Michal Marvan

On integrability of Weingarten surfaces: a forgotten class

Rediscovered by a systematic search, a forgotten class of integrable surfaces is shown to disprove the Finkel–Wu conjecture. The associated integrable nonlinear partial differential equation possesses a zero curvature representation, a third-order symmetry and a nonlocal transformation to the sine-Gordon equation ξη = sin . We leave open the problem of finding a Backlund autotransformation and a recursion operator that would produce a local hierarchy.

Sufficient Set of Integrability Conditions of an Orthonomic System

Every orthonomic system of partial differential equations is known to possess a finite number of integrability conditions sufficient to ensure the validity of them all. Here we show that a redundancy-free sufficient set of integrability conditions can be constructed in a time proportional to the number of equations cubed.

Scalar second-order evolution equations possessing an irreducible sl2-valued zero-curvature representation

We find all scalar second-order evolution equations that possess an sl2-valued zero-curvature representation irreducible to a proper subalgebra of sl2. None of these zero-curvature representations depends on a parameter that could serve as the spectral parameter.

Classification of integrable Weingarten surfaces possessing an \mathfrak{sl} (2)-valued zero curvature representation

In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an (2)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary differential equation to govern the functional dependence of the principal curvatures. Employing the scaling and translation (offsetting) symmetry, we give a general solution of the governing equation in terms of elliptic integrals. We show that the instances when the elliptic integrals degenerate to elementary functions were known to nineteenth-century geometers. Finally, we characterize the associated normal congruences.

Differential invariants of generic hyperbolic monge-ampére equations

In this paper basic differential invariants of generic hyperbolic Monge-Ampère equations with respect to contact transformations are constructed and the equivalence problem for these equations is solved.

A Reciprocal Transformation for the Constant Astigmatism Equation

We introduce a nonlocal transformation to generate exact solutions of the con- stant astigmatism equation zyy + (1=z)xx + 2 = 0. The transformation is related to the special case of the famous Backlund transformation of the sine-Gordon equation with the Backlund parameter = 1. It is also a nonlocal symmetry.

Some Local Properties of Bäcklund Transformations

For Bäcklund transformations, treated as relations in the categoryof diffieties, local conditions of effectivity and normality are introduced,having implications for the solution generating properties. We check themfor the pKdV, the sine-Gordon, and the Tzitzéica equation.

Coverings and Integrability of the Gauss–Mainardi–Codazzi Equations

Using the covering theory approach (zero-curvature representations with the gauge group SL), we insert the spectral parameter into the Gauss–Mainardi–Codazzi equations in Chebyshev and geodesic coordinates. For each choice, four integrable systems are obtained.

Nonlocal conservation laws of the constant astigmatism equation

Abstract For the constant astigmatism equation, we construct a system of nonlocal conservation laws (an abelian covering) closed under the reciprocal transformations. The corresponding potentials are functionally independent modulo a Wronskian type relation.

On construction of symmetries and recursion operators from zero-curvature representations and the Darboux–Egoroff system

Abstract The Darboux–Egoroff system of PDEs with any number n ≥ 3 of independent variables plays an essential role in the problems of describing n -dimensional flat diagonal metrics of Egoroff type and Frobenius manifolds. We construct a recursion operator and its inverse for symmetries of the Darboux–Egoroff system and describe some symmetries generated by these operators. The constructed recursion operators are not pseudodifferential, but are Backlund autotransformations for the linearized system whose solutions correspond to symmetries of the Darboux–Egoroff system. For some other PDEs, recursion operators of similar types were considered previously by Papachristou, Guthrie, Marvan, Pobořil, and Sergyeyev. In the structure of the obtained third and fifth order symmetries of the Darboux–Egoroff system, one finds the third and fifth order flows of an ( n − 1 ) -component vector modified KdV hierarchy. The constructed recursion operators generate also an infinite number of nonlocal symmetries. In particular, we obtain a simple construction of nonlocal symmetries that were studied by Buryak and Shadrin in the context of the infinitesimal version of the Givental–van de Leur twisted loop group action on the space of semisimple Frobenius manifolds. We obtain these results by means of rather general methods, using only the zero-curvature representation of the considered PDEs.