Existence of ground states for a one-dimensional relativistic schrödinger equation

Relativistic Schrödinger equation with a nonlinear potential interaction describes the dynamics of a particle, with rest mass m, travelling to a significant fraction |v| < 1 of the light speed c = 1. At first, we deal with the local and global existence of solutions of the flux, and in the second term, and according to the relativistic nature of the problem, we look for boosted solitons as ψ(x, t) = eiμtφv(x - vt), where the profile φ v ∈ H 1/2 (R{double-struck}) is a minimizer of a suitable variational problem. Our proof uses a concentration-compactness-type argument. Stability results for the boosted solitons are established.Fil: Borgna, Juan Pablo. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Rial, Diego Fernando. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

On the existence of nematic-superconducting states in the Ginzburg-Landau regime

In this article, we investigate the existence of nematic-superconducting states in the Ginzburg-Landau regime, both analytically and numerically. From the configurations considered, a slab and a cylinder with a circular cross-section, we demonstrate the existence of geometrical thresholds for the obtention of non-zero nematic order parameters. Within the frame of this constraint, the numerical calculations on the slab reveal that the competition or collaboration between nematicity and superconductivity is a complex energy minimization problem, requiring the accommodation of the Ginzburg-Landau parameters of the decoupled individual systems, the sign of the bi-quadratic potential energy relating both order parameters and the magnitude of the applied magnetic field. Specifically, the numerical results show the existence of a parameter regime for which it is possible to find mixed nematic-superconducting states. These regimes depend strongly on both the applied magnetic field and the potential coupling parameter. Since the proposed model corresponds to the weak coupling regime, and since it is a condition on these parameters, we design a test to decide whether this condition is fulfilled.Comment: 18 pages, 8 figures, the complete final version will be published in Chaos, Solitons and Fractals (Elsevier) vol 179, February 202

Molecular response for nematic superconducting media in a hollow cylinder: A numerical approach

In the context of the Ginzburg Landau formalism proposed by Barci et al. in 2016 for nematic superconductivity, and by performing a numerical treatment based on the Shooting method, we analyze the behaviour of the radial distribution of the nematic order parameter when the superconducting order parameter reaches the typical non trivial saddle point. We consider the cases of a hollow cylindrical domain, with a disk or an annular domain as its cross section, whether the order parameter is subjected to Newmann or Dirichlet boundary conditions. We conclude that depending on the corresponding situation a non trivial solution holds if certain relations between the radii are satisfied. Moreover, we observe a saturation effect on each instances that constitutes a purely geometrical consequence on the relation between the typical sizes and shapes of the samples. These conclusions can be useful for further experimental realizations and extensions to the interaction of the nematic (superconducting) order parameters with electromagnetic fields.Fil: de Leo, Mariano Fernando. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; ArgentinaFil: Ovalle, Diego García. Centre National de la Recherche Scientifique; FranciaFil: Borgna, Juan Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Ciencias Físicas. - Universidad Nacional de San Martín. Instituto de Ciencias Físicas; Argentin

Orbital stability of solitons in hamiltonian equations

Esta tesis está dedicada al estudio de la estabilidad orbital de un par de ecuaciones diferenciales que tienen en común el hecho de poseer una estructura hamiltoniana, ellas son la ecuación de Schrödinger y la de Klein-Gordon, ambas con término no lineal del tipo potencial y dato de borde periodico. En primer lugar probamos que para este tipo de problema existe solución fundamental, con su correspondiente perfil solitón. Luego centramos el estudio de la estabilidad del flujo de la ecuación entorno de esta solución fundamental, para lo que construimos una conveniente función de Lyapunov usando las cantidades conservadas por el flujo de las ecuación. Para el caso de la ecuación de Schrödinger probamos la estabilidad orbital de las soluciones, bajo algunas condiciones sobre los parámetros de la ecuación. Los resultados alcanzados son comparables con los ya conocidos para esta misma ecuación con dominio espacial no acotado. Debido al comportamiento estable de las soluciones que inician cercanas al perfil solit´on, tambi´en introdujimos un m´etodo num´erico para la simulaci´on de la din´amica, para esto realizamos el estudio completo de la existencia y unicidad de los solitones discretos y del comportamiento estable de las soluciones discretas. Esto mismo nos permitió probar la convergencia del método. Por último, los resultados previos conocidos determinaban la inestabilidad de las soluciones de la ecuación de Klein-Gordon con dominio espacial no acotado, por lo tanto no era esperable obtener algo distinto en nuestro caso. La aplicación del método aquí dado, si bien no nos da la estabilidad, nos permite caracterizar la dirección responsable de la inestabilidad, en el caso que la haya.This thesis is devoted to the study of the orbital stability in a pair of hamiltonian type differential equations: Schrödinger and Klein-Gordon equation, both with potential nonlinear term and periodic boundary data. At first we proved for this kind of problems there exists ground state, with its corresponding profile soliton. Then we centered our attention in to study the stability of the equation flux when the initial data is closed to soliton, for this we built a suitable Lyapunov function using conservation laws. For the Schrödinger equation we proved orbital stability of the solutions, under some conditions over parameters of the equation. We obtained comparable results with the well-known ones for spatial non bound domain. Due to stable behavior of solutions with initial data closed to soliton profile, we introduced a numeric method for the dynamic simulation, for this we studied existence and unicity of the discrete solitons and stable behavior of discrete solutions. With this result we proved convergence of the method. At last, for Klein-Gordon equation with spatial non bound domain instability previous results were given, then we could not expect a different result for our case. The application of the method developed here, although it does not give us the stability, it allows us to characterize the instability direction, in the case that is it.Fil: Borgna, Juan Pablo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

Properties of some breather solutions of a nonlocal discrete NLS equation

We present results on breather solutions of a discrete nonlinear Schrödinger equation with a cubic Hartree-type nonlinearity that models laser light propagation in waveguide arrays that use a nematic liquid crystal substratum. A recent study of that model by Ben et al [R.I. Ben, L. Cisneros Ake, A.A. Minzoni, and P. Panayotaros, Phys. Lett. A, 379:1705C-1714, 2015] showed that nonlocality leads to some novel properties such as the existence of orbitaly stable breathers with internal modes, and of shelf-like configurations with maxima at the interface. In this work we present rigorous results on these phenomena and consider some more general solutions. First, we study energy minimizing breathers, showing existence as well as symmetry and monotonicity properties. We also prove results on the spectrum of the linearization around one-peak breathers, solutions that are expected to coincide with minimizers in the regime of small linear intersite coupling. A second set of results concerns shelf-type breather solutions that may be thought of as limits of solutions examined in [R.I. Ben, L. Cisneros Ake, A.A. Minzoni, and P. Panayotaros, Phys. Lett. A, 379:1705C-1714, 2015]. We show the existence of solutions with a non-monotonic front-like shape and justify computations of the essential spectrum of the linearization around these solutions in the local and nonlocal cases.Fil: Ben, Roberto Ignacio. Universidad Nacional de General Sarmiento. Instituto del Desarrollo Humano; ArgentinaFil: Borgna, Juan Pablo. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Panayotaros, Panayotis. Universidad Nacional Autonoma de Mexico. Facultad de Ciencias; Méxic

Fréedericksz transition on a phenomenological model for a nematic inhomogeneous superfluid in presence of an electric field

In this article we derive a Ginzburg–Landau energy functional for a nematic inhomogeneous superfluid in presence of an electric field. The molecules occupy an infinite cylinder Ω with cross section D. We suppose vacuum in R3∖Ω, with the possibility that an external electric field can be applied parallel to D. The Helmholtz free energy is obtained by taking the London limit of a Ginzburg–Landau nematic superconducting model in absence of magnetic fields, and by including an appropriate contribution of the electric potential energy. We show that the critical parameter inside Ω, which defines the Fréedericksz transition on the molecular alignment, is not only influenced by the effects of the electric field in the sample, but also by the additional contribution of the superfluid current. We take a particular solution for the Ginzburg–Landau equations, where the superfluid phase does not have circulation. Then, we demonstrate that the corresponding Fréedericksz threshold can be calculated, on an arbitrary domain, by using the notion of the first positive eigenvalue of the Laplacian. This eigenvalue depends on the chosen geometry and the boundary conditions on the nematic phase in the sample. Next, we apply our results in an infinite slab and in an infinite cylinder with circular cross section, where the nematic superfluid system is subjected to Dirichlet or Neumann boundary conditions in each case. We deduce a modified Fréedericksz threshold, for each configuration mentioned before, in a uniform electric field. In these instances we notice the remarkable fact that, for specific values and regimes of the intrinsic parameters, the critical fields are different than the ones obtained in the pure nematic case. Finally, we also study a Fréedericksz type threshold in a long hollow cylinder with uniform charge density, where molecules are reoriented by the electric field produced only by the internal charges of the sample. This setting suggests that, if molecules are oriented radially at the boundary of the region, a Fréedericksz type threshold appears in order to maintain the radial molecular distribution, which varies with the typical radii of the domain.Fil: García Ovalle, Diego. Pontificia Universidad Católica de Chile. Facultad de Física; ChileFil: Borgna, Juan Pablo. Universidad Nacional de San Martin. Escuela de Ciencia y Tecnología. Centro de Matemática Aplicada; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Instituto de Ciencias Fisicas. - Universidad Nacional de San Martin. Instituto de Ciencias Fisicas.; ArgentinaFil: de Leo, Mariano Fernando. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentin

Optical solitons in nematic liquid crystals: Arbitrary deviation angle model

We study a coupled Schrödinger-elliptic evolution system that describes the propagation of a laser beam in nematic liquid crystals. The elliptic equation describes the effects of the beam electric field on the local orientation (director field) of the nematic liquid crystal and has an important regularizing effect, seen experimentally and understood theoretically in related models. In the present work we propose a new nonlinear elliptic equation for the director field that makes no assumption on the size of the director field angle. The analysis of this elliptic equation leads to an upper bound for the size of the director angle that we believe is optimal and physically relevant, and that implies that the elastic response of the medium prevents a complete alignment between the electric field and the orientation of the liquid crystal. The results on the elliptic problem are combined with arguments from dispersive wave theory to show the local and global well-posedness of the evolution problem and the decay of small initial conditions. We also show the existence of constrained minimizers of the Hamiltonian, assuming sufficiently large optical power ($L^2$-norm of the laser field). These minimizers are solitons with radial, monotonically decreasing profiles.Fil: Borgna, Juan Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de San Martin. Escuela de Ciencia y Tecnología. Centro de Matemática Aplicada; ArgentinaFil: Panayotaros, Panayotis. Universidad Nacional Autónoma de México; MéxicoFil: Rial, Diego Fernando. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Sánchez de la Vega, Constanza. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

Optimal Control for Optical Solitons in Nematic Liquid Crystals

We study an optimal control problem for a coupled Schrödinger-elliptic evolution system that describes the propagation of a laser beam in nematic liquid crystals. We consider a bilinear control related to an electric field depending on the optical axis acting on the sample. This problem arises from the study of an optimal way to transform the input signal into a target signal by modifying a system parameter related to the bias electric field. We prove well-posedness, existence, and first-order necessary conditions for an optimal solution.Fil: Sanchez Fernandez de la Vega, Constanza Mariel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Borgna, Juan Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Instituto de Ciencias Físicas. - Universidad Nacional de San Martín. Instituto de Ciencias Físicas; ArgentinaFil: Rial, Diego Fernando. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

Optical solitons in nematic liquid crystals: Model with saturation effects

We study a 2D system that couples a Schrödinger evolution equation to a nonlinear elliptic equation and models the propagation of a laser beam in a nematic liquid crystal. The nonlinear elliptic equation describes the response of the director angle to the laser beam electric field. We obtain results on well-posedness and solitary wave solutions of this system, generalizing results for a well-studied simpler system with a linear elliptic equation for the director field. The analysis of the nonlinear elliptic problem shows the existence of an isolated global branch of solutions with director angles that remain bounded for arbitrary electric field. The results on the director equation are also used to show local and global existence, as well as decay for initial conditions with sufficiently small L 2-norm. For sufficiently large L 2-norm we show the existence of energy minimizing optical solitons with radial, positive and monotone profiles.Fil: Borgna, Juan Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de San Martín; ArgentinaFil: Panayotaros, Panayotis. Universidad Nacional Autónoma de México; MéxicoFil: Rial, Diego Fernando. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Sanchez Fernandez de la Vega, Constanza Mariel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin