Related papers: Equivalence groupoids and group classification of …
We consider the problem of performing the preliminary "symmetry classification'' of a class of quasi-linear PDE's containing one or more arbitrary functions: we provide an easy condition involving these functions in order that nontrivial…
This paper is concerned with the existence of normalized solutions of the nonlinear Schr\"odinger equation \[ -\Delta u+V(x)u+\lambda u = |u|^{p-2}u \qquad\text{in $\mathbb{R}^N$} \] in the mass supercritical and Sobolev subcritical case…
Conditional and Lie symmetries of semi-linear 1D Schr\"odinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear…
We consider a wide class of nonlinear canonical quantum systems described by a one-particle Schroedinger equation containing a complex nonlinearity. We introduce a nonlinear unitary transformation which permits us to linearize the…
We carry out a detailed Lie point symmetry group classification of the Li\'enard type equation, $\ddot{x}+f(x)\dot{x}+g(x) = 0$, where $f(x)$ and $g(x)$ are arbitrary smooth functions of $x$. We divide our analysis into two parts. In the…
The purpose of this paper is to establish the existence of solutions with prescribed norm to a class of nonlinear equations involving the mixed fractional Laplacians. This type of equations arises in various fields ranging from biophysics…
By using Lie symmetry methods, we identify a class of second order nonlinear ordinary differential equations invariant under at least one dimensional subgroup of the symmetry group of the Ermakov-Pinney equation. In this context, nonlinear…
The enhanced group classification of a semi-linear generalization of a general bond-pricing equation is carried out by employing the underlying equivalence and additional equivalence transformations. The knowledge of the sub classes with…
A class of solvable (systems of) nonlinear evolution PDEs in multidimensional space is discussed. We focus on a rotation-invariant system of PDEs of Schr\"odinger type and on a relativistically-invariant system of PDEs of Klein-Gordon type.…
In this paper we consider a generalized Kuramoto-Sivashinsky equation. The equivalence group of the class under consideration has been constructed. This group allows us to perform a comprehensive study and a clear and concise formulation of…
In this paper, we prove a multiplicity result of solutions for the following stationary Schr\"odinger-Poisson-Slater equations \begin{equation}\label{eq-abstract} -\Delta u - \lambda u + (\left | x \right |^{-1}\ast \left | u \right |^2) u…
We develop an algebraic approach to studying the spectral properties of the stationary Schr\"odinger equation in one dimension based on its high order conditional symmetries. This approach makes it possible to obtain in explicit form…
We establish sharp-in-time kernel and dispersive estimates for the Schr\"odinger equation on non-compact Riemannian symmetric spaces of any rank. Due to the particular geometry at infinity and the Kunze-Stein phenomenon, these properties…
We analyze initial-boundary value problems for an integrable generalization of the nonlinear Schr\"odinger equation formulated on the half-line. In particular, we investigate the so-called linearizable boundary conditions, which in this…
We investigate the existence of normalized ground states for Schr\"odinger equations on noncompact metric graphs in presence of nonlinear point defects, described by nonlinear $\delta$-interactions at some of the vertices of the graph. For…
We study the following one-dimensional cubic nonlinear Schr\"{o}dinger system: \[ u_i''+2\Big(\sum_{k=1}^Nu_k^2\Big)u_i=-\mu_iu_i \ \,\ \mbox{in}\, \ \mathbb{R} , \ \ i=1, 2, \cdots, N, \] where $\mu_1\leq\mu_2\leq\cdots\leq\mu_N<0$ and…
We show that symmetric polynomials previously introduced by the author satisfy a certain differential equation. After a change of variables, it can be written as a non-stationary Schr\"odinger equation with elliptic potential, which is…
Analytical solutions of variable coefficient nonlinear Schr\"odinger equations having four-dimensional symmetry groups which are in fact the next closest to the integrable ones occurring only when the Lie symmetry group is five-dimensional…
In this paper our objective is to investigate the existence of multiple normalized solutions to the logarithmic Schr\"{o}dinger equation given by \begin{align*} \left\{ \begin{aligned} &-\epsilon^2 \Delta u+V( x)u=\lambda u+u \log u^2,…
In this paper, we investigate the one-dimensional derivative nonlinear Schr\"odinger equations of the form $iu_t-u_{xx}+i\lambda\abs{u}^k u_x=0$ with non-zero $\lambda\in \Real$ and any real number $k\gs 5$. We establish the local…