In this paper, a new approach is applied to study the self-similar solutions of 2×2 systems of nonlinear hyperbolic conservation laws. A notion of characteristic directions is introduced and then used to construct local and smooth solutions of the associated Riemann problem

By using a generalization of the Lie–Poisson brackets for dual Maxwell and Born–Infeld field strength tensors, we construct gauge invariant axial-vector conserved currents Heisenberg–Euler nonlinear electrodynamics in 4-dimensional Minkowski space-time. The infinite hierarchies given by Lie generally covariant vector axial are established. These upon action gravitational fields, but conservatio...

Abstract. Bridges and Reich suggested the idea of multi-symplectic spectral discretization on Fourier space [4]. Based on their theory, we investigate the multi-symplectic Fourier pseudospectral discretization of the nonlinear Schrödinger equation (NLS) on real space. We show that the multi-symplectic semi-discretization of the nonlinear Schrödinger equation with periodic boundary conditions ha...

Recent results on the local and global properties of multisymplectic discretizations of Hamiltonian PDEs are discussed. We consider multisymplectic (MS) schemes based on Fourier spectral approximations and show that, in addition to a MS conservation law, conservation laws related to linear symmetries of the PDE are preserved exactly. We compare spectral integrators (MS vs. non-symplectic) for t...

For one-dimensional kinetic BGK models, regarded as relaxation models for scalar conservation laws with genuinely nonlinear fluxes, existence of small amplitude travelling waves is proven. Dynamic stability of these kinetic shock profiles is shown by extending a classical energy method for viscous regularizations of conservation laws.

We consider the generalization of scalar subcell resolution schemes to systems of viscous conservation laws. For this purpose we use a weakly nonlinear geometrical optics approximation for parabolic perturbations of hyperbolic conservation laws and the Roe-type field by field decomposition. Computations of the reactive Navier-Stokes equations are presented as an application.