The method of multiple scales for nonlinear kleingordon. 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. The description of nature is essentially probabilistic, with the probability of an. Suppose for example that our initial wave form is a highly peaked gaussian.
An introduction to semilinear evolution equations, oxford lecture series in mathematics and its applications, oxford university press, oxford, 1998. Therefore, the solution of the 3d schrodinger equation is obtained by multiplying the solutions of the three 1d schrodinger equations. Several anomalies are highlighted and resolutions proposed. Schrodinger equation with a cubic nonlinearity eqworld. A copublication of the ams and the courant institute. An introduction to nonlinear schrodinger equations thierry cazenave. We study local well posedness in classical sobolev space hsof the associated initial value problem and periodic boundary value problem. In principle, the methods presented apply to a large class of dispersive semilinear equations. 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. The schrodinger map equation is also related to the ishimori equation see for some recent results on this equation in one dimension local well posedness is known for smooth data by the general theory of derivative nonlinear schrodinger equations, however this is not yet established in higher dimensions. Semilinear schrodinger equations courant lecture notes in mathematics uk ed. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus.
From the mathematical point of view, schrodingers equation is a delicate problem, possessing a mixture of the properties of parabolic and elliptic equations. Derivation of the nonlinear schrodinger equation from first principles theodore bodurov eugene, oregon, usa, email. Schrodinger schrodinger equation with a cubic nonlinearity. Semilinear schrodinger equations pdf free download epdf. This paper presents the derivation of the schrodinger, kleingordon and dirac equations of particle physics, for free particles, using classical methods. The energy method 1 problems for lecture 1 10 lecture 2.
Derivation of the nonlinear schrodinger equation from. A brief introduction to classical, statistical, and quantum mechanics. Consider the cauchy problem for semilinear schrodinger equations in. The schrodinger equation the previous the chapters were all about kinematics how classical and relativistic particles, as well as waves, move in free space. We will indicate possible extensions and references. Semilinear schrodinger equations courant lecture notes in mathematics by cazenave, thierry and a great selection of related books, art and collectibles available now at. Semilinear schrodinger equations courant lecture notes in. Mathematical and computational methods for semiclassical schrodinger equations shi jin department of mathematics, university of wisconsin, madison, wi 53706. Solutions of semilinear schrodinger equations in hs numdam. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. Semilinear schrodinger equations thierry cazenave download. A system is completely described by a wave function.
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. This monograph contains all the material taht was treated in a course of lectures given by dr. Low regularity exponentialtype integrators for semilinear schr. An introduction to semilinear evolution equations revised edition thierry cazenave cnrsand university of paris vi, france and alain haraux cnrs and university of paris vi, france translated by yvan martel university of cergypontoise, france clarendon press oxford 1998. It should go without saying that this work is a major contribution to mathematics. We consider the cubic nonlinear schrodinger equation on the spatial domain r.
The main questions we investigate are longtime behavior. Modified scattering for the cubic schrodinger equation on product. Szeftel, almost global existence for hamiltonian semilinear. In particular, this means that semilinear equations are ones in which the coe. To our knowledge, there are no previous results of this type. The simple harmonic oscillator recall our rule for setting up the quantum mechanical problem. It studies both problems of local nature local existence of solutions, uniqueness, regularity, smoothing effect and problems of global nature finitetime blowup, global existence, asymptotic behavior of solutions. The cauchy problem for the quasilinear schrodinger. It is also a good model dispersive equation, since it is often technically simpler than other dispersive equations, such as the wave or kortewegde vries equation. Numerous and frequentlyupdated resource results are available from this search. Buy semilinear schrodinger equations courant lecture notes in mathematics on free shipping on qualified orders. Picture below is from the 56th page of lyapunov stability of ground states of nonlinear dispersive evolution equations. The schrodingers schrodingers equation is the basic equation of quantum mechanics. We will show that the schrodinger and dirac equations are spacetime versions of the averaged energy relations.
American mathematical society 201 charles street providence, rhode island 0290422 4014554000 or 8003214267 ams, american mathematical society, the tricolored ams logo, and advancing research, creating connections, are trademarks and services marks of the american mathematical society and registered in the u. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Part i 3 and then, by superposition, the solution to our wave equation will be ux. Analytical and numerical results for some classes of. Analytical and numerical results for some classes of nonlinear schr odinger equations xiao liu doctor of philosophy graduate department of mathematics university of toronto 20 this thesis is devoted to the study of nonlinear dispersive partial di erential equations of schr odinger type. The cauchy problem for the quasilinear schrodinger equation following kenigponcevega 1 lecture 1. Then in the case of the linear advection equation all the fourier modes travel together at the same speed and. Find all the books, read about the author, and more. For the sake of simplicity, we will mostly consider two model equations. It is also a good model dispersive equation, since it is often technically simpler than other dispersive equations, such as the wave or the kortewegde vries equation. This paper is concerned with the initial value problem ivp associated.
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