Related papers: Conservation Laws in Higher-Order Nonlinear Optica…
Conservation laws related to the gauge invariance of Lagrangians and Euler-Lagrange operators in finite and infinite order Lagrangian formalisms are analyzed.
The solution of time dependent differential equations with neural networks has attracted a lot of attention recently. The central idea is to learn the laws that govern the evolution of the solution from data, which might be polluted with…
We develop an approach describing nonlinear-optical processes in the strong-field domain characterized by the nonperturbative field-with-matter interaction. The polarization of an isolated atom in the external field calculated via the…
We propose a nonlinear modification of the Schr\"{o}dinger equation that possesses the main properties of this equation such as the Galilean invariance, the weak separability of composite systems, and the homogeneity in the wave function.…
Conservation laws are discussed in conjunction with quantum-mechanical indeterminacies of the corresponding observables. The considered examples show that the connections between energy and its indeterminacy may be quite intricate. The…
We investigate the following inhomogeneous nonlinear Schr\"odinger equation in the radial regime, featuring a focusing energy-critical nonlinearity and a defocusing perturbation: $$ i\partial_t u +\Delta u =|x|^{-a} |u|^{p-2} u - |x|^{-b}…
We propose a class of numerical methods for the nonlinear Schr\"odinger (NLS) equation that conserves mass and energy, is of arbitrarily high-order accuracy in space and time, and requires only the solution of a scalar algebraic equation…
We investigate the global well-posedness and modified scattering for the one-dimensional Schr\"odinger equation with gauge-invariant polynomial nonlinearity. For small localized initial data of finite energy in a low-regularity class, we…
It is shown that the Kadanoff-Baym equations at consistent first-order gradient approximation reveal exact rather than approximate conservation laws related to global symmetries of the system. The conserved currents and energy-momentum…
The higher-order dispersive and nonlinear effects (alias {\it the perturbation terms}) like the third-order dispersion, the self-steepening, and the self-frequency shift play important roles in the study of the ultra-short optical pulse…
Conserved quantities increasingly underpin the inference of physical models. Recently new conserved quantities have been found in this context, that currently lack an interpretation. Here, we show that irreversible reactions in CRNs and…
We indicate that the nonlinear Schr\"odinger equation with white noise dispersion possesses stochastic symplectic and multi-symplectic structures. Based on these structures, we propose the stochastic symplectic and multi-symplectic methods,…
The discrete nonlinear Schr\"odinger equation on \(\Z^d\), \(d \geq 1\) is an example of a dispersive nonlinear wave system. Being a Hamiltonian system that conserves also the \(\ell^2(\Z^d)\)-norm, the well-posedness of the corresponding…
Noether's theorem connects symmetries to invariants in continuous systems, however its extension to discrete systems has remained elusive. Recognizing the lowest-order finite difference as the foundation of local continuity, a viable method…
We introduce a method to construct conservation laws for a large class of linear partial differential equations. In contrast to the classical result of Noether, the conserved currents are generated by any symmetry of the operator, including…
Reduced order models of nonlinear conservation laws in fluid dynamics do not typically inherit stability properties of the full order model. We introduce projection-based hyper-reduced models of nonlinear conservation laws which are…
Oblique propagation of magnetohydrodynamic waves in warm plasmas is described by a modified vector derivative nonlinear Schroedinger equation, if charge separation in Poisson's equation and the displacement current in Ampere's law are…
The reaction diffusion equation arises in physical situations in problems from population growth, genetics and physical sciences. We consider the generalised Fisher equation in cylindrical coordinates from Lie theory stand point. An…
English version of abstract: The dynamic optimization problems treated by the calculus of variations are usually solved with the help of the 2nd order Euler-Lagrange differential equations. These equations are, generally speaking,…
We examine the validity of the principle of mass conservation for solutions of some typical equations in the theory of nonlinear diffusion, including equations in standard differential form and also their fractional counterparts. In Part 1,…