Darboux Integrable Systems Research Articles

Miura-Type Transformations for Integrable Lattices in 3D

This article studies a class of integrable semi-discrete equations with one continuous and two discrete independent variables. At present, in the literature there are nine integrable equations of the form un+1,xj=f(un,xj,unj+1,unj,un+1j,un+1j−1) up to point transformations. An efficient method based on some relation that generalizes the notion of the local conservation law is proposed for searching for Miura-type transformations relating to semi-discrete equations in 3D. The efficiency of the method is illustrated with the equations from the list. For one of the equations, which is little studied, the continuum limit is calculated. For this equation, the problem of finite-field reductions in the form of Darboux integrable systems of equations of a hyperbolic type is discussed. For reductions of small orders, N=1 and N=2, complete sets of characteristic integrals are presented. Note that the existence of characteristic integrals makes it possible to construct particular solutions to the original lattice. For the case N=1, an explicit solution was found in this paper. A new semi-discrete equation is found that lies beyond the considered class. For this equation, the Lax pair is presented.

Darboux integrability for diagonal systems of hydrodynamic type

We prove that (1) diagonal systems of hydrodynamic type are Darboux integrable if and only if the corresponding systems for commuting flows are Darboux integrable, (2) systems for commuting flows are Darboux integrable if and only if the Laplace transformation sequences terminate, (3) Darboux integrable systems are necessarily semihamiltonian. We give geometric interpretation for Darboux integrability of such systems in terms of congruences of lines and in terms of solution orbits with respect to symmetry subalgebras, discuss known and new examples.

Application of invariants of characteristics to construction of solutions without gradient catastrophe

The one-dimensional system of equations of isentropic gas dynamics is considered. First-order invariants of characteristics of this system are classified. Second-order invariants of characteristics are classified for polytropic processes. The infinite sequence of Darboux integrable systems is described. The approach to construction of smooth solutions without gradient catastrophe is proposed. Examples of solutions without gradient catastrophe are presented.

Symmetries of a Certain Periodic Chain

We consider a periodic closure of a nonlinear integrable two-dimensional three-point chain. Integrability is understood in the sense that the chain admits a wide class of reductions, which are nonlinear hyperbolic Darboux integrable systems with two independent variables.We consider a system obtained as a period-2 periodic closure of one of two-dimensional three-point chains found within this framework. For this system, a second-order higher symmetry depending on two arbitrary functions is constructed.

Characteristic Lie algebras of integrable differential-difference equations in 3D

The purpose of this article is to discuss an algebraic method for studying integrable differential-difference lattices with two discrete and one continuous independent variables. The main idea is based on the hypothesis that any integrable equation of this type admits an infinite sequence of reductions, which are Darboux integrable systems of differential-difference equations with two independent variables. To detect the Darboux integrability property of the reductions the authors use the characteristic Lie–Rinehart algebras. The hypothesis is confirmed by integrable three-dimensional lattices from the list given by Ferapontov et al (2015 Int. Math. Res. Not. 2015 4933). The required reductions are found for these lattices. Characteristic algebras corresponding to small-order reductions are described, integrals are calculated in both directions, which proves Darboux integrability. The approach might be useful for the integrable classification in 3D.

Finite cyclicity of the contact point in slow-fast integrable systems of Darboux type

Using singular perturbation theory and family blow-up we prove that nilpotent contact points in deformations of slow-fast Darboux integrable systems have finite cyclicity. The deformations are smooth or analytic depending on the region in the parameter space. This article is a natural continuation of [1,3], where one studies limit cycles in polynomial deformations of slow-fast Darboux integrable systems, around the ``integrable" direction in the parameter space. We extend the existing finite cyclicity result of the contact point to analytic deformations, and under some assumptions we prove that the contact point has finite cyclicity around the ``slow-fast" direction in the parameter space.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/90/abstr.html

On the discretization of Darboux Integrable Systems

We study the discretization of Darboux integrable systems. The discretization is done using x-, y-integrals of the considered continuous systems. New examples of semi-discrete Darboux integrable systems are obtained.

Алгебраические свойства квазилинейных двумеризованных цепочек, связанные с интегрируемостью

In the article a classification method for nonlinear integrable equations with three independent variables is discussed based on the notion of the integrable reductions. We call the equation integrable if it admits a large class of reductions being Darboux integrable systems of hyperbolic type equations with two independent variables. The most natural and convenient object to be studied within the frame of this scheme is the class of two dimensional lattices generalizing the well-known Toda lattice. In the present article we deal with the quasilinear lattices of the form $u_{n,xy}=\alpha(u_{n+1} ,u_n,u_{n-1} )u_{n,x}u_{n,y} + \beta(u_{n+1},u_n,u_{n-1})u_{n,x}+\gamma(u_{n+1} ,u_n,u_{n-1} )u_{n,y}+\delta(u_{n+1} ,u_n,u_{n-1})$. We specify the coefficients of the lattice assuming that there exist cutting off conditions which reduce the lattice to a Darboux integrable hyperbolic type system of the arbitrarily high order. Under some extra assumption of nondegeneracy we described the class of the lattices integrable in the sense indicated above. There are new examples in the obtained list of chains.

Bäcklund transformations for Darboux integrable differential systems: Examples and applications

In this article we demonstrate a new symmetry based method for constructing Bäcklund transformations by finding explicit Bäcklund transformations between Darboux integrable systems. This results in a number of new examples of Bäcklund transformations which are quite different in character than those typically found in the literature. The relation between the intermediate integrals for Darboux integrable systems and the differential invariants of the Vessiot group is also illustrated. We then show that a well known class of Bäcklund transformations between a Darboux integrable Monge–Ampère system and the wave equation always arises by this method. The results of this paper build upon the presentation of Darboux integrable systems as quotients of differential systems by symmetry groups.

Darboux integrability and algebraic limit cycles for a class of polynomial differential systems

This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems $$\mathop x\limits^. $$ = λ x−y +P n+1(x, y)+xF 2n (x, y), $$\mathop y\limits^. $$ = x+λ y +Q n+1(x, y)+yF 2n (x, y), where P i (x, y), Q i (x, y) and F i (x, y) are homogeneous polynomials of degree i. Within this class, we identify some new Darboux integrable systems having either a focus or a center at the origin. For such Darboux integrable systems having degrees 5 and 9 we give the explicit expressions of their algebraic limit cycles. For the systems having degrees 3, 5, 7 and 9 and restricted to a certain subclass we present necessary and sufficient conditions for being Darboux integrable.

Pseudo-abelian integrals on slow-fast Darboux systems

We study pseudo-Abelian integrals associated with polynomial deformations of slow-fast Darboux integrable systems. Under some assumptions we prove local boundedness of the number of their zeros.

Integrability, degenerate centers, and limit cycles for a class of polynomial differential systems

We consider the class of polynomial differential equations x˙ P n ( x,y)+ P n+1 ( x,y)+ P n+2 ( x,y), y˙= Q n ( x,y)+ Q n+1 ( x,y)+ Q n+2 ( x,y), for n ≥ 1 and where P i and Q i are homogeneous polynomials of degree i These systems have a linearly zero singular point at the origin if n > 2. Inside this class, we identify a new subclass of Darboux integrable systems, and some of them having a degenerate center, i.e., a center with linear part identically zero. Moreover, under additional conditions such Darboux integrable systems can have at most one limit cycle. We provide the explicit expression of this limit cycle.

A family of isochronous foci with Darboux first integral

We consider the class of polynomial differential equations ? = ?x-y +Pn(x,y)+P2n-1(x,y), ? = x + ?y + Qn(x,y) + Q2n-1(x,y) with n = 2, where Pi and Qi are homogeneous polynomials of degree i. These systems have a focus at the origin if ?0, and have either a center or a focus if ? = 0. Inside this class we identify a new subclass of Darboux integrable systems having either a focus or a center at the origin. Under generic conditions such Darboux integrable systems can have at most two limit cycles, and when they exist are algebraic. For the case n = 2 and n = 3 we present new classes of Darboux integrable systems having a focus.

Linearization of Isochronous Centers

In this paper we study isochronous centers of polynomial systems. We first discuss isochronous centers of quadratic systems, cubic symmetric systems and reduced Kukles system. All these systems have rational first integrals. We give a unified proof of the isochronicity of these systems, by constructing algebraic linearizing changes of coordinates. We then study two other classes of systems with isochronous centers, namely the class of "complex"systems ż = i P( z), and the class of cubic systems symmetric with respect to a line and satisfying Θ = 1. Both classes consist of Darboux integrable systems. We discuss their geometric properties and construct the linearizing changes of coordinates. We show that the class of polynomial isochronous systems carries a very rich geometry. Finally, we discuss the geometry of the linearizing changes of coordinates in the complex plane.