11 research outputs found
Nonlinear Impurity Modes in Homogeneous and Periodic Media
We analyze the existence and stability of nonlinear localized waves described
by the Kronig-Penney model with a nonlinear impurity. We study the properties
of such waves in a homogeneous medium, and then analyze new effects introduced
by periodicity of the medium parameters. In particular, we demonstrate the
existence of a novel type of stable nonlinear band-gap localized states, and
also reveal an important physical mechanism of the oscillatory wave
instabilities associated with the band-gap wave resonances.Comment: 11 pages, 3 figures; To be published in: Proceedings of the NATO
Advanced Research Workshop "Nonlinearity and Disorder: Theory and
Applications" (Tashkent, 2-6 Oct, 2000) Editors: P.L. Christiansen and F.K.
Abdullaev (Kluwer, 2001
Solitary waves in the Nonlinear Dirac Equation
In the present work, we consider the existence, stability, and dynamics of
solitary waves in the nonlinear Dirac equation. We start by introducing the
Soler model of self-interacting spinors, and discuss its localized waveforms in
one, two, and three spatial dimensions and the equations they satisfy. We
present the associated explicit solutions in one dimension and numerically
obtain their analogues in higher dimensions. The stability is subsequently
discussed from a theoretical perspective and then complemented with numerical
computations. Finally, the dynamics of the solutions is explored and compared
to its non-relativistic analogue, which is the nonlinear Schr{\"o}dinger
equation. A few special topics are also explored, including the discrete
variant of the nonlinear Dirac equation and its solitary wave properties, as
well as the PT-symmetric variant of the model
Discrete breathers in and related models
We touch upon the wide topic of discrete breather formation with a special
emphasis on the the model. We start by introducing the model and
discussing some of the application areas/motivational aspects of exploring time
periodic, spatially localized structures, such as the discrete breathers. Our
main emphasis is on the existence, and especially on the stability features of
such solutions. We explore their spectral stability numerically, as well as in
special limits (such as the vicinity of the so-called anti-continuum limit of
vanishing coupling) analytically. We also provide and explore a simple, yet
powerful stability criterion involving the sign of the derivative of the energy
vs. frequency dependence of such solutions. We then turn our attention to
nonlinear stability, bringing forth the importance of a topological notion,
namely the Krein signature. Furthermore, we briefly touch upon linearly and
nonlinearly unstable dynamics of such states. Some special aspects/extensions
of such structures are only touched upon, including moving breathers and
dissipative variations of the model and some possibilities for future work are
highlighted
Multicomplex Wave Functions for Linear And Nonlinear Schrödinger Equations
We consider a multicomplex Schrödinger equation with general scalar potential, a generalization of both the standard Schrödinger equation and the bicomplex Schrödinger equation of Rochon and Tremblay, for wave functions mapping onto (Formula presented.). We determine the equivalent real-valued system in recursive form, and derive the relevant continuity equations in order to demonstrate that conservation of probability (a hallmark of standard quantum mechanics) holds in the multicomplex generalization. From here, we obtain the real modulus and demonstrate the generalized multicomplex version of Bornâs formula for the probability densities. We then turn our attention to possible generalizations of the multicomplex Schrödinger equation, such as the case where the scalar potential is replaced with a multicomplex-valued potential, or the case where the potential involves the real modulus of the wave function, resulting in a multicomplex nonlinear Schrödinger equation. Finally, in order to demonstrate the solution methods for such equations, we obtain several particular solutions to the multicomplex Schrödinger equation. We interpret the generalized results in the context of the standard results from quantum mechanics