix for those Neumann type systems that yield the integrals of motion, elliptic variables, and a hyperelliptic curve of Riemann surface. Then, we attain the Liouville integrability for a hierarchy of Neumann type systems in view of a Lax equation on (M, ω²) and a set of quasi‐Abel–Jacobi variables. We also specify the relationship between Neumann type systems and INLEEs, where the involutive solutions of Neumann type systems give rise to the finite parametric solutions of INLEEs and the Neumann map cuts out a finite dimensional invariant subspace for INLEEs. Under the Abel–Jacobi variables, the Neumann type flows, the 1+1, and 2+1 dimensional flows are integrated with Abel–Jacobi solutions; as a result, the finite‐gap solutions expressed by Riemann theta functions for some 1+1 and 2+1 dimensional INLEEs are achieved through the Jacobi inversion with the aid of the Riemann theorem.     

The dressing method and nonlocal Riemann-Hilbert problems

Summary We consider equations in 2+1 solvable in terms of a nonlocal Riemann-Hilbert problem and show that for such an equation there exists a unified dressing method which yields: (i) a Lax pair suitable for obtaining solutions that are perturbations of an arbitrary exact solution of the given equation; (ii) certain integrable generalizations of the given equation. Using this generalized dressing method large classes of solutions of these equations, including dromions and line dromions, can be obtained. The method is illustrated by using theN-wave interactions, the Davey-Stewartson I, and the Kadomtsev-Petviashvili I equations. We also show that a careful application of the usual dressing method yields a certain generalization of theN-wave interactions.     

Differential-geometric structures in the theory of two-dimensional integrable equations

The gauge representations of the integrable generalization of the Heisenberg magnet in (2+1)-dimensional space-time are interpreted in terms of topological charge. Restrictions on the class of solutions to the equation for a two-dimensional magnet are described for which it becomes gauge-equivalent to the Davy-Stuartson equation.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 189, pp. 75–81, 1991.The author is grateful to P. P. Kulish for stimulating discussions.     

Einige Bemerkungen über die Regularität von stationären Punkten gewisser geometrischer Variationsintegrale

We discuss the regularity properties of weak solutions of certain degenerate elliptic systems arising from problems in geometry. For example we prove C¹-regularity of conformal m-harmonic maps from the m-ball into m-dimensional oriented submanifolds of R^m+1 as well as a regularity theorem for weak H-surfaces in R^m+1. Our arguments make strong use of a version of the isoperimetric inequality.     

Non-commutative integrability, paths and quasi-determinants

In previous work, we showed that the solution of certain systems of discrete integrable equations, notably Q and T-systems, is given in terms of partition functions of positively weighted paths, thereby proving the positive Laurent phenomenon of Fomin and Zelevinsky for these cases. This method of solution is amenable to generalization to non-commutative weighted paths. Under certain circumstances, these describe solutions of discrete evolution equations in non-commutative variables: Examples are the corresponding quantum cluster algebras (Berenstein and Zelevinsky (2005) [3]), the Kontsevich evolution (Di Francesco and Kedem (2010) [10]) and the T-systems themselves (Di Francesco and Kedem (2009) [8]). In this paper, we formulate certain non-commutative integrable evolutions by considering paths with non-commutative weights, together with an evolution of the weights that reduces to cluster algebra mutations in the commutative limit. The general weights are expressed as Laurent monomials of quasi-determinants of path partition functions, allowing for a non-commutative version of the positive Laurent phenomenon. We apply this construction to the known systems, and obtain Laurent positivity results for their solutions in terms of initial data.     

Hamiltonization and Integrability of the Chaplygin Sphere in ℝ n

This paper studies a natural n-dimensional generalization of the classical nonholonomic Chaplygin sphere problem. We prove that for a specific choice of the inertia operator, the restriction of the generalized problem onto a zero value of the SO(n−1)-momentum mapping becomes an integrable Hamiltonian system after an appropriate time reparametrization.     

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.     

On q-Gaussian Integrable Hamiltonian Reductions in Anisentropic Magneto-gasdynamics

Integrable substructure in 2+1-dimensional anisentropic magneto-gasdynamics is investigated via a general elliptic vortex ansatz. The procedure involves introduction of a q-Gaussian density representation. Thermodynamically consistent relations are isolated associated with certain integrable Hamiltonian reductions.     

Jacobi Elliptic Function Solutions and Traveling Wave Solutions of the (2 + 1)-Dimensional Gardner-KP Equation

The fully integrable KP equation is one of the models that describes the evolution of nonlinear waves, the expansion of the well-known KdV equation, where the impacts of surface tension and viscosity are negligible. This paper uses the Modified Extended Direct Algebraic (MEDA) method to build fresh exact, periodic, trigonometric, hyperbolic, rational, triangular and soliton alternatives for the (2 + 1)-dimensional Gardner KP equation. These solutions that we discover in this article will help us understand the phenomena of the (2 + 1)-dimensional Gardner KP equation. Comparing the study in this paper and existing work, we find more exact solutions with soliton and periodic structures and the rational function solution in this paper is more general than the rational solution in existing literature. Most of the Jacobi elliptic function solutions and the mixed Jacobi elliptic function solutions to the (2 + 1)-dimensional Gardner KP equation discovered in this paper, to the best of our highest understanding are not seen in any existing paper until now.     

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.     

Damping of Periodic Waves in Physically Significant Wave Systems

Damping of periodic waves in the classically important nonlinear wave systems—nonlinear Schrödinger, Korteweg–deVries (KdV), and modified KdV—is considered here. For small damping, asymptotic analysis is used to find an explicit equation that governs the temporal evolution of the solution. These results are then confirmed by direct numerical simulations. The undamped periodic solutions are given in terms of Jacobi elliptic functions. The damping structure is found as a function of the elliptic function modulus, m=m(t) . The damping rate of the maximum amplitude is ascertained and is found to vary smoothly from the linear solution when m= 0 to soliton waves when m= 1 .     

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.     

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.