Nonlinear Systems and Their Remarkable Mathematical Structures, Volume 1 aims to
describe the recent progress in nonlinear differential equations and nonlinear dynamical …

We utilize the invariant energy quadratization approach to transform the nonlinear
Hamiltonian ODE into an equivalent form which has a quadratic energy. Then the …

It is well known that the Allen-Cahn equation satisfies a nonlinear stability property, ie, the
free-energy functional decreases in time. Linearly implicit integrators have been developed …

A unified framework of invariant-conserving explicit Runge-Kutta schemes for the nonlinear
Hamiltonian ODEs and PDEs are proposed by utilizing the invariant energy quadratization …

There has been considerable recent work on the development of energy conserving one-
step methods that are not symplectic. Here we extend these ideas to stochastic Hamiltonian …

We present several novel examples of integrable quadratic vector fields for which Kahanʼs
discretization method preserves integrability. Our examples include generalized Suslov and …

In this paper, we present two different approaches for constructing reduced‐order models
(ROMs) for the two‐dimensional shallow water equation (SWE). The first one is based on the …

Kahan's method and a two-step generalisation of the discrete gradient method are both
linearly implicit methods that can preserve a modified energy for Hamiltonian systems with a …

The main result of this paper is the discretization of second-order Hamiltonian systems of the
form $\ddot x=-K\nabla W (x) $, where K is a constant symmetric matrix and …

We propose a geometric construction of three-dimensional birational maps that preserve two
pencils of quadrics. The maps act as compositions of involutions, which, in turn, act along …