In this article, we develop a singular perturbation theory for describing the long time cumulative effects of weak perturbations on solitons. Osa continuum generation by perturbation of soliton. Marzuola, sarah raynor, and gideon simpson abstract. Stochastic perturbation of parabolic law optical solitons. The theory is applied to the description of soliton evolution in the presence of permanent perturbation. In order to describe soliton evolution under perturbations, a soliton perturbation theory is required. In this paper, there are a few such perturbed nonlinear wave equations that will be studied by the aid of soliton perturbation theory. Modern versions of perturbation theory for the rh problem have been published in a series of papers 4447.
This motivates a careful examination of the dynamics of. P7 if we could guess some good linear combinations and, then we can just use nondegenerate perturbation theory. This leads to the long term behaviour of the soliton amplitude as lim t. With only the lightest octets of pseu doscalar mesons and spin12 baryons, pt is orderby. The klein gordon equation kge appears in theoretical physics. The direct perturbation theory is another form of soliton perturbation theory, which develops on the basis of the perturbation solution expansion into the square eigenfunction of the linearized soliton equation. Because of the complexity of many physical problems, very few can be solved exactly unless they involve only small hilbert spaces. The formalism is based on using the riemannhilbert problem and provides the means to analytically calculate evolution of the soliton parameters. The adiabatic parameter dynamics of the solitons in presence of the perturbation terms are obtained.
Soliton perturbation theory for the compound kdv equation. In addition, energy radiation can be excited which can affect the solitons evolution in nontrivial ways. Pdf soliton perturbation theory for the generalized. The direct perturbation theory is another form of soliton perturbation theory, which develops on the basis of the perturbation solution expansion into the square eigenfunction of the linearized soliton. The soliton perturbation theory is used to study the adiabatic parameter dynamics of solitons due to the generalized fifthorder kdv equation in presence of perturbation terms. We begin our treatment of free energy differences by examining the problem of transforming a system from one thermodynamic state to another. Introduction to regular perturbation theory very often, a mathematical problem cannot be solved exactly or, if the exact solution is available, it exhibits such an intricate dependency in the parameters that it is hard to use as such. Nls with such a nonlinearity is known to possess a minimal mass soliton. A critical feature of the technique is a middle step that breaks the problem into solvable and perturbation parts.
The complete theory perturbation theory basic idea go beyond adiabatic approximation the complete soliton consists of. Siam journal on applied mathematics siam society for. Soliton perturbation theory for nonlinear wave equations. Soliton perturbation theory for the gardner equation mhikari. Soliton perturbation theory for the generalized fifthorder kdv equation.
Pt 911 is a low momentum effective field theory for qcd written as an expansion in small momenta and quark masses, and it has become an invaluable tool for subatomic physics. Integrable equations can support soliton solutions which travel without change of shape. Lumped perturbations of a soliton, such as gain, filtering, and amplitude and phase modulation, are considered. Soliton perturbation theory for the general modified. Soliton perturbation theory 67 where a0 is the initial amplitude of the soliton. Horikis perturbation theory for the defocusing nls 15 38. Soliton perturbation theory for the gardner equation article pdf available in advanced studies in theoretical physics 2 january 2008 with 105 reads how we measure reads. F j 0, and the f i are independent, so the df i are linearly independent at each point 2m. The adiabatic parameter dynamics of solitons due to gardners equa tion is obtained, in presence of the perturbation terms, by the aid of soliton perturbation. This theory is used to establish the adiabatic parameter dynamics of the soliton energy. Herman rl 2005 exploring the connection between quasistationary and squared eigenfunction expansion techniques in soliton perturbation theory. Also, it is shown that the velocity undergoes a slow change due to these perturbation terms. Revisiting quasistationary perturbation theory for.
The dynamics is studied for the case of polarizationpreserving fibers, while the types of pulses that are considered here are gaussian, supergaussian and supersech. Pdf soliton perturbation theory for the gardner equation. Soliton perturbation theory for the generalized fifth. We will focus mainly on kink solitons in one space dimension, on gauge theory vortices in two dimensions, and on skyrmions in three dimensions.
Soliton perturbation theory for the modified kawahara equation. However the vast majority of systems in nature cannot be solved exactly, and we need. Perturbation theory is applicable if the problem at hand cannot be solved exactly, but can be formulated by. A system of odes for a perturbation of a minimal mass soliton. Timedependent perturbation theory so far, we have focused on quantum mechanics of systems described by hamiltonians that are timeindependent. Perturbation theory is a very broad subject with applications in many areas of the physical sciences. A perturbation theory for soliton systems sciencedirect. Soliton perturbation theories and applications nonlinear waves. In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one.
Pdf soliton perturbation theory for the generalized fifthorder. We develop a perturbation theory for bright solitons of the f1 integrable spinor boseeinstein condensate bec model. The second time is the perturbation time t, corresponding to a signifi cant change of soliton amplitude. Thus, from 3 and 4, the relation between the amplitude and the width of the soliton is given by.
We study soliton solutions to the nonlinear schr odinger equation nls with a saturated nonlinearity. Soliton perturbation theory for nonlinear wave equations some of these equations are the kortewegde vries kdv equation, boussinesq equation, regularized long wave equation, gardners equation, kaupkupershmidt equation, kadomtsevpetviashvili kp. This theory gives the ability to compute the adiabatic variation of the soliton energy and hence the adiabatic variation of the soliton amplitude. In the presence of perturbation terms, these integrals of. Twodimensional harmonic oscilator 3 timedependent perturbation theory 4 literature igor luka cevi c perturbation theory. As each of the f i is a conserved quantity, the motion of the system is con ned to a submanifold of phase space determined by the initial. Nondegenerate time independent perturbation theory if the solution to an unperturbed system is known, including eigenstates. A method for approximating dark soliton solutions of the nonlinear. That is why we could solve separately for each order of. The soliton perturbation theory is used to study the solitons that are governed by the compound korteweg devries equation in presence of perturbation terms. The adiabatic variation of the soliton parameters, due to the presence of perturbation terms, is obtained.
Development and applications of soliton perturbation theory. In such cases, time dependence of wavefunction developed through. An exact expression for the continuum in the far field is obtained. Detailed evaluation of few integrals using residue theorem is given in appendix. Pdf soliton perturbation theory for the compound kdv. Pdf soliton perturbation theory for the compound kdv equation. Though solitons to the left and right of this minimum can readily be classi. Conclusions in this paper, soliton perturbation theory is used to study the perturbed mke. Perturbation theory comprises mathematical methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. Soliton evolution in the presence of perturbation iopscience. Perturbation theory for bright spinor boseeinstein. It is well known that these nonlinear evolution equations have a number conserved quantities. It may be the case, however, that a parameter can be identi.
Perturbation theory algebraic equations ordinary di erential equations the nonlinear spring. In this paper, soliton perturbation theory is used to study the peturbed gardner equation that arises in the context of fluid dynamics. In all cases, the solitons behave in a similar fashion to either relativistic or newtonian particles or nonlinear oscillators under the influence of external forces. Indeed, it is almost more a philosophy than a theory. Solovev institute for terrestrial magnetism, ionosphere and radiowave propagation izmiran moscow region, 142092, ussr a simplified perturbational approach appropriate for systems of solitons governed by the perturbed integrable equations is described. It is shown that small perturbation leads to three main effects. This paper is going to study one of the classical nonlinear evolution equations that is.
Pdf the adiabatic parameter dynamics of 1soliton solution of the gener alized fifthorder nonlinear equation is obtained by virtue of the. The idea is to start with a simple system for which a mathematical solution is known, and add an additional perturbing hamiltonian representing a weak disturbance to the system. The soliton perturbation theory is used to study the solitons that are governed by the modified nonlinear schrodingers equation. We will assume the rst of these is the hamiltonian. At the microscopic level, these two states are characterized by. Fundamental result of degenerate perturbation theory.
Perturbation theory and celestial mechanics in this last chapter we shall sketch some aspects of perturbation theory and describe a few of its applications to celestial mechanics. Yu h, yan j 2006 direct approach of perturbation theory for kink solitons. Soliton perturbation theory for the modified nonlinear. A new formalism is developed to compute the continuum generated by these perturbations through the use of adjoint functions. Solitons as particles, oscillators, and in slowly changing. Horikis perturbation theory for the defocusing nls 15. Perturbation theory as applied to solitons which decay at infinity, i. Perturbation theory for propagating magnetic droplet solitons. Details about the method can be found in almost any book on soliton theory, 1, 2, 4 here we outline the procedure. In particular, the nonlinear gain damping and the coefficient of finite conductivity are treated as perturbation terms. Institute for terrestrial magnetism, the ionosphere, and propagation of radiowaves, ussr.
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