Related papers: Collective Coordinate Methods and Their Applicabil…
We use a one-scale similarity analysis which gives specific relations between the velocity, amplitude and width of localized solutions of nonlinear differential equations, whose exact solutions are generally difficult to obtain. We also…
We present applications of variational -- wavelet approach to three different models of nonlinear beam motions with underlying collective behaviour: Vlasov-Maxwell-Poisson systems, envelope dynamics, beam-beam model. We have the…
This paper focuses on coordinate update methods, which are useful for solving problems involving large or high-dimensional datasets. They decompose a problem into simple subproblems, where each updates one, or a small block of, variables…
In order to solve a system of nonlinear rate equations one can try to use some soliton methods. The procedure involves three steps: (1) Find a `Lax representation' where all the kinetic variables are combined into a single matrix $\rho$,…
We investigate the interaction of solitons with an external periodic field within the framework of the modified Korteweg-de Vries (mKdV) equation. In the case of small perturbation a simple dynamical system is used to describe the soliton…
Birds in a flock move in a correlated way, resulting in large polarization of velocities. A good understanding of this collective behavior exists for linear motion of the flock. Yet observing actual birds, the center of mass of the group…
We investigate bright solitons in the one-dimensional Schr\"odinger equation in the framework of an extended variational approach. We apply the latter to the stationary ground state of the system as well as to coherent collisions between…
We study the transformation of the statistical mechanics of N particles to the statistical mechanics of fields, that are the collective coordinates, describing the system. We give an explicit expression for the functional Fourier transform…
We use the Fokker-Planck equation and its moment equations to study the collective behavior of interacting particles in unsteady one-dimensional flows. Particles interact according to a long-range attractive and a short-range repulsive…
This paper is devoted to the study of the asymptotic behavior of solutions to multi-order fractional cooperative systems. First, we demonstrate the boundedness of solutions to fractional-order systems under certain conditions imposed on the…
We use a class of trial wave functions which are generalizations of gaussians to study single soliton approximate analytic solutions to the KdV equations. The variational parameters obey a Hamiltonian dynamics obtained from the Principle of…
In this report, the various 1D single soliton and multi-soliton solutions of the Sine-Gordon equation are explored. First the topological kink solitons and their properties for the Sine-Gordon, as well as the $\phi^{4}$ model are…
We develop a new framework for the study of complex continuous time dynamical systems based on viewing them as collections of interacting control modules. This framework is inspired by and builds upon the groupoid formalism of Golubitsky,…
The Casimir energy for the classically stable configurations of the topological solitons in 2D quantum antiferromagnets is studied by performing the path-integral over quantum fluctuations. The magnon fluctuation around the solitons…
The study of molecular polaritons beyond simple quantum emitter ensemble models (e.g., Tavis-Cummings) is challenging due to the large dimensionality of these systems (the number of molecular emitters is $N\approx 10^{6}-10^{10}$) and the…
In this work, families of kinks are analytically identified in multifield theories with either polynomial or deformed sine-Gordon-type potentials. The underlying procedure not only allows us to obtain analytical solutions for these models,…
The aim of this work is to ascertain the characterization of the existence of coexistence states for a class of cooperative systems supported by the study of an associated non--local equation through classical variational methods. Thanks to…
Interactions between solitary waves have been pivotal to understanding nonlinear phenomena across various disciplines. The dynamics of rarefaction solitary waves holds great potential, yet their fundamental characteristics and interactions…
The collective dynamics of solitons with a coset space G/H as moduli space is studied. It is shown that the collective band for a vibrational state is given by the inequivalent coset space quantization corresponding to the representation of…
Matching dynamical systems, through different forms of conjugacies and equivalences, has long been a fundamental concept, and a powerful tool, in the study and classification of nonlinear dynamic behavior (e.g. through normal forms). In…