Related papers: Algebraic entropy computations for lattice equatio…
We study the growth of degrees in many autonomous and non-autonomous lattice equations defined by quad rules with corner boundary values, some of which are known to be integrable by other characterisations. Subject to an enabling…
We study complexity in terms of degree growth of one-component lattice equations defined on a $3\times 3$ stencil. The equations include two in Hirota bilinear form and the Boussinesq equations of regular, modified and Schwarzian type.…
In this paper we investigate the integrability of two-dimensional partial difference equations using the newly developed techniques of study of the degree of the iterates. We show that while for generic, nonintegrable equations, the degree…
While it is known that one can consider the Cauchy problem for evolution equations with Caputo derivatives, the situation for the initial value problems for the Riemann-Liouville derivatives is less understood. In this paper we propose new…
We consider initial value problems for differential-algebraic equations in a possibly infinite-dimensional Hilbert space. Assuming a growth condition for the associated operator pencil, we prove existence and uniqueness of solutions for…
We confront two integrability criteria for rational mappings. The first is the singularity confinement based on the requirement that every singularity, spontaneously appearing during the iteration of a mapping, disappear after some steps.…
The theory of degree growth and algebraic entropy plays a crucial role in the field of discrete integrable systems. However, a general method for calculating degree growth for lattice equations (partial difference equations) is not yet…
Completely integrable finite dimensional Hamiltonian systems are well understood thanks to the work of Liouville and Arnold. On the other hand, the Lax Pair formulation of the KdV equation marks the beginning of the extension of the…
We introduce a family of extensions of the Hietarinta-Viallet equation to a multi-term recurrence relation via a reduction from the coprimeness-preserving extension to the discrete KdV equation. The recurrence satisfies the irreducibility…
We propose dynamical systems defined on algebra of lattices, which we call `lattice equations'. We give exact general solutions of initial value problems for a class of lattice equations, and evaluate the complexity of the solutions.…
The energy method can be used to identify well-posed initial boundary value problems for quasi-linear, symmetric hyperbolic partial differential equations with maximally dissipative boundary conditions. A similar analysis of the discrete…
We study initial value problem for a system consisting of an integer order and distributed-order fractional differential equation describing forced oscillations of a body attached to a free end of a light viscoelastic rod. Explicit form of…
In this paper, we examine the space of initial values for integrable lattice equations, which are lattice equations classified by Adler {\em et al} (2003), known as ABS equations. By considering the map which iterates the solution along…
We conjecture recurrence relations satisfied by the degrees of some linearizable lattice equations. This helps to prove linear growth of these equations. We then use these recurrences to search for lattice equations that have linear growth…
In this thesis we consider so-called linear evolutionary problems, a class of linear partial differential equations covering classical elliptic, parabolic and hyperbolic equations from mathematical physics as well as classes of…
The Learning with Errors (LWE) problem underlies modern lattice-based cryptography and is assumed to be quantum hard. Recent results show that estimating entanglement entropy is as hard as LWE, creating tension with quantum gravity and…
This work is an extension of the study into statistical mechanics of the early Universe that has been the subject in prior works of the author, the principal approach being the density matrix deformation. In the work it is demonstrated that…
We investigate the initial-value problem of the non-linear Liouville hierarchy. For the general form of the interaction potential we construct an explicit solution in terms of an expansion over particle clusters whose evolution is described…
In this work, we study the initial value problem associated with an abstract integrodifferential equation in interpolation scales. We prove local-in-time existence, uniqueness, continuation, and a blow-up alternative for regular mild…
We examine initial-boundary value problems for diffusion equations with distributed order time-fractional derivatives. We prove existence and uniqueness results for the weak solution to these systems, together with its continuous dependency…