By applying lous direct perturbation method to perturbed nonlinear schrodinger equation and the critical nonlinear schrodinger equation with a small dispersion, their approximate analytical. E, statistical physics, plasmas, fluids, and related interdisciplinary topics 616 pt b. Via the bifurcation theory and qualitative theory, equilibrium points for the twodimensional planar dynamic system are obtained. By a variational approach, we construct a solution u.
The sc hr o ding er w av e equati on macquarie university. Many exact traveling wave solutions, such as trigonometric, hyperbolic, rational, soliton and complex function solutions, are characterized. In quantum mechanics, a hamiltonian is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system this addition is the total energy of the system in most of the cases under analysis. These results were announced at the icm in zurich 50. The schrodinger equation is a linear partial differential equation that describes the wave function or state function of a quantummechanical system 12 it is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. When the diffusion is replaced by its bounded fourier truncation, li, mclaughlin, shatah, and wiggins 26 proved the existence of homoclinic orbits for the perturbed equation. However the vast majority of systems in nature cannot be solved exactly, and we need. Invariant manifolds and fibrations for perturbed nonlinear. By employing the sub equation method, we also construct its complexitons solutions.
Modeling relativistic soliton interactions in overdense. We distinguish between integrability in the painleve sense and integrability in the lax sense. We find optical solitons with high dispersion order for nonlinear perturbed schrodinger equations of the fourth, sixth, eighth, tenth and twelfth orders. The actual derivation, however, involves copious amounts of tedious algebra. The description of nature is essentially probabilistic, with the probability of an. In this paper we introduce a systematic perturbation theory for analyzing the effect of additional small terms on selffocusing, in which the perturbed critical nls is reduced to a simpler system of modulation equations that do not depend on the spatial variables transverse to the beam axis. Itisthusaveryinteresting questionhowtomodifythecubicnlsapproximationtocapture. Pdf downloads 30 html views 0 cited by 4 other articles by authors. New exact solutions of the perturbed nonlinear fractional. From the point of view of existence and differentiability of invariant manifolds, and their persistence under perturbation, neutrally stable modes in the linear approximation pose more analytical difficulties than exponentially growing and decaying modes. Ejde2018126 perturbed fractional schrodinger equation 3 a3 there exists a sequence t i i.
It is reasonable to suppose that the solutions for the perturbed problem will. Modified schrodinger equation, its analysis and experimental. Solutions of perturbed schrodinger equations with electromagnetic fields and critical nonlinearity volume 54 issue 1 sihua liang, jihui zhang. The perturbed nlse depict the quantic nonlinearity effects on promulgation of the ultrashort optical pulses in. A perturbed nonlinear schrodinger equation framework. Cautionary note in our derivation of the firstorder formula for the shift in energy in. Optical solitons, complexitons and power series solutions. Pdf chaos in a perturbed nonlinear schrodinger equation. Selffocusing in the perturbed and unperturbed nonlinear. Today, it provides the material for a large fraction of most introductory quantum mechanics courses. Potential vladimir georgiev and mirko tarulli abstract. It is shown that a nonlinear schroedinger soliton behaves stochastically with random phases in both time and space in the presence of small external oscillating fields and emits smallamplitude. Perturbed nonlinear fractional schrodinger equation.
Let us discuss the solution of the timeindependent schrodinger equation for the full. Let us discuss the solution of the timeindependent schrodinger equation for. Soliton interactions in perturbed nonlinear schroedinger. H0 is the hamiltonian of the unperturbed system, is a small. Its spectrum is the set of possible outcomes when one. Hence, this regularized laplacian acts in such a way that it. Bifurcation behaviour of the travelling wave solutions of the perturbed nonlinear schrodinger equation with kerr law nonlinearity. We consider some scale invariant generalizations of the smoothing estimates for the free schro. The physical structures corresponding to nonsingular complex wave solutions are found out bright soliton waves, stable bright periodic. On a singularly perturbed grosspitaevskii equation 5 the space x2 of functions which have locally the h2r regularity and satisfy at 0 the jump condition 1. Applying these estimates and using appropriate commutator. Traveling wave and exact solutions for the perturbed. Diffusivescaling of position moments anda centrallimit theoremareobtained for the mean position of a quantum particle hopping on a cubic lattice and subject to a ran.
Perturbations of the schroedinger equation by potentials with small. Hence, this regularized laplacian acts in such a way that it smooths the dissipation at short wavelengths. One can now substitute these expressions into the full 3d schrodinger equation and see that they solve it even at the points r where r 0. Therefore, the solution of the 3d schrodinger equation is obtained by multiplying the solutions of the three 1d schrodinger equations. This paper retrieves, bright, dark and singular optical solitons with the help of extended trial equation scheme.
Solitons and elliptic function solutions of higherorder. It is suggested that the same analysis can be applied to other evolution equations and for solving the perturbed. Siam journal on applied mathematics volume 60, issue 3. On the integrability properties of a perturbed nonlinear. Pdf in this paper, the modified trigonometric function series method is employed to solve the perturbed nonlinear schrodingers equation nlse with. Time dependent perturbation theory of weak field ionization.
Before we take the giant leap into wonders of quantum mechanics, we shall start with a brief. Pdf optical soliton for perturbed nonlinear fractional. Pdf new exact solutions to the perturbed nonlinear schrodingers. An analysis of the perturbed planewave solutions of the cubic schroedinger equation is presented. The physical structures corresponding to nonsingular complex wave solutions are found out bright soliton waves, stable bright periodic waves and. However, there is no soliton whose amplitude is stationary in time. Rigorous derivation of the grosspitaevskii equation.
Unperturbed and perturbed nonlinear schrodinger system for optical fiber solitons. Homoclinic degeneracy for perturbed nonlinear schrodinger equation. Solving the schrodinger equation timedependent schrodinger equation the wave function of a particle undergoing a force fx is the solution to the schrodinger equation. Stephen wiggins the nonlinear schroedinger nls equation is a fundamental nonlinear partial differential equation pde that arises in many areas and engineering, e. Consider the schrodinger equation for a single free particle. Apr 15, 2020 the main idea of the method is that we use expressions of the dependent variable and its derivatives in the differential equation the polynomial form of the solitary wave. This book is an outgrowth of our work over the past nine years concerning homoclinic chaos in the perturbed nonlinear schrodinger equation. When, the terms and are perturbatively damping terms. The hartreefock method may therefore be regarded as a first step toward the construction of atomic wave functions. Multiplicity of solutions for a perturbed fractional schrodinger equation involving oscillatory terms chao ji, fei fang communicated by binlin zhang abstract.
Soliton solutions of the perturbed resonant nonlinear schrodingers equation. Perturbation theory for the nonlinear schrodinger equation. Schrodinger equation nlse 1 with a random potential term on a lattice. Pdf spectral theory of the nonstationary schrodinger. By substituting this expansion into the schrodinger equation and collect. We consider kerr law, power law and log law of nonlinearity. The theorems in this book provide key building blocks for much of that work. In the instability region, we also specify where the solitary waves decay, collapse, develop moving fronts, or evolve into a stable spatially localized and.
We consider perturbations of the timedependent schroedinger equation in l2 rs by potentials having small support in an appropriate sense. Comparisons between sinegordon and perturbed nonlinear. It is reasonable to suppose that the solutions for the perturbed. The schrodinger equation of linear potential solution. Persistent homoclinic orbits for a perturbed nonlinear.
The schrodinger equation the previous the chapters were all about kinematics how classical and relativistic particles, as well as waves, move in free space. The 1d model of linear potential that is known as a quantum bouncer, which is define as. Derive an equation which, at least formally, determines the energy eigenvalues. Each soliton has a definite velocity and pulsates in time with a definite period. The modified auxiliary equation method to investigate.
Siam journal on applied mathematics siam society for. Solutions of perturbed schrodinger equations with electromagnetic fields and critical nonlinearity volume 54 issue 1 sihua liang, jihui zhang skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Spectral theory of the nonstationary schrodinger equation with a twodimensionally perturbed arbitrary onedimensional potential. A system is completely described by a wave function. Newtons laws, the schrodinger equation does not give the trajectory of a particle, but rather the wave function of the quantum system, which carries information about the wave nature of the particle, which allows us to only discuss the probability of nding the. Pdf unperturbed and perturbed nonlinear schrodinger system. Zaiyun zhang, xiangyang gan, and deming yu college of mathematics, hunan institute of science and technology, yueyang 414006, hunan, p. Write down the general solution to the time independent schrodinger equation in each of the above three regions.
Several integration schemes are used to construct solitons to the model. This is a short mathematical exercise to get to the root of what the equation is really describing, and how to go through the math step by step to derive information about the simplest of the atoms, hydrogen. Schrodinger equation, and a term h, the perturbation. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle. These images show the ground state of a perturbed double well. We consider perturbations of the timedependent schroedinger equation in l2rs by potentials having small support in an appropriate sense. Perturbation theory is an important tool for describing real quantum systems, as it turns out to be very difficult to find exact solutions to the schrodinger equation for hamiltonians of even moderate complexity. Physicad23920101120 14 1121 smoothsolutionspersistforever25. The rest of this article deals with this form of perturbed vector nonlinear schrodinger equations. Highly dispersive solitary wave solutions of perturbed. The time independent schrodinger equation looks like. Perturbation theory is extremely successful in dealing with those cases that.
Singularly perturbed vector and scalar nonlinear schr. The solution of the perturbed equation, with the perturbing term depending only on time, is found by direct integration. In order to do this, we s tart with an equation to describe a plane wave moving through. We find the solutions in terms of jacobi elliptic functions and in. Bifurcation behaviour of the travelling wave solutions of the. In this article we study the perturbed fractional schr odinger equation involving oscillatory terms. The sc hr o ding er w av e equati on so far, w e ha ve m ad e a lot of progr ess con cerni ng th e prop erties of, an d inte rpretation of th e w ave fu nction, bu t as yet w e h ave h ad very little to sa y ab out ho w the w ave fu nction ma y b e deriv ed in a general situ ation, th at is to say, w e d o not h ave on han d a ow ave. Soliton solutions of the perturbed resonant nonlinear. Finally, we can conclude that, new optical solutions of the spacetime fractional nonlinear perturbed schrodinger equations arising in nonlinear optical fibers with higherdispersion is obtained in this paper with the assist of atanganabaleanu conformable derivative. It is shown that a nonlinear schroedinger soliton behaves stochastically with random phases in both time and space in the presence of small external oscillating fields. It is usually denoted by, but also or to highlight its function as an operator.
Approximation methods for complex atomic systems the hydrogen atom wavefunctions and energies, we have seen, are determined as a combination of the various quantum dynamical analogues of. Timeindependent perturbation theory 86 the waveequation because it was constructed in analogy to maxwells equations for electromagnetic waves. A construction of the fundamental solution of the schr. Optical soliton solutions for a spacetime fractional. Our methods are based on infinite dimensional versions of invariant manifolds, geometric singular perturbation. The two forms of nonlinearity that are studied in detail are power law and dual power law, while. Singular complex wave solutions and nonsingular complex wave solutions are investigated for the nonlinear perturbed schrodinger equation with kerr law nonlinearity via the modified auxiliary equation method. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. An important result of our derivation is that the 1st order correction e1 n is linear in the. Some notes on understanding the schrodinger wave equation. It is shown that the control of the perturbative term allows one to decrease the velocity of soliton to zero and conserve a positive value of its amplitude. Without the details of derivation, we shall list a number of basic. Chapter four gives the proofs of the main results on persistence and smoothness of fibrations of invariant manifolds.
Optical soliton for perturbed nonlinear fractional. However, formatting rules can vary widely between applications and fields of interest or study. However, it had to come from somewhere, and it is indeed possible to derive the schr odinger equation using. The objective of this paper was to obtain soliton solutions for a perturbed nonlinear schrodinger equation with kerr law nonlinearity using the modified simple equation method and subordinary differential equation method. The perturbed nonlinear schrodinger equation springerlink.
Timeindependent perturbation theory cornell university. Soliton solutions of perturbed nonlinear schrodinger. The inversescattering method is used, and the asymptotic state is represented by a series of solitons, noting that no soliton exists whose amplitude is stationary in time. Numerous and frequentlyupdated resource results are available from this search. By considering the ansatz method, we obtain the bright and dark soliton solutions of the equation.
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