The adiabatic variation of the soliton parameters, due to the presence of perturbation terms, is obtained. 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. Herman rl 2005 exploring the connection between quasistationary and squared eigenfunction expansion techniques in soliton 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. Pdf soliton perturbation theory for the gardner equation. Soliton perturbation theory for the generalized fifthorder kdv equation. Soliton perturbation theory for the gardner equation. Perturbation theory for bright spinor boseeinstein. Solitons as particles, oscillators, and in slowly changing. 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. The second time is the perturbation time t, corresponding to a signifi cant change of soliton amplitude.
Perturbation theory for the defocusing nonlinear schr. Nls with such a nonlinearity is known to possess a minimal mass soliton. Herman university of north carolina at wilmington, wilmington, nc abstract we revisit quasistationary perturbation theory for integrable systems and compare the solutions to those obtained through eigenfunction expansion methods. Pdf the adiabatic parameter dynamics of 1soliton solution of the gener alized fifthorder nonlinear equation is obtained by virtue of the. Siam journal on applied mathematics siam society for. 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. It is well known that these nonlinear evolution equations have a number conserved quantities. Details about the method can be found in almost any book on soliton theory, 1, 2, 4 here we outline the procedure. 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.
Conclusions in this paper, soliton perturbation theory is used to study the perturbed mke. The complete theory perturbation theory basic idea go beyond adiabatic approximation the complete soliton consists of. We study soliton solutions to the nonlinear schr odinger equation nls with a saturated nonlinearity. A method for approximating dark soliton solutions of the nonlinear. P7 if we could guess some good linear combinations and, then we can just use nondegenerate perturbation theory.
That is why we could solve separately for each order of. Soliton perturbation theory for the compound kdv equation. Soliton perturbation theories and applications nonlinear waves. Perturbation theory comprises mathematical methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. At the microscopic level, these two states are characterized by. 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.
Perturbation theory for propagating magnetic droplet solitons. Development and applications of soliton perturbation theory. F j 0, and the f i are independent, so the df i are linearly independent at each point 2m. 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. 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. Stochastic perturbation of parabolic law optical solitons. In order to describe soliton evolution under perturbations, a soliton perturbation theory is required. Soliton perturbation theory for the modified kawahara equation. In this paper, soliton perturbation theory is used to study the peturbed gardner equation that arises in the context of fluid dynamics. Soliton perturbation theory for the generalized fifth.
Institute for terrestrial magnetism, the ionosphere, and propagation of radiowaves, ussr. Perturbation theories for single solitons have been developed for many integrable equations. A critical feature of the technique is a middle step that breaks the problem into solvable and perturbation parts. Pdf soliton perturbation theory for the generalized fifthorder. Indeed, it is almost more a philosophy than a theory. Detailed evaluation of few integrals using residue theorem is given in appendix. Horikis perturbation theory for the defocusing nls 15 38. 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. A new formalism is developed to compute the continuum generated by these perturbations through the use of adjoint functions. This theory gives the ability to compute the adiabatic variation of the soliton energy and hence the adiabatic variation of the soliton amplitude. An exact expression for the continuum in the far field is obtained.
The adiabatic parameter dynamics of the solitons in presence of the perturbation terms are obtained. 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. This motivates a careful examination of the dynamics of. The formalism is based on using the riemannhilbert problem and provides the means to analytically calculate evolution of the soliton parameters. In the presence of perturbation terms, these integrals of. Soliton perturbation theory for the modified nonlinear. 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. Kawahara equation, perturbation, soliton ams codes. Pdf soliton perturbation theory for the compound kdv.
Modern versions of perturbation theory for the rh problem have been published in a series of papers 4447. We develop a perturbation theory for bright solitons of the f1 integrable spinor boseeinstein condensate bec model. The dynamics is studied for the case of polarizationpreserving fibers, while the types of pulses that are considered here are gaussian, supergaussian and supersech. In this article, we develop a singular perturbation theory for describing the long time cumulative effects of weak perturbations on solitons.
Perturbation theory algebraic equations ordinary di erential equations the nonlinear spring. Integrable equations can support soliton solutions which travel without change of shape. Modern versions of perturbation theory for the rh problem have been published in a series of papers 44,45,46,47. Yu h, yan j 2006 direct approach of perturbation theory for kink solitons. Soliton perturbation theory for the gardner equation mhikari. This leads to the long term behaviour of the soliton amplitude as lim t. Introduction the theory of nonlinear evolution equations is an ongoing topic of research for decades 1to10. This theory is used to establish the adiabatic parameter dynamics of the soliton energy.
Perturbation theories is in many cases the only theoretical technique that we have to handle various complex systems quantum and classical. We will assume the rst of these is the hamiltonian. In such cases, time dependence of wavefunction developed through. It may be the case, however, that a parameter can be identi. A system of odes for a perturbation of a minimal mass soliton. Perturbation theory is applicable if the problem at hand cannot be solved exactly, but can be formulated by.
The soliton perturbation theory is used to study the solitons that are governed by the compound korteweg devries equation in presence of perturbation terms. Timedependent perturbation theory so far, we have focused on quantum mechanics of systems described by hamiltonians that are timeindependent. In addition, energy radiation can be excited which can affect the solitons evolution in nontrivial ways. However the vast majority of systems in nature cannot be solved exactly, and we need.
Revisiting quasistationary perturbation theory for. We will focus mainly on kink solitons in one space dimension, on gauge theory vortices in two dimensions, and on skyrmions in three dimensions. Twodimensional harmonic oscilator 3 timedependent perturbation theory 4 literature igor luka cevi c perturbation theory. Osa continuum generation by perturbation of soliton. Marzuola, sarah raynor, and gideon simpson abstract. 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. Soliton perturbation theory 67 where a0 is the initial amplitude of the soliton.
Thus, from 3 and 4, the relation between the amplitude and the width of the soliton is given by. Soliton perturbation theory for the general modified. The klein gordon equation kge appears in theoretical physics. Soliton evolution in the presence of perturbation iopscience. Also, it is shown that the velocity undergoes a slow change due to these perturbation terms. Pdf soliton perturbation theory for the compound kdv equation. In this paper, there are a few such perturbed nonlinear wave equations that will be studied by the aid of soliton perturbation theory. The theory is applied to the description of soliton evolution in the presence of permanent perturbation. Lumped perturbations of a soliton, such as gain, filtering, and amplitude and phase modulation, are considered. Though solitons to the left and right of this minimum can readily be classi. 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. These include dropletdroplet and dropletboundary interactions, spatial magnetic.
With only the lightest octets of pseu doscalar mesons and spin12 baryons, pt is orderby. Soliton perturbation theory for nonlinear wave equations. The soliton perturbation theory is used to study the solitons that are governed by the modified nonlinear schrodingers equation. In particular, the nonlinear gain damping and the coefficient of finite conductivity are treated as perturbation terms. Perturbation theory is a very broad subject with applications in many areas of the physical sciences. This paper is going to study one of the classical nonlinear evolution equations that is. Nondegenerate time independent perturbation theory if the solution to an unperturbed system is known, including eigenstates.
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