S. Yu. Sakovich

Coupled KdV Equations of Hirota-Satsuma Type

Abstract It is shown that the system of two coupled Korteweg-de Vries equations passes the Painleve; test for integrability in nine distinct cases of its coefficients. The integrability of eight cases is verified by direct construction of Lax pairs, whereas for one case it remains unknown.

Cyclic Bases of Zero-Curvature Representations: Five Illustrations to One Concept

The paper contains five examples of using cyclic bases of zero-curvature representations in studies of weak and strong Lax pairs, hierarchies of evolution systems, and recursion operators.

On Integrability of a (2+1)-Dimensional Perturbed KdV Equation

Abstract A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma and B. Fuchssteiner, is proven to pass the Painleve; test for integrability well, and its 4×4 Lax pair with two spectral parameters is found. The results show that the Painleve; classification of coupled KdV equations by A. Karasu should be revised.

Bäcklund transformation and special solutions for the Drinfeld-Sokolov-Satsuma-Hirota system of coupled equations

Using the Weiss method of truncated singular expansions, we construct an explicit Btransformation of the Drinfeld-Sokolov-Satsuma-Hirota system into itself. Then we find all the special solutions generated by this transformation from the trivial zero solution of this system.

Symmetrically coupled higher-order nonlinear Schrödinger equations: singularity analysis and integrability

The integrability of a system of two symmetrically coupled higher-order nonlinear Schrodinger equations with parameter coefficients is tested by means of singularity analysis. It is proven that the system passes the Painleve test for integrability only in ten distinct cases, of which two are new. For one of the new cases, a Lax pair and a multi-field generalization are obtained; for the other one, the equations of the system are uncoupled by a nonlinear transformation.

On conservation laws and zero-curvature representations of the Liouville equation

Applying the first Noether theorem to the Liouville equation uxy=exp u, we find all (namely, a continuum of) non-trivial conservation laws of this equation. Then we find five new zero-curvature representations of the Liouville equation (by 2*2 traceless matrices) which contain, respectively, 1, 1, 2, 2 and 3 essential parameters. Finally, we show that all known zero-curvature representations of the Liouville equation are equivalent (in a definite sense) to matrices of conservation laws.

On Miura transformations of evolution equations

The general Miura transformation (t,x,u(t,x)) to (s,y,v(s,y)): v=a(t,x,u,. . ., delta ru/ delta xr), y=b(t,x,u,. . ., delta ru/ delta xr), s=c(t,x,u,. . ., delta ru/ delta xr) is considered which connects two evolution equations ut=f(t,x,u,. . ., delta nu/ delta xn) and vs=g(t,x,u,. . ., delta mu/ delta xm). The conditions c=c(t) and m=n are proven to be necessary. It is shown that every Miura transformation, admitted by a constant separant equation ut=f, consists of the following three transformations: (i) (t,x,u) to (t,x,w), where w=a(t,x,u,. . .,ux. . .x); (ii) (t,x,w) to (t,y,v), where y=x and v=w, or y=w and v=wx, or y=wx and v=wxx; (iii) a transformation of time s=c(t) and a contact transformation of (y,v). As an example, the Korteweg-de Vries equation is transformed to three new nonlinear equations, of which two have neither nontrivial algebra of generalized symmetries nor infinite set of conserved densities.

The Miura transformation and Lie-Bäcklund algebras of exactly solvable equations

Abstract All the one-dimensional one-component local evolution equations connected via the Miura transformation are found. Exactly solvable equations and their Lie-Backlund algebras are shown to generate interesting transformations of infinite classes of evolution equations.

A System of Four ODEs: The Singularity Analysis

Abstract The singularity analysis is carried out for a system of four first-order quadratic ODEs with a parameter, which was proposed recently by Golubchik and Sokolov. A transformation of dependent variables is revealed by the analysis, after which the transformed system possesses the Painlevé property and does not contain the parameter.

Painleve analysis and Backlund transformations of Doktorov-Vlasov equations

The singularity analysis of the system of nonlinear equations iat=axx+aa2a*-ip, px+i beta p+ar=0, rx= 1/2 (ap*+a*p) (where * denotes the complex conjugation, functions a and p are complex, function r and constants alpha and beta are real) indicates that the system has the Painleve property at a= 1/2 only. This analytic exclusiveness of the case a= 1/2 agrees with results by Doktorov and Vlasov (1983) who selected the same case by a modification of the Wahlquist-Estabrook method and found a corresponding Lax pair. In the integrable case, the method of truncating Weiss-Tabor-Carnevale expansions determines a Backlund auto-transformation which, unfortunately, violates the condition of complex conjugateness between a and a*. Another Backlund auto-transformation, compatible with this condition, is found by a technique of Miura transformations.