Search results for "Schrödinger equation"
showing 10 items of 196 documents
Numerical study of the long wavelength limit of the Toda lattice
2014
We present the first detailed numerical study of the Toda equations in $2+1$ dimensions in the limit of long wavelengths, both for the hyperbolic and elliptic case. We first study the formal dispersionless limit of the Toda equations and solve initial value problems for the resulting system up to the point of gradient catastrophe. It is shown that the break-up of the solution in the hyperbolic case is similar to the shock formation in the Hopf equation, a $1+1$ dimensional singularity. In the elliptic case, it is found that the break-up is given by a cusp as for the semiclassical system of the focusing nonlinear Schr\"odinger equation in $1+1$ dimensions. The full Toda system is then studie…
Deformations of third-order Peregrine breather solutions of the nonlinear Schrödinger equation with four parameters
2013
We present a new representation of solutions of the one-dimensional nonlinear focusing Schr\"odinger equation (NLS) as a quotient of two determinants. This formulation gives in the case of the order 3, new solutions with four parameters. This gives a very efficient procedure to construct families of quasirational solutions of the NLS equation and to describe the apparition of multirogue waves. With this method, we construct analytical expressions of four-parameters solutions; when all these parameters are equal to 0, we recover the Peregrine breather of order 3. It makes possible with this four-parameters representation, to generate all the types of patterns for the solutions, like the tria…
Efficient numerical method for simulating static and dynamic properties of superfluid helium
2004
Density functional theory (DFT) offers computationally affordable way of describing static and dynamic properties of superfluid 4He. In general, the DFT models yield single particle-like Schrodinger equations with a nonlinear potential term that accounts for all the many-body interactions. The resulting equations can be solved for small amplitude plane wave excitations in the bulk whereas fully numerical solution must be sought in more complicated cases. In this paper we propose a numerical method that can be used in solving the time-dependent nonlinear Schrodinger equation in both real and imaginary times. The method is based on operator splitting technique where each component operator is…
Solution of time-independent Schrödinger equation by the imaginary time propagation method
2007
Numerical solution of eigenvalues and eigenvectors of large matrices originating from discretization of linear and non-linear Schrodinger equations using the imaginary time propagation (ITP) method is described. Convergence properties and accuracy of 2nd and 4th order operator-splitting methods for the ITP method are studied using numerical examples. The natural convergence of the method is further accelerated with a new dynamic time step adjustment method. The results show that the ITP method has better scaling with respect to matrix size as compared to the implicitly restarted Lanczos method. An efficient parallel implementation of the ITP method for shared memory computers is also demons…
Generation and characterization of 06-THz polarization domain-wall trains in an ultralow-birefringence spun fiber
2007
Polarization domain-wall (PDW) trains have been generated at a repetition rate of 0.6 THz in an ultralow-birefringence spun optical fiber and measured by use of an adapted frequency-resolved optical gating technique. Characterization of the intensity and the phase of the PDW train shows complete switching between adjacent domains of counterrotating circular polarizations and directly confirms predictions based on numerical simulations of the incoherently coupled nonlinear Schrödinger equations.
Impact of optical and structural aging in As_2S_3 microstructured optical fibers on mid-infrared supercontinuum generation
2014
We analyze optical and structural aging in As2S3 microstructured optical fibers (MOFs) that may have an impact on mid-infrared supercontinuum generation. A strong alteration of optical transparency at the fundamental OH absorption peak is measured for high-purity As2S3 MOF stored in atmospheric conditions. The surface evolution and inherent deviation of corresponding chemical composition confirm that the optical and chemical properties of MOFs degrade upon exposure to ambient conditions because of counteractive surface process. This phenomenon substantially reduces the optical quality of the MOFs and therefore restrains the spectral expansion of generated supercontinuum. This aging process …
Condensation of classical optical waves beyond the cubic nonlinear Schrodinger equation
2012
International audience; A completely classical nonlinear wave is known to exhibit a process of condensation whose thermodynamic properties are analogous to those of the genuine Bose-Einstein condensation. So far this phenomenon of wave condensation has been studied essentially in the framework of the nonlinear Schrodinger (NLS) equation with a pure cubic Kerr nonlinearity. We study wave condensation by considering two representative generalizations of the NLS equation that are relevant to the context of nonlinear optics, the nonlocal nonlinearity and the saturable nonlinearity. For both cases we derive analytical expressions of the condensate fraction in the weakly and the strongly nonlinea…
On universality of critical behavior in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the Tritronquée solution to the…
2008
We argue that the critical behavior near the point of “gradient catastrophe” of the solution to the Cauchy problem for the focusing nonlinear Schrodinger equation \(i\epsilon \varPsi _{t}+\frac{\epsilon^{2}}{2}\varPsi _{xx}+|\varPsi |^{2}\varPsi =0\) , e ≪1, with analytic initial data of the form \(\varPsi (x,0;\epsilon)=A(x)e^{\frac{i}{\epsilon}S(x)}\) is approximately described by a particular solution to the Painleve-I equation.
Purification of Lindblad dynamics, geometry of mixed states and geometric phases
2015
We propose a nonlinear Schr\"odinger equation in a Hilbert space enlarged with an ancilla such that the partial trace of its solution obeys to the Lindblad equation of an open quantum system. The dynamics involved by this nonlinear Schr\"odinger equation constitutes then a purification of the Lindbladian dynamics. This nonlinear equation is compared with other Schr\"odinger like equations appearing in the theory of open systems. We study the (non adiabatic) geometric phases involved by this purification and show that our theory unifies several definitions of geometric phases for open systems which have been previously proposed. We study the geometry involved by this purification and show th…
Theory of Nuclear Quantum Dynamics Simulations
2016
In Chap. 2, we have seen that the theoretical study of a molecular system is, in a vast majority of cases, separated in two steps. In a first step, the electronic structure of the system is studied by solving the electronic Schrodinger equation with fixed nuclei. This approach, combined with geometry optimization techniques, allows one to locate the important features of the various potential energy surfaces (PESs) of the electronic states of interest. In the context of photochemistry, as seen in Chap. 3, this approach allows one to characterize the various decay pathways of the molecule after photoexcitation. This information can then be used to interpret the various decay time constants o…