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. 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. 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. Transfer of energy to high frequencies in the cubic defocusing nonlinear schrodinger equation. If ux,t ux, then the schroedinger equation becomes. Pdf the initial value problem for some defocusing coupled nonlinear schrodinger equations is investigated. It will turn out that it is still possible for the other cases p 3 to have the same lower bound. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to boseeinstein condensates confined to highly anisotropic cigarshaped traps, in the meanfield regime. Pdf the defocusing nonlinear schrodinger equation with periodic. 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. These dark solitons have been used to carry information for a great distance 3. Horikis university of ioannina in collaboration with. On timesplitting spectral approximations for the schrodinger.
Bourgain, scattering in the energy space and below in 3d nls, journal danalyse mathematique, 4 1998, 267. Onthespectrumofthediracoperatorandtheexistence of discrete. 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. Wigner distribution converges to the solution of a radiative transport equation. It will turn out that it is still possible for the other cases p 3 to have the same. We study the defocusing nonlinear schr\odinger equation in three space dimensions. Defocusing nonlinear schrodinger equations request pdf. Pdf we consider solutions of the defocusing nonlinear schr\odinger nls equation on the halfline whose dirichlet and neumann boundary. It summarizes stateoftheart knowledge on the defocusing. Pdf least action nodal solutions for a quasilinear. The defocusing nonlinear schrodinger equation is a broad study of nonlinear excitations in selfdefocusing nonlinear media. Soliton solutions of cubicquintic nonlinear schrodinger. The sc hr o ding er w av e equati on macquarie university. This behavior is quantified by the growth of higher sobolev norms.
The inverse scattering transform for the defocusing. Here the commutative diagram for the inversescattering transform yields a procedure less explicit, perhaps for. Apr 21, 2010 we consider the cubic defocusing nonlinear schrodinger equation on the two dimensional torus. Global wellposedness of defocusing critical nonlinear schrodinger equation in the. Inverse scattering transform for the focusing nonlinear. The defocusing energycritical nonlinear schrodinger equation. In theoretical physics, the onedimensional nonlinear schrodinger equation nlse is a nonlinear variation of the schrodinger equation. It is the success of this equation in describing the experimentally ob served quantum mechanical phenomena correctly, that justi. The argument is inspired by the recent work of dodson global wellposedness and scattering for the defocusing, l2critical, nonlinear schrodinger equation when d 3.
Suppose that pw,z and qw,z are polynomials in cw,z. Darboux transformation and multidark soliton for ncomponent. An exact solution to equation 3 is then obtained by solving this equation. Bourgain, refinements of strichartz inequality and applications to 2dnls with critical nonlinearity, international mathematical research notices, 5 1998, 253. Pdf the radial defocusing nonlinear schrodinger equation. Boundary value problems for the defocusing nonlinear. Universal dynamics for the defocusing logarithmic schrodinger equation remi carles and isabelle gallagher abstract. But classical mechanics can be rederived from the schroedinger equation in some limit.
Recreational mathematics, mathematics, differential and integral equations, dynamical systems and control theory. For each of the focusing and defocusing nls equations there exists an inverse scattering transform ist 30, 32. By exactly the same arguments as in the defocusing case involving writing di erential equations in. The defocusing nonlinear schrodinger equation is a broad study of nonlinear excitations in self defocusing nonlinear media. This regime is described by the defocusing nonlinear schrodinger equation dnlse. We consider the periodic defocusing cubic nonlinear schr odinger nls equation 1. Wavevortex interactions in the nonlinear schrodinger. It summarizes stateoftheart knowledge on the defocusing nonlinear schrodingertype models in a single volume and contains a wealth of resources, including over 800 references to relevant articles and monographs and a. On the whitham equations for the defocusing nonlinear.
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. Szeftel, where we studied the problem of global regularity for a defocusing supercritical schrodinger equation. The propagation of wave energy in a scattering medium is described phenomenologically by radiative transport theory 8 as follows. In this paper we study timesplitting spectral approximations for the linear schr. The trajectory, the positioning, and the energy of these systems can be retrieved by solving the schrodinger equation. We exhibit smooth solutions for which the support of the conserved energy moves to higher fourier modes. 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. Schrodinger equation reading french and taylor, chapter 3 quantum mechanics sets probabilities outline wave equations from. Band gaps and lattice solitons for the higherorder nonlinear. Supercritical defocusing schrodinger equations igor rodnianski we will discuss recent work with f. The equation for rcan be simpli ed in form by substituting ur rrr. The same result without radial condition was obtained by miao, xu and zhao 17, for n 9. 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.
For the defocusing nonlinear schrodinger equation 1 on the halfline. For nonrelativistic quantum physics the basic equation to be solved is the schr odinger equation. The defocusing nonlinear schrodinger equation society. 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. 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. The corresponding problem had been settled in the a rmative in a long series of works in. Transfer of energy to high frequencies in the cubic. 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. The kdv equation provides a purely dispersive regularization of the hopf equation, and its solutions. Elliptic solutions of the defocusing nls equation are stable. In fact, this particular case will cover most of the problems that well encounter in ee 439. The schrodinger equation the previous the chapters were all about kinematics how classical and relativistic particles, as well as waves, move in free space.
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. Pdf global wellposedness of defocusing critical nonlinear. 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. Wavevortex interactions in the nonlinear schrodinger equation.
Consider the defocusing nonlinear schrodinger equation. Schrodinger equation with timedependent potential and to show that the associated average. Band gaps and lattice solitons for the higherorder. Introduction in this paper, we show that the initial value problem ivp for the nonlinear schr. Schrodinger nls equations numerically by implementing the inverse scattering transform. The ist can be thought of as a nonlinear generalization of the fourier transform on r. Perturbation theory for the defocusing nonlinear schr.
The schroedinger equation can not be derived from classical mechanics. Cambridge core abstract analysis defocusing nonlinear schrodinger equations by. Recall that we did not derive the tise, we simple constructed a differential equation that is consistent with the freeparticle wave function. Numerical inverse scattering for the focusing and defocusing. Vortex solutions of the defocusing discrete nonlinear. Nonlinear schrodinger equations, morawetz estimates, scattering. Pdf transfer of energy to high frequencies in the cubic.
Pdf on defocusing coupled nonlinear schrodinger equations. The defocusing energycritical nonlinear schrodinger equation in dimensions. This implies there are no smooth soliton solutions with spatial decay for the defocusing nls equation. Initialvalue problems for some nonlinear wave equations can be treated similarly thanks to the inverse scattering transform. Chapter 4 schroedinger equation mit opencourseware. In this regime, the equation propagates oscillations with a wavelength of o. We consider the nonlinear schr odinger equation with a logarithmic nonlinearity, whose sign is such that no nontrivial stationary solution exists. The defocusing nonlinear schrodinger equation society for. In particular, the dynamics of constantamplitude initial conditions with one or more frequency jumps i. Defocusing nonlinear schrodinger equations by benjamin dodson.
Explicit computations show that in the case of gaussian initial data. Vortex solutions of the defocusing discrete nonlinear schrodinger equation j. 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. 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. Perturbation theory for the defocusing nonlinear schrodinger equation. Mixed soliton solutions of the defocusing nonlocal nonlinear. As proposed in the introduction and appendix a, the solution to the wave function for a free particle possessing mass is. Consider the semilinear schrodinger equation nls in arbitrary dimensions. Soliton, rational, and periodic solutions for the in.
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