Related papers: Bifurcations of relative periodic orbits in NLS/GP…

We examine the dynamics of solutions to nonlinear Schrodinger/Gross-Pitaevskii equations that arise due to Hamiltonian Hopf (HH) bifurcations--the collision of pairs of eigenvalues on the imaginary axis. To this end, we use inverse…

We focus on the existence and persistence of families of saddle periodic orbits in a four-dimensional Hamiltonian reversible ordinary differential equation derived using a travelling wave ansatz from a generalised nonlinear Schr{\"o}dinger…

We rigorously analyze the bifurcation of stationary so called nonlinear Bloch waves (NLBs) from the spectrum in the Gross-Pitaevskii (GP) equation with a periodic potential, in arbitrary space dimensions. These are solutions which can be…

We consider a class nonlinear Schr\"odinger / Gross-Pitaevskii equations (NLS/GP) with a focusing (attractive) nonlinear potential and symmetric double well linear potential. NLS/GP plays a central role in the modeling of nonlinear optical…

We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but…

We consider a class of nonlinear Schrodinger / Gross-Pitaveskii (NLS-GP) equations, i.e. NLS with a linear potential. We obtain conditions for a symmetry breaking bifurcation in a symmetric family of states as N, the squared L^2 norm…

The nonlinear Schrodinger (NLS) equation is considered on a periodic metric graph subject to the Kirchhoff boundary conditions. Bifurcations of standing localized waves for frequencies lying below the bottom of the linear spectrum of the…

We study the existence and stability of localized states in the two-dimensional (2D) nonlinear Schrodinger (NLS)/Gross-Pitaevskii equation with a symmetric four-well potential. Using a fourmode approximation, we are able to trace the…

We develop a detailed rigorous analysis of edge bifurcations of standing waves in the nonlinear Schr\"odinger (NLS) equation on a tadpole graph (a ring attached to a semi-infinite line subject to the Kirchhoff boundary conditions at the…

We address the Gross--Pitaevskii (GP) equation with a periodic linear potential and a periodic sign-varying nonlinearity coefficient. Contrary to the claims in the previous works of Abdullaev {\em et al.} [PRE {\bf 77}, 016604 (2008)] and…

This article is a review of results on the nonlinear Schroedinger / Gross-Pitaevskii equation (NLS / GP). Nonlinear bound states and aspects of their stability theory are discussed from variational and bifurcation perspectives. Nonlinear…

We implement the geometric method proposed in ([9], [3], [16]) to analytically predict the sequence of bifurcations leading to a change of stability and/or the appearance of new periodic orbits in the secular 3D planetary three body…

We consider the bifurcations of standing wave solutions to the nonlinear Schr\"odinger equation (NLS) posed on a quantum graph consisting of two loops connected by a single edge, the so-called dumbbell, recently studied by Marzuola and…

We investigate the KdV-Burgers and Gardner equations with dissipation and external perturbation terms by the approach of dynamical systems and Shil'nikov's analysis. The stability of the equilibrium point is considered, and Hopf…

This paper gives an analysis of the periodic solutions of a ring of $n$ oscillators coupled to their neighbors. We prove the bifurcation of branches of such solutions from a relative equilibrium, and we study their symmetries. We give…

For piecewise-smooth ordinary differential equations, the occurrence of a Hopf bifurcation on a switching surface is known as a boundary Hopf bifurcation. Boundary Hopf bifurcations are codimension-two, so occur at points in two-parameter…

The stationary Gross-Pitaevskii equation in one dimension is considered with a complex periodic potential satisfying the conditions of the PT (parity-time reversal) symmetry. Under rather general assumptions on the potentials we prove…

In this work we employ a recently proposed bifurcation analysis technique, the deflated continuation algorithm, to compute steady-state solitary waveforms in a one-component, two dimensional nonlinear Schr\"odinger equation with a parabolic…

The dynamics of single carrier wavepackets in nonlinear wave problems over periodic structures can be often formally approximated by the constant coefficient nonlinear Schr\"odinger equation (NLS) as an effective model for the wavepacket…

Following Part~I, we consider a class of reversible systems and study bifurcations of homoclinic orbits to hyperbolic saddle equilibria. Here we concentrate on the case in which homoclinic orbits are symmetric, so that only one control…