MULTIPLICITY OF POSITIVE SOLUTIONS FOR A CLASS OF NONLINEAR SCHRODINGER EQUATIONS

This paper proves the multiplicity of positive solutions for the following class of quasilinear problems: {-epsilon(p)Delta(p)u+(lambda A(x) + 1)vertical bar u vertical bar(p-2)u = f(u), R(N) u(x)>0 in R(N), where Delta(p) is the p-Laplacian operator, N > p >= 2, lambda and epsilon are positive parameters, A is a nonnegative continuous function and f is a continuous function with subcritical growth. Here, we use variational methods to get multiplicity of positive solutions involving the Lusternick-Schnirelman category of intA(-1)(0) for all sufficiently large lambda and small epsilon.

Effective nonlinear Schrodinger equations for cigar-shaped and disc-shaped Fermi superfluids at unitarity

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Radial sign-changing solutions to biharmonic nonlinear Schrodinger equations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Studies on some nonlinear schrodinger type equations describing pulse propagation through optical fibers

The discovery of the soliton is considered to be one of the most significant events of the twentieth century. The term soliton refers to special kinds of waves that can propagate undistorted over long distances and remain unaffected even after collision with each other. Solitons have been studied extensively in many fields of physics. In the context of optical fibers, solitons are not only of fundamental interest but also have potential applications in the field of optical fiber communications. This thesis is devoted to the theoretical study of soliton pulse propagation through single mode optical fibers.

Two-dimensional supersonic nonlinear Schrodinger flow past an extended obstacle

Supersonic flow of a superfluid past a slender impenetrable macroscopic obstacle is studied in the framework of the two-dimensional (2D) defocusing nonlinear Schroumldinger (NLS) equation. This problem is of fundamental importance as a dispersive analog of the corresponding classical gas-dynamics problem. Assuming the oncoming flow speed is sufficiently high, we asymptotically reduce the original boundary-value problem for a steady flow past a slender body to the one-dimensional dispersive piston problem described by the nonstationary NLS equation, in which the role of time is played by the stretched x coordinate and the piston motion curve is defined by the spatial body profile. Two steady oblique spatial dispersive shock waves (DSWs) spreading from the pointed ends of the body are generated in both half planes. These are described analytically by constructing appropriate exact solutions of the Whitham modulation equations for the front DSW and by using a generalized Bohr-Sommerfeld quantization rule for the oblique dark soliton fan in the rear DSW. We propose an extension of the traditional modulation description of DSWs to include the linear ""ship-wave"" pattern forming outside the nonlinear modulation region of the front DSW. Our analytic results are supported by direct 2D unsteady numerical simulations and are relevant to recent experiments on Bose-Einstein condensates freely expanding past obstacles.

Studies on integrability of some perturbed nonlinear evolution equations

Usually typical dynamical systems are non integrable. But few systems of practical interest are integrable. The soliton concept is a sophisticated mathematical construct based on the integrability of a class ol' nonlinear differential equations. An important feature in the clevelopment. of the theory of solitons and of complete integrability has been the interplay between mathematics and physics. Every integrable system has a lo11g list of special properties that hold for integrable equations and only for them. Actually there is no specific definition for integrability that is suitable for all cases. .There exist several integrable partial clillerential equations( pdes) which can be derived using physically meaningful asymptotic teclmiques from a very large class of pdes. It has been established that many 110nlinear wa.ve equations have solutions of the soliton type and the theory of solitons has found applications in many areas of science. Among these, well-known equations are Korteweg de-Vries(KdV), modified KclV, Nonlinear Schr6dinger(NLS), sine Gordon(SG) etc..These are completely integrable systems. Since a small change in the governing nonlinear prle may cause the destruction of the integrability of the system, it is interesting to study the effect of small perturbations in these equations. This is the motivation of the present work.

Asymptotic soliton train solutions of the defocusing nonlinear Schrodinger equation

Asymptotic behavior of initially large and smooth pulses is investigated at two typical stages of their evolution governed by the defocusing nonlinear Schrodinger equation. At first, wave breaking phenomenon is studied in the limit of small dispersion. A solution of the Whitham modulational equations is found for the case of dissipationless shock wave arising after the wave breaking point. Then, asymptotic soliton trains arising eventually from a large and smooth initial pulse are studied by means of a semiclassical method. The parameter varying along the soliton train is calculated from the generalized Bohr-Sommerfeld quantization rule, so that the distribution of eigenvalues depends on two functions-intensity rho(0)(x) of the initial pulse and its initial chirp v(0)(x). The influence of the initial chirp on the asymptotic state is investigated. Excellent agreement of the numerical solution of the defocusing NLS equation with predictions of the asymptotic theory is found.

Nonlinear Schrodinger equation for a superfluid Bose gas from weak coupling to unitarity: Study of vortices

We introduce a nonlinear Schrodinger equation to describe the dynamics of a superfluid Bose gas in the crossover from the weak-coupling regime, where an(1/3)<<1 with a the interatomic s-wave scattering length and n the bosonic density, to the unitarity limit, where a ->+infinity. We call this equation the unitarity Schrodinger equation (USE). The zero-temperature bulk equation of state of this USE is parametrized by the Lee-Yang-Huang low-density expansion and Jastrow calculations at unitarity. With the help of the USE we study the profiles of quantized vortices and vortex-core radius in a uniform Bose gas. We also consider quantized vortices in a Bose gas under cylindrically symmetric harmonic confinement and study their profile and chemical potential using the USE and compare the results with those obtained from the Gross-Pitaevskii-type equations valid in the weak-coupling limit. Finally, the USE is applied to calculate the breathing modes of the confined Bose gas as a function of the scattering length.

SOLITON SOLUTIONS TO SYSTEMS OF COUPLED SCHRODINGER EQUATIONS OF HAMILTONIAN TYPE

We study the existence of positive solutions of Hamiltonian-type systems of second-order elliptic PDE in the whole space. The systems depend on a small parameter and involve a potential having a global well structure. We use dual variational methods, a mountain-pass type approach and Fourier analysis to prove positive solutions exist for sufficiently small values of the parameter.

On positive solutions of nonlinear elliptic equations involving concave and critical nonlinearities

We study the existence of nonnegative solutions of elliptic equations involving concave and critical Sobolev nonlinearities. Applying various variational principles we obtain the existence of at least two nonnegative solutions.

Semidiscretization and long-time asymptotics of nonlinear diffusion equations

We review several results concerning the long time asymptotics of nonlinear diffusion models based on entropy and mass transport methods. Semidiscretization of these nonlinear diffusion models are proposed and their numerical properties analysed. We demonstrate the long time asymptotic results by numerical simulation and we discuss several open problems based on these numerical results. We show that for general nonlinear diffusion equations the long-time asymptotics can be characterized in terms of fixed points of certain maps which are contractions for the euclidean Wasserstein distance. In fact, we propose a new scaling for which we can prove that this family of fixed points converges to the Barenblatt solution for perturbations of homogeneous nonlinearities for values close to zero.

A mixed finite element method for nonlinear diffusion equations

We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model.

Phenomenological approach to nonlinear Langevin equations

In this paper we address the problem of consistently constructing Langevin equations to describe fluctuations in nonlinear systems. Detailed balance severely restricts the choice of the random force, but we prove that this property, together with the macroscopic knowledge of the system, is not enough to determine all the properties of the random force. If the cause of the fluctuations is weakly coupled to the fluctuating variable, then the statistical properties of the random force can be completely specified. For variables odd under time reversal, microscopic reversibility and weak coupling impose symmetry relations on the variable-dependent Onsager coefficients. We then analyze the fluctuations in two cases: Brownian motion in position space and an asymmetric diode, for which the analysis based in the master equation approach is known. We find that, to the order of validity of the Langevin equation proposed here, the phenomenological theory is in agreement with the results predicted by more microscopic models

Group-invariant solutions of semilinear Schrodinger equations in multi-dimensions

Symmetry group methods are applied to obtain all explicit group-invariant radial solutions to a class of semilinear Schr¨odinger equations in dimensions n = 1. Both focusing and defocusing cases of a power nonlinearity are considered, including the special case of the pseudo-conformal power p = 4/n relevant for critical dynamics. The methods involve, first, reduction of the Schr¨odinger equations to group-invariant semilinear complex 2nd order ordinary differential equations (ODEs) with respect to an optimal set of one-dimensional point symmetry groups, and second, use of inherited symmetries, hidden symmetries, and conditional symmetries to solve each ODE by quadratures. Through Noether’s theorem, all conservation laws arising from these point symmetry groups are listed. Some group-invariant solutions are found to exist for values of n other than just positive integers, and in such cases an alternative two-dimensional form of the Schr¨odinger equations involving an extra modulation term with a parameter m = 2−n = 0 is discussed.

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.