Solving large linear algebraic systems in the context of integrable non-abelian Laurent ODEs

Abstract The paper reports on a computer algebra program LSSS (Linear Selective Systems Solver) for solving linear algebraic systems with rational coefficients. The program is especially efficient for very large sparse systems that have a solution in which many variables take the value zero. The program is applied to the symmetry investigation of a non-abelian Laurent ODE introduced recently by M. Kontsevich. The computed symmetries confirmed that a Lax pair found for this system earlier generates all first integrals of degree at least up to 14

Painleve equations and applications

The theme running throughout this thesis is the Painlevé equations, in their differential, discrete and ultra-discrete versions. The differential Painlevé equations have the Painlevé property. If all solutions of a differential equation are meromorphic functions then it necessarily has the Painlevé property. Any ODE with the Painlevé property is necessarily a reduction of an integrable PDE. Nevanlinna theory studies the value distribution and characterizes the growth of meromorphic functions, by using certain averaged properties on a disc of variable radius. We shall be interested in its well-known use as a tool for detecting integrability in difference equations—a difference equation may be integrable if it has sufficiently many finite-order solutions in the sense of Nevanlinna theory. This does not provide a sufficient test for integrability; additionally it must satisfy the well-known singularity confinement test. [Continues.

A Lie symmetry analysis and explicit solutions of the two-dimensional ∞-Polylaplacian

In this work, we consider the Lie point symmetry analysis of a strongly nonlinear partial differential equation of third order, the ∞‐Polylaplacian, in two spatial dimensions. This equation is a higher order generalization of the ∞‐Laplacian, also known as Aronsson's equation, and arises as the analog of the Euler–Lagrange equations of a second‐order variational principle in L∞. We obtain its full symmetry group, one‐dimensional Lie subalgebras and the corresponding symmetry reductions to ordinary differential equations. Finally, we use the Lie symmetries to construct new invariant ∞‐Polyharmonic functions

Whitham Deformations of the Korteweg-de Vries Equation

In this thesis, a new independent approach to obtain a dispersionless version of the KdV hierarchy is presented and used to describe a class of solutions that are accessible by the generalized hodograph method. The Korteweg-de Vries (KdV) equation is a dispersive and non-linear partial differential equation (PDE) in one spatial dimension and time. It originates from the observation and description of solitary shallow-water waves and later became famous in the context of the Fermi–Pasta–Ulam–Tsingou problem. An aspect that has drawn a lot of attention is that the KdV equation possesses infinitely many conserved quantities and corresponding symmetries – giving rise to the structure of an integrable hierarchy. The focus of the thesis is on a dispersionless version of the integrable KdV hierarchy which is obtained by applying adiabatic theory from classical mechanics. The resulting KdV Whitham hierarchy is again integrable, but in a more general way. While the dispersive equation admits stable solitary waves as solutions, on the dispersionless side breaking waves occur. The theoretical description of the dispersionless hierarchy yields algebraic-geometric and differential-geometric structures which are defined on the space of the conserved quantities of the dispersive hierarchy. In particular, Euler–Poisson–Darboux equations, known from classical differential geometry, can be used to characterize solutions of the KdV Whitham hierarchy