Related papers: Solitons, Surfaces, Curves, and the Spin Descripti…
In this report, fundamental educational concepts of linear and non-linear equations and solutions of nonlinear equations from the book High-Temperature Superconductivity: The Nonlinear Mechanism and Tunneling Measurements (Kluwer Academic…
In the spirit of the many recent simple models of evolution inspired by statistical physics, we put forward a simple model of the evolution of such models. Like its objects of study, it is (one supposes) in principle testable and capable of…
We provide a detailed description of solutions of Curve Shortening in $\R^n$ that are invariant under some one-parameter symmetry group of the equation, paying particular attention to geometric properties of the curves, and the asymptotic…
The properties of the evolution equation have been analyzed. The uniqueness and the existence of solution for the evolution equation with special value of parameter characterizing intensity of change of external conditions, of the…
We present a collection of results on the evolution by curvature of networks of planar curves. We discuss in particular the existence of a solution and the analysis of singularities.
Separate spin evolution quantum hydrodynamics is generalized to include the Coulomb exchange interaction. The Coulomb exchange interaction is considered as the interaction between the spin-down electrons being in the quantum states occupied…
In recent years, many papers have been devoted to the regularity of doubly nonlinear singular evolution equations. Many of the proofs are unnecessarily complicated, rely on superfluous assumptions or follow an inappropriate approximation…
An analytical formula is obtained to describe the evolution of the average populations of spin components of spin-1 atomic gases. The formula is derived from the exact time-dependent solution of the Hamiltonian $H_{S}=c mathbf{S}^{2}$…
In this article, we review the different models and methods to understand and estimate correlated evolution within the same genome, individual or species. We describe correlated evolution among traits, among genetic components and finally…
We prove short-time existence of smooth solutions for a class of nonlinear, and in general spatially nonlocal, Hamiltonian evolution equations that describe the self-interaction of weakly nonlinear scale-invariant waves. These equations…
We present the explicit formulae, describing the structure of symmetries and formal symmetries of any scalar (1+1)-dimensional evolution equation. Using these results, the formulae for the leading terms of commutators of two symmetries and…
How to effectively solve the eigen solutions of the nonlinear spinor field equation coupling with some other interaction fields is important to understand the behavior of the elementary particles. In this paper, we derive a simplified form…
A brief review of recent research on soliton and black hole solutions of Einstein's equations with nonlinear field sources is presented and some open questions are pointed out.
In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions…
In this chapter we briefly review the basics of the theory of stellar evolution.
We show that the mean curvature flow for a closed and rotationally symmetric surface can be formulated as an evolution problem consisting of an evolution equation for the square of the function whose graph is rotated and two ODEs describing…
We consider several ways of how one could classify the various types of soliton solutions related to nonlinear evolution equations which are solvable by the inverse scattering method. In doing so we make use of the fundamental analytic…
Two essential methods, the symmetry analysis and of the singularity analysis, for the study of the integrability of nonlinear ordinary differential equations are discussed. The main similarities and differences of these two different…
Motivated by recent proposals (Bialynicki-Birula, Mycielski; Haag, Bannier; Weinberg; Doebner, Goldin) for nonlinear quantum mechanical evolution equations for pure states some principal difficulties in the framework of usual quantum…
In this paper, we solve the eigen solutions to some nonlinear spinor equations, and compute several functions reflecting their characteristics. The numerical results show that, the nonlinear spinor equation has only finite meaningful eigen…