arXiv subjects
Publications and source records attributed to S. Ya. Startsev.
A new family of Darboux integrable partial differential equations is constructed. The minimal orders of integrals in both characteristics can simultaneously be arbitrarily high for equations within this family. This suggests that the complete list of Darboux integrable equations may contain substantially more equations than are currently known.
The autonomous Hietarinta equation is a well-known example of the quad-graph discrete equation which is consistent around the cube. In a recent work, it was conjectured that this equation is Darboux integrable (i.e., for each of two independent discrete variables there exist non-trivial functions that remain unchanged on solutions of the equation after the shift in this discrete variable). We demonstrate that this conjecture is not true for generic values of the equation coefficients. To do this, we employ two-point invertible transformations introduced by R.I.~Yamilov. We prove that an autonomous difference equation on the quad-graph cannot be Darboux integrable if a transformation of the above type maps solutions of this equation into its solutions again. This implies that the generic Hietarinta equation is not Darboux integrable since the Hietarinta equation in the general case possesses the two-point invertible auto-transformations. Along the way, all Darboux integrable subcases of the Hietarinta equation are found. All of them are reduced by point transformations to already known integrable equations. At the end of the article, we also briefly describe another way to prove the Darboux non-integrability of the Hietarinta equation. This alternative way is based on the known fact that a difference substitution relates this equation to a linear one. Thus, the Hietarinta equation gives us an example of a quad-graph equation that is linearizable but not Darboux integrable.
This text is devoted to hyperbolic equations admitting differential operators that map any function of one independent variable into a symmetry of the corresponding equation. We use the term `symmetry driver' for such operators and prove that any symmetry driver of the smallest order is pre-Hamiltonian (i.e., the image of the driver is closed with respect to the standard bracket). This allows us to prove that the composition of a symmetry driver with the Fréchet derivative of an integral is also pre-Hamiltonian (in a new set of the variables) if both the symmetry driver and the integral have the smallest orders.
A direct generalization of Laplace invariants to the case of hyperbolic partial differential systems is considered. The proof of the following statement is given: the determinant of a Laplace invariant vanishes if the corresponding system admits an integral.
The paper is devoted to the conjecture that an equation is Darboux integrable if and only if it possesses symmetries depending on arbitrary functions. We note that results of previous works together prove this conjecture for scalar partial differential equations of the form $u_{xy}=F(x,y,u,u_x,u_y)$. For autonomous semi-discrete and discrete analogues of these equations we prove that the sequence of Laplace invariants is terminated by zero for an equation if this equation admits an operator mapping any function of one independent variable into a symmetry of the equation. The vanishing of an Laplace invariant allows us to construct a formal integral, i.e. an operator that maps symmetries into integrals (including, generally speaking, trivial integrals). This and results of previous works together prove a `formal' version of the aforementioned conjecture in the semi-discrete and pure discrete cases.
All Darboux integrable difference equations on the quad-graph are described in the case of the equations that possess autonomous first-order integrals in one of the characteristics. A generalization of the discrete Liouville equation is obtained from a subclass of these equation via a non-point transformation. The general proposition on the symmetry structure of the quad-graph equations is proved as an auxiliary result.
The notion of Laplace invariants is transferred to the lattices and discrete equations which are difference analogs of hyperbolic PDE's with two independent variables. The sequence of Laplace invariants satisfy the discrete analog of twodimensional Toda lattice. The terminating of this sequence by zeroes is proved to be the necessary condition for existence of the integrals of the equation under consideration. The formulae are presented for the higher symmetries of the equations possessing integrals. The general theory is illustrated by examples of difference analogs of Liouville equation.
An algorithm is constructed which allows to express conserved flows of hyperbolic equations in terms of corresponding conserved densities and to eliminate these flows from conservation laws of hyperbolic equations. The application of this algorithm to canonical conservation laws gives constructive necessary integrability conditions of hyperbolic equations in terms of the generalized Laplace invariants of these equations.
There are considered differential substitutions of the form $v=P(x,u,u_{x})$ for which there exists a differential operator $H=\sum^{k}_{i=0} α_{i} D^{i}_{x}$ such that the differential substitution maps the equation $u_{t}=H[s(x,P,D_{x}(P),...,D^{k}_{x}(P))]$ into an evolution equation for any function $s$ and any nonnegative integer $k$. All differential substitutions of the form $v=P(x,u,u_{x})$ known to the author have this property. For example, the well-known Miura transformation $v=u_{x}-u^{2}$ maps any equation of the form $$u_{t}=(D^{2}_{x}+2uD_{x}+2u_{x}) [s(x,u_{x}-u^{2},D_{x}(u_{x}-u^{2}),...,D^{k}_{x}(u_{x}-u^{2}))]$$ into the equation $$v_{t}=(D^{3}_{x}+4vD_{x}+2v_{x})[s(x,v,{{\partial v}\over{\partial x }},...,{{\partial^{k} v}\over{\partial x^{k}}})].$$ The complete classification of such differential substitutions is given. An infinite set of the pairwise nonequivalent differential substitutions with the property mentioned above is constructed. Moreover, a general result about symmetries and invariant functions of hyperbolic equations is obtained.