Related papers: Approximate Analytic Solutions to Coupled Nonlinea…
By constructing the commutative operators chain, we derive the integrable conditions for solving the eigenfunctions of Dirac equation and Schr\"odinger equation. These commutative relations correspond to the intrinsic symmetry of the…
We prove, by a shooting method, the existence of infinitely many solutions of the form $\psi(x^0,x) = e^{-i\Omega x^0}\chi(x)$ of the nonlinear Dirac equation {equation*} i\underset{\mu=0}{\overset{3}{\sum}} \gamma^\mu \partial_\mu \psi-…
We obtain time dependent $q$-Gaussian wave-packet solutions to a non linear Schr\"odinger equation recently advanced by Nobre, Rego-Montero and Tsallis (NRT) [Phys. Rev. Lett. 106 (2011) 10601]. The NRT non-linear equation admits plane…
We consider a Dirac operator with short range potential and with eigenvalues. We add a nonlinear term and we show that the small standing waves of the corresponding nonlinear Dirac equation (NLD) are attractors for small solutions of the…
We find the exact bound-state solutions and normalization constant for the Dirac equation with scalar-vector-pseudoscalar interaction terms for the generalized Hulth\'{e}n potential in the case where we have a particular mass function…
For the first time exact analytical solutions to the eikonal equations in (1+1) dimensions with a refractive index being a saturated function of intensity are constructed. It is demonstrated that the solutions exhibit collapse; an explicit…
We demonstrate that neutral Dirac particles in external electric fields, which are equivalent to generalized Dirac oscillators, are physical examples of quasi-exactly solvable systems. Electric field configurations permitting quasi-exact…
From the mathematical side, nonlinear Schr\"odinger equations are usually investigated via variational methods, that cease to work in higher dimensions. This thesis tries to overcome this problem by focusing on spherically symmetric…
We consider nonlinear Schr\"{o}dinger equations, $i\partial_t \psi = H_0 \psi + \lambda |\psi|^2\psi$ in $\mathbb{R}^3 \times [0,\infty)$, where $H_0 = -\Delta + V$, $\lambda=\pm 1$, the potential $V$ is radial and spatially decaying, and…
We obtain periodic solutions for nonlinear Dirac equations with a nonlinear term that is not necessarily coercive.This amounts to study the equation on a three-dimensional torus.The Palais-Smale condition is enhanced by involving a coercive…
Quasi-periodic solutions of a nonlinear periodic polyharmonic equation in $\R^n$, $n>1$, are studied. It is proven that there is an extensive "non-resonant" set ${\mathcal G}\subset \R^n$ such that for every $\vec k\in \mathcal G$ there is…
We introduce a notion of approximate viscosity solution for a class of nonlinear path-dependent PDEs (PPDEs), including the Hamilton-Jacobi-Bellman type equations. Existence, comparaison and stability results are established under fairly…
The interaction between two co-propagating electrostatic wavepackets characterized by arbitrary carrier wavenumber is considered. A one-dimensional (1D) non-magnetized plasma model is adopted, consisting of a cold inertial ion fluid…
Three classes of higher-order nonlinear parabolic hyperbolic, and nonlinear dispersion equations are shown to admit exact blow-up or compacton solutions, which are induced by elliptic equations with non-Lipschitz nonlinearities. Variational…
Exact solutions are found to the Dirac equation for a combination of Lorentz scalar and vector Coulombic potentials with additional non-Coulombic parts. An appropriate linear combination of Lorentz scalar and vector non-Coulombic…
Using symmetry analysis we systematically present a higher-dimensional similarity transformation reducing the (3+1)-dimensional inhomogeneous nonlinear Schrodinger (NLS) equation with variable coefficients and parabolic potential to the…
We study the point spectrum of the linearization at a solitary wave solution $\phi_\omega(x)e^{-\mathrm{i}\omega t}$ to the nonlinear Dirac equation in $\mathbb{R}^n$, $n\ge 1$, with the nonlinear term given by $f(\psi^*\beta\psi)\beta\psi$…
We discuss stationary solutions of the discrete nonlinear Schr\"odinger equation (DNSE) with a potential of the $\phi^{4}$ type which is generically applicable to several quantum spin, electron and classical lattice systems. We show that…
Let ${\mathsf D}$ and ${\mathsf H}$ be the self-adjoint, one-dimensional Dirac and Schr\"odinger operators in $L^{2}(\mathbb{R};\mathbb{C}^{2})$ and $L^{2}(\mathbb{R};\mathbb{C})$ respectively. It is well known that, in absence of an…
We study the dynamics of the collision of two solitary waves for the Zakharov-Kuznetsov equation in dimension $2$ and $3$. We describe the evolution of the solution behaving as a sum of $2$-solitary waves of nearly equal speeds at time…