Related papers: Multisymplectic variational integrators and space/…
One-dimensional scattering mediated by non-Hermitian Hamiltonians is studied. A schematic set of models is used which simulate two point interactions at a variable strength and distance. The feasibility of the exact construction of the…
Recent successes in image generation, model-based reinforcement learning, and text-to-image generation have demonstrated the empirical advantages of discrete latent representations, although the reasons behind their benefits remain unclear.…
In contact Hamiltonian systems, the so-called dissipated quantities are akin to conserved quantities in classical Hamiltonian systems. In this paper, we prove a Noether's theorem for non-autonomous contact Hamiltonian systems,…
Recently, Lobb and Nijhoff initiated the study of variational (Lagrangian) structure of discrete integrable systems from the perspective of multi-dimensional consistency. In the present work, we follow this line of research and develop a…
The Lagrange problem is established in the discrete field theory subject to constraints with values in a Lie group. For the admissible sections that satisfy a certain regularity condition, we prove that the critical sections of such…
Symplectic integrators that preserve the geometric structure of Hamiltonian flows and do not exhibit secular growth in energy errors are suitable for the long-term integration of N-body Hamiltonian systems in the solar system. However, the…
A broad class of nonlinear acoustic wave models possess a Hamiltonian structure in their dissipation-free limit and a gradient flow structure for their dissipative dynamics. This structure may be exploited to design numerical methods which…
In order to perform numerical studies of long-term stability in nonlinear Hamiltonian systems, one needs a numerical integration algorithm which is symplectic. Further, this algorithm should be fast and accurate. In this paper, we propose…
The standard techniques of variational calculus are geometrically stated in the ambient of fiber bundles endowed with a (pre)multisymplectic structure. Then, for the corresponding variational equations, conserved quantities (or, what is…
This paper deals with the application of probabilistic time integration methods to semi-explicit partial differential-algebraic equations of parabolic type and its semi-discrete counterparts, namely semi-explicit differential-algebraic…
We reveal the symplectic nature of parameter-drift maps by embedding them into extended phase space. Applying the embedding to the parameter-drift standard nontwist map, our construction yields an autonomous symplectic map in extended phase…
Differential flatness serves as a powerful tool for controlling continuous time nonlinear systems in problems such as motion planning and trajectory tracking. A similar notion, called difference flatness, exists for discrete-time systems.…
Long-term stability studies of nonlinear Hamiltonian systems require symplectic integration algorithms which are both fast and accurate. In this paper, we study a symplectic integration method wherein the symplectic map representing the…
The first part of this paper develops a geometric setting for differential-difference equations that resolves an open question about the extent to which continuous symmetries can depend on discrete independent variables. For general…
Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for…
We consider the geometric numerical integration of Hamiltonian systems subject to both equality and "hard" inequality constraints. As in the standard geometric integration setting, we target long-term structure preservation. We…
Evolutionary partial differential equations play a crucial role in many areas of science and engineering. Spatial discretization of these equations leads to a system of ordinary differential equations which can then be solved by numerical…
The success of symplectic integrators for Hamiltonian ODEs has led to a decades-long program of research seeking analogously structure-preserving numerical methods for Hamiltonian PDEs. In this paper, we construct a large class of such…
We present the symplectic algorithm in the Lagrangian formalism for the Hamiltonian systems by virtue of the noncommutative differential calculus with respect to the discrete time and the Euler--Lagrange cohomological concepts. We also show…
We develop a geometric framework for the numerical integration of mechanical systems evolving on manifolds. After briefly reviewing classical numerical methods and highlighting their limitations and shortcomings in non-flat (non-Euclidean)…