Related papers: Linear superposition in the general heavenly equat…

The theory of superposition rules for solutions of a Lie system of first-order differential equations is extended to deal with analogous systems of second-order and the theory is illustrated with the very rich example of Ermakov-like…

Based on the analysis of biquaternion quadratic forms of field, it is shown that Maxwell equations arise as a consequence of the principle of conservation of the energy-momentum flow of field in space-time. It turns out that this principle…

The superposition principle is a very basic ingredient of quantum theory. What may come as a surprise to many students, and even to many practitioners of the quantum craft, is tha superposition has limitations imposed by certain…

The goal of this article is to discuss a recent conjecture of the two authors, which aims to describe the long time behavior of solutions to one-dimensional dispersive equations with cubic and higher nonlinearities. These problems arguably…

Clarifying the nature of the quantum state $|\Psi\rangle$ is at the root of the problems with insight into counter-intuitive quantum postulates. We provide a direct-and math-axiom free-empirical derivation of this object as an element of a…

The mathematical properties of a nonlinear parabolic equation arising in the modelling of non-newtonian flows are investigated. The peculiarity of this equation is that it may degenerate into a hyperbolic equation (in fact a linear…

We revisit and present new linear spaces of explicit solutions to incompressible Euler and Navier-Stokes equations on $\mathbb{R}^n$, as well as the rotating Boussinesq equations on $\mathbb{R}^3$. We cast these solutions are superpositions…

Black holes and gravitational waves are consequences of the nonlinear character of the Einstein equations. Yet, the remarkable properties of General Relativity point to the existence of other effects. Here we uncover new nonlinear facets of…

Superposition rules form a class of functions that describe general solutions of systems of first-order ordinary differential equations in terms of generic families of particular solutions and certain constants. In this work we extend this…

Linear superposition of gravitational fields is shown to be possible for a large class of spacetimes, in some specific coordinates. Explicit examples are presented.

We prove that if a linear equation, whose coefficients are continuous rational functions on a nonsingular real algebraic surface, has a continuous solution, then it also has a continuous rational solution. This is known to fail in higher…

I consider an extension of General Relativity by an auxiliary non-dynamical dimension that enables our space-time to acquire an extrinsic curvature. Obtained gravitational equations, without or with a cosmological constant, have a…

In this paper, we show the so-called "combined effect" of two different kinds of nonlinear terms for semilinear wave equations in one space dimension. Such a special phenomenon appears only in the case that the total integral of the initial…

This article complements recent results of the papers [J. Math. Phys. 41 (2000), 480; 45 (2004), 336] on the symmetry classification of second-order ordinary difference equations and meshes, as well as the Lagrangian formalism and…

Many physical systems can be described by nonlinear eigenvalues and bifurcation problems with a linear part that is non-selfadjoint e.g. due to the presence of loss and gain. The balance of these effects is reflected in an antilinear…

Nonlinear perturbation of Fuchsian systems are studied in a region including two singularities. It is proved that such systems are generally not analytically equivalent to their linear part (they are not linearizable) and the obstructions…

We review some recent results on nonlinear Schrodinger equations with potential, with emphasis on the case where the potential is a second order polynomial, for which the interaction between the linear dynamics caused by the potential, and…

We consider a new variant of cosmological perturbation theory that has been designed specifically to include non-linear density contrasts on scales 100 Mpc, while still allowing for linear fluctuations on larger scales. This theory is used…

We consider the forced surface quasi-geostrophic equation with supercritical dissipation. We show that linear instability for steady state solutions leads to their nonlinear instability. When the dissipation is given by a fractional…

The work presents two examples of simple mathematical formulas which are natural nonlinear modifications (one being a generalization) of Gielis' formula. These formulas involve a comparable number of parameters and provide non-Platonic…