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We study one third-order nonlinear evolution equation, recently introduced by Chou and Qu in a problem of plane curve motions, and find its transformation to the modified Korteweg - de Vries equation, its zero-curvature representation with an essential parameter, and its second-order recursion operator., Comment: 10 pages

For zero-curvature representations (ZCRs) At-Bx-AB+BA=0 of evolution equations ut=f(x, u, ux, ..., ux...x), we develop a description which is invariant under gauge transformations A'=SAS-1-SxS-1 and B'=SBS-1-StS-1, where A, B and S are matrix functions of x, u, ux, uxx, ... . We prove that every fixed matrix A of any dimension and order (in ux...x) determines a continual class of evolution equations which admit ZCRs with this A. Then we quote examples illustrating how a dependence of A on an essential parameter restricts classes of represented equations. One of our examples shows that some non-integrable systems can admit parametric Lax pairs and infinitely many non-trivial conservation laws.

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.

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.

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.

Soliton solutions are obtained for the modified Manakov equations which are the integrable version of equations proposed recently by Hisakado, Iizuka and Wadati to describe the femtosecond pulse propagation in a birefringent optical fiber.

We perform the singularity analysis of four new generalized bilinear equations by Hu (1993), of which two possess two-soliton solutions and the other two possess even three-soliton solutions. All four equations fail to pass the Painleve test for integrability, exhibiting non-dominant movable logarithmic singularities at highest resonances of generic branches.

A recursion operator is constructed for a new integrable system of coupled Korteweg - de Vries equations by the method of gauge-invariant description of zero-curvature representations. This second-order recursion operator is characterized by unusual structure of its nonlocal part., Comment: 12 pages, final version

The integrability of a coupled KdV-mKdV system is tested by means of singularity analysis. The true Lax pair associated with this system is obtained by the use of prolongation technique., Comment: 9 pages

The gauge-invariant description of zero-curvature representations of evolution equations is applied to the problem of how to distinguish the fake Lax pairs from the true Lax pairs. The main difference between the true Lax pairs and the fake ones is found in the structure of their cyclic bases., 5 pages; v2: sections, discussion, references added

Using the Weiss method of truncated singular expansions, we construct an explicit Backlund transformation 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., LaTeX, 5 pages

Integrability of the differential constraints arising from the singularity analysis of two (1+1)-dimensional second-order evolution equations is studied. Two nonlinear ordinary differential equations are obtained in this way, which are integrable by quadratures in spite of very complicated branching of their solutions., Comment: arxiv version is already offcial

The integrability of a system of two symmetrically coupled higher-order nonlinear Schr\"{o}dinger equations with parameter coefficients is tested by means of the singularity analysis. It is proven that the system passes the Painlev\'{e} 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., Comment: 12 pages, LaTeX2e, IOP style, final version, to appear in J.Phys.A:Math.Gen

The polynomial Miura transformation ϕ = ω x + a ( ω ) ( a ( ω ) is an n th-power polynomial) connects continual classes of local evolution equations at n n ⩾3 it turns out to connect only the trivial equations ϕ t = ϕ x and ω t = ω x or some equivalent to them.

The variable coefficient Korteweg-de Vries (KdV) equation u 1 +α t n uu x +β t m u xxx =0 is shown to be exceptional in the two cases m = n and m = 2 n +1 by showing that they are equivalent to the plane KdV and cylindrical KdV equations respectively. Thus the integrability depends on the two parameters m and n . Here we also consider the numerical solutions of the equation in different cases, viz ., the integrability condition is satisfied and not satisfied. Only when the equation is equivalent to the plane KdV, the profile is seen to be propagating as a stable solitary-wave. In all other cases an oscillatory tail appear on the left of the wave. In the case of interactions of two waves also oscillatory tails appear. The momentum and energy are conserved in all the cases.

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.

The applicability of the Painleve test of the complete integrability of the one-component nonlinear Klein-Fock-Gordon equations in an arbitrary Riemannian space, in the formulation of Weiss et al. (J. Math. Phys., vol.25, p.13, 1984) is discussed. Three infinite series of these equations are found in the flat two-dimensional space which possess the Painleve property and include, as a special case, the Liouville, sine-Gordon, and Dodd-Bullough equations. It is pointed out that the approach of Weiss et al. to select integrable nonlinear equations is not sufficiently reliable and needs some strengthening.