Band gaps and lattice solitons for the higherorder nonlinear. Recreational mathematics, mathematics, differential and integral equations, dynamical systems and control theory. The equation for rcan be simpli ed in form by substituting ur rrr. The defocusing energycritical nonlinear schrodinger equation in dimensions. These dark solitons have been used to carry information for a great distance 3. Pdf the initial value problem for some defocusing coupled nonlinear schrodinger equations is investigated.
The defocusing nonlinear schrodinger equation society. With this consideration at n 2, we in this paper construct two new types of exponentialandrational mixed soliton solutions for the defocusing nonlocal nls equation. In fact, this particular case will cover most of the problems that well encounter in ee 439. Supercritical defocusing schrodinger equations igor rodnianski we will discuss recent work with f. The defocusing daveystewartson ii equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its onedimensional reduction, the defocusing nonlinear schrodinger equation, namely the. Defocusing nonlinear schrodinger equations request pdf. If ux,t ux, then the schroedinger equation becomes. This is now referred to as the radial wave equation, and would be identical to the onedimensional schr odinger equation were it not for the term r 2 added to v, which pushes the particle away.
The propagation of wave energy in a scattering medium is described phenomenologically by radiative transport theory 8 as follows. The timeindependent schroedinger equation a very important special case of the schroedinger equation is the situation when the potential energy term does not depend on time. Perturbation theory for the defocusing nonlinear schrodinger equation. Recall that we did not derive the tise, we simple constructed a differential equation that is consistent with the freeparticle wave function. In this paper we study timesplitting spectral approximations for the linear schr. In theoretical physics, the onedimensional nonlinear schrodinger equation nlse is a nonlinear variation of the schrodinger equation. The defocusing energycritical nonlinear schrodinger equation. We consider the nonlinear schr odinger equation with a logarithmic nonlinearity, whose sign is such that no nontrivial stationary solution exists. Cambridge core abstract analysis defocusing nonlinear schrodinger equations by.
By exactly the same arguments as in the defocusing case involving writing di erential equations in. Wigner distribution converges to the solution of a radiative transport equation. Soliton, rational, and periodic solutions for the in. Elliptic solutions of the defocusing nls equation are stable. 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 particle in di erent regions of space. In this regime, the equation propagates oscillations with a wavelength of o. The ist can be thought of as a nonlinear generalization of the fourier transform on r. Initialvalue problems for some nonlinear wave equations can be treated similarly thanks to the inverse scattering transform. This regime is described by the defocusing nonlinear schrodinger equation dnlse. The defocusing nonlinear schrodinger equation is a broad study of nonlinear excitations in self defocusing nonlinear media. Mixed soliton solutions of the defocusing nonlocal nonlinear. Wavevortex interactions in the nonlinear schrodinger. The kdv equation provides a purely dispersive regularization of the hopf equation, and its solutions.
Szeftel, where we studied the problem of global regularity for a defocusing supercritical schrodinger equation. Global wellposedness of defocusing critical nonlinear schrodinger equation in the. Global wellposedness and scattering for the defocusing. Schrodinger nls equations numerically by implementing the inverse scattering transform. Soliton solutions of cubicquintic nonlinear schrodinger. Universal dynamics for the defocusing logarithmic schrodinger equation remi carles and isabelle gallagher abstract. Schrodinger equation with timedependent potential and to show that the associated average. Pdf least action nodal solutions for a quasilinear. In this paper, we exploit the integrability of the nls equation to establish the spectral stability of all such stationary solutions, this time by explicitly. Miller and zhenyun qin initialboundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of socalled uni. The defocusing energycritical nonlinear schrodinger. Apr 21, 2010 we consider the cubic defocusing nonlinear schrodinger equation on the two dimensional torus. We study the defocusing nonlinear schr\odinger equation in three space dimensions. Pdf global wellposedness of defocusing critical nonlinear.
The corresponding problem had been settled in the a rmative in a long series of works in. Bourgain, refinements of strichartz inequality and applications to 2dnls with critical nonlinearity, international mathematical research notices, 5 1998, 253. The defocusing nonlinear schrodinger equation is a broad study of nonlinear excitations in selfdefocusing nonlinear media. This implies there are no smooth soliton solutions with spatial decay for the defocusing nls equation. Perturbation theory for the defocusing nonlinear schr. Apr 06, 2020 the schrodinger equation also known as schrodingers wave equation is a partial differential equation that describes the dynamics of quantum mechanical systems via the wave function. We consider the periodic defocusing cubic nonlinear schr odinger nls equation 1. As proposed in the introduction and appendix a, the solution to the wave function for a free particle possessing mass is. Boundary value problems for the defocusing nonlinear. Bourgain, scattering in the energy space and below in 3d nls, journal danalyse mathematique, 4 1998, 267.
Nonlinear schrodinger equations, morawetz estimates, scattering. The schroedinger equation can not be derived from classical mechanics. Defocusing nonlinear schrodinger equations by benjamin dodson. The behavior of solutions of the finitegenus whitham equations for the weak dispersion limit of the defocusing nonlinear schrodinger equation is investigated analytically and numerically for piecewiseconstant initial data. Wavevortex interactions in the nonlinear schrodinger equation. Vortex solutions of the defocusing discrete nonlinear. For nonrelativistic quantum physics the basic equation to be solved is the schr odinger equation. For the defocusing nonlinear schrodinger equation 1 on the halfline.
Inverse scattering transform for the focusing nonlinear. Explicit computations show that in the case of gaussian initial data. For the onecomponent system, many studies have been carried out 4 which demonstrate that the corresponding scalar nls equation admits bright solitons 4, a breather 65 and rogue wave, 7, 8 in the focusing case, and dark solitons 9 in the defocusing case. In this paper, we will obtain a rigorous derivation of the defocusing cubic nonlinear schr odinger equation on the threedimensional torus t3 from the manybody limit of interacting bosonic. Suppose that pw,z and qw,z are polynomials in cw,z. In particular, the dynamics of constantamplitude initial conditions with one or more frequency jumps i. Pdf the radial defocusing nonlinear schrodinger equation. Pdf the defocusing nonlinear schrodinger equation with periodic. On the whitham equations for the defocusing nonlinear.
Consider the semilinear schrodinger equation nls in arbitrary dimensions. Pdf on defocusing coupled nonlinear schrodinger equations. The sc hr o ding er w av e equati on macquarie university. It is the success of this equation in describing the experimentally ob served quantum mechanical phenomena correctly, that justi. The inverse scattering transform for the defocusing. The trajectory, the positioning, and the energy of these systems can be retrieved by solving the schrodinger equation. It will turn out that it is still possible for the other cases p 3 to have the same. Jun 29, 2006 the behavior of solutions of the finitegenus whitham equations for the weak dispersion limit of the defocusing nonlinear schrodinger equation is investigated analytically and numerically for piecewiseconstant initial data. We exhibit smooth solutions for which the support of the conserved energy moves to higher fourier modes.
The argument is inspired by the recent work of dodson global wellposedness and scattering for the defocusing, l2critical, nonlinear schrodinger equation when d 3. A rigorous derivation of the defocusing cubic nonlinear schrodinger equation on t3 from the dynamics of manybody quantum systems vedran sohinger abstract. Onthespectrumofthediracoperatorandtheexistence of discrete. But classical mechanics can be rederived from the schroedinger equation in some limit. An exact solution to equation 3 is then obtained by solving this equation. Band gaps and lattice solitons for the higherorder. It summarizes stateoftheart knowledge on the defocusing. Numerical inverse scattering for the focusing and defocusing. For each of the focusing and defocusing nls equations there exists an inverse scattering transform ist 30, 32. In this short note, we present a new proof of the global wellposedness and scattering result for the defocusing energycritical nonlinear schrodinger equation nls in four space dimensions. Darboux transformation and multidark soliton for ncomponent.
The defocusing nonlinear schrodinger equation society for. 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. Horikis university of ioannina in collaboration with. Chapter 4 schroedinger equation mit opencourseware. Introduction in this paper, we show that the initial value problem ivp for the nonlinear schr. Vortex solutions of the defocusing discrete nonlinear schrodinger equation j. The same result without radial condition was obtained by miao, xu and zhao 17, for n 9.
On timesplitting spectral approximations for the schrodinger. Consider the defocusing nonlinear schrodinger equation. We prove that any radial solution that remains bounded in the critical sobolev space must be global and scatter. Transfer of energy to high frequencies in the cubic defocusing nonlinear schrodinger equation. We reveal that the first type of solution can display a large variety of elastic interactions, in which there are in general two exponential solitons and two rational solitons. Pdf we consider solutions of the defocusing nonlinear schr\odinger nls equation on the halfline whose dirichlet and neumann boundary.
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