Related papers: Explicit symplectic approximation of nonseparable …
Symplectic integrators are the established standard for long-term simulations of nearly-integrable Hamiltonian systems due to their preservation of geometric structures. However, they suffer from an inherent limitation: secular phase-shift…
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…
The non-Hermitian models, which are symmetric under parity (P) and time-reversal (T) operators, are the cornerstone for the fabrication of new ultra-sensitive optoelectronic devices. However, providing the gain in such systems usually…
An explicit high-order noncanonical symplectic algorithm for ideal two-fluid systems is developed. The fluid is discretized as particles in the Lagrangian description, while the electromagnetic fields and internal energy are treated as…
Explicit and semi-explicit geometric integration schemes for dissipative perturbations of Hamiltonian systems are analyzed. The dissipation is characterized by a small parameter $\epsilon$, and the schemes under study preserve the…
Discrete Hamiltonian variational integrators are derived from Type II and Type III generating functions for symplectic maps, and in this paper we establish a variational error analysis result that relates the order of accuracy of the…
Symplectic mappings are discrete-time analogs of Hamiltonian systems. They appear in many areas of physics, including, for example, accelerators, plasma, and fluids. Integrable mappings, a subclass of symplectic mappings, are equivalent to…
The polysymplectic analysis of the Short Pulse Equation known in nonlinear optics is used in order to construct a geometric polysymplectic integrator for it. The proposed scheme turns out to be much more effective than other standard…
Symplectic N-body integrators are widely used to study problems in celestial mechanics. The most popular algorithms are of 2nd and 4th order, requiring 2 and 6 substeps per timestep, respectively. The number of substeps increases rapidly…
On this paper, we have proposed an approach to observe the time-centered difference scheme for dissipative mechanical systems from a Hamiltonian perspective and to introduce the idea of symplectic algorithm to dissipative systems. The…
We prove that a class of A-stable symplectic Runge--Kutta time semidiscretizations (including the Gauss--Legendre methods) applied to a class of semilinear Hamiltonian PDEs which are well-posed on spaces of analytic functions with analytic…
The aim of this paper is to introduce a class of Hamiltonian autonomous systems in dimension 4 which are completely integrable and their dynamics is described in all details. They have an equilibrium point which is stable for some rare…
This paper investigates a class of non-autonomous highly oscillatory ordinary differential equations characterized by a linear component inversely proportional to a small parameter $\varepsilon$, with purely imaginary eigenvalues, and an…
Using a Newtonian model of the Solar System with all 8 planets, we perform extensive tests on various symplectic integrators of high orders, searching for the best splitting scheme for long term studies in the Solar System. These…
The effort to generate matrix exponentials and associated differentials, required to determine the time evolution of quantum systems, frequently constrains the evaluation of problems in quantum control theory, variational circuit…
Symplectic integrators are the tool of choice for many researchers studying dynamical systems because of their good long-term energy conservation properties. For systems with a dominant central mass, symplectic integrators are also highly…
We prove symplectic non-squeezing (in the sense of Gromov) for the cubic nonlinear Schr\"odinger equation on $\R^2$. This is the first symplectic non-squeezing result for a Hamiltonian PDE in infinite volume. As the underlying symplectic…
Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. Variational integrators are an important class of geometric integrators. The general idea…
Geometric numerical integration has recently been exploited to design symplectic accelerated optimization algorithms by simulating the Lagrangian and Hamiltonian systems from the variational framework introduced in Wibisono et al. In this…
A variational formulation of accelerated optimization on normed spaces was recently introduced by considering a specific family of time-dependent Bregman Lagrangian and Hamiltonian systems whose corresponding trajectories converge to the…