Phononic Rogue Waves

We present a theoretical study of extreme events occurring in phononic lattices. In particular, we focus on the formation of rogue or freak waves, which are characterized by their localization in both spatial and temporal domains. We consider two examples. The first one is the prototypical nonlinear mass-spring system in the form of a homogeneous Fermi-Pasta-Ulam-Tsingou (FPUT) lattice with a polynomial potential. By deriving an approximation based on the nonlinear Schroedinger (NLS) equation, we are able to initialize the FPUT model using a suitably transformed Peregrine soliton solution of the NLS, obtaining dynamics that resembles a rogue wave on the FPUT lattice. We also show that Gaussian initial data can lead to dynamics featuring rogue wave for sufficiently wide Gaussians. The second example is a diatomic granular crystal exhibiting rogue wave like dynamics, which we also obtain through an NLS reduction and numerical simulations. The granular crystal (a chain of particles that interact elastically) is a widely studied system that lends itself to experimental studies. This study serves to illustrate the potential of such dynamical lattices towards the experimental observation of acoustic rogue waves.Comment: 9 pages, 4 figure

Skyrmions, Rational Maps & Scaling Identities

Starting from approximate Skyrmion solutions obtained using the rational map ansatz, improved approximate Skyrmions are constructed using scaling arguments. Although the energy improvement is small, the change of shape clarifies whether the true Skyrmions are more oblate or prolate.Comment: 13 pages, 3 figure

Deflation-based Identification of Nonlinear Excitations of the 3D Gross--Pitaevskii equation

We present previously unknown solutions to the 3D Gross--Pitaevskii equation describing atomic Bose-Einstein condensates. This model supports elaborate patterns, including excited states bearing vorticity. The discovered coherent structures exhibit striking topological features, involving combinations of vortex rings and multiple, possibly bent vortex lines. Although unstable, many of them persist for long times in dynamical simulations. These solutions were identified by a state-of-the-art numerical technique called deflation, which is expected to be applicable to many problems from other areas of physics.Comment: 9 pages, 11 figure

Lattice Three Dimensional Skyrmions Revisited

In the continuum a skyrmion is a topological nontrivial map between Riemannian manifolds, an a stationary point of a particular energy functional. This paper describes lattice analogues of the aforementioned skyrmions, namely a natural way of using the topological properties of the three dimensional continuum Skyrme model to achieve topological stability on the lattice. In particular, using fixed point iterations, numerically exact lattice skyrmions are constructed: and their stability under small perturbation sis explored by means of linear stability analysis. While stable branches of such solutions are identified, it is also shown that they possess a particularly delicate bifurcation structure, especially so in the vicinity of the continuum limit. The corresponding bifurcation diagram is elucidated and a prescription for selecting the branch asymptoting to the well known continuum limit is given. Finally, the robustness of the spectrally stable solutions is corroborated by the virtue of direct numerical simulations

Self-similar blow-up solutions in the generalized Korteweg-de Vries equation: Spectral analysis, normal form and asymptotics

In the present work we revisit the problem of the generalized Korteweg-de Vries equation parametrically, as a function of the relevant nonlinearity exponent, to examine the emergence of blow-up solutions, as traveling waveforms lose their stability past a critical point of the relevant parameter $p$, here at $p=5$. We provide a {\it normal form} of the associated collapse dynamics and illustrate how this captures the collapsing branch bifurcating from the unstable traveling branch. We also systematically characterize the linearization spectrum of not only the traveling states, but importantly of the emergent collapsing waveforms in the so-called co-exploding frame where these waveforms are identified as stationary states. This spectrum, in addition to two positive real eigenvalues which are shown to be associated with the symmetries of translation and scaling invariance of the original (non-exploding) frame features complex patterns of negative eigenvalues that we also fully characterize. We show that the phenomenology of the latter is significantly affected by the boundary conditions and is far more complicated than in the corresponding symmetric Laplacian case of the nonlinear Schr{\"o}dinger problem that has recently been explored. In addition, we explore the dynamics of the unstable solitary waves for $p>5$ in the co-exploding frame.Comment: 33 pages, 16 figure

Kuznetsov-Ma breather-like solutions in the Salerno model

The Salerno model is a discrete variant of the celebrated nonlinear Schr\"odinger (NLS) equation interpolating between the discrete NLS (DNLS) equation and completely integrable Ablowitz-Ladik (AL) model by appropriately tuning the relevant homotopy parameter. Although the AL model possesses an explicit time-periodic solution known as the Kuznetsov-Ma (KM) breather, the existence of time-periodic solutions away from the integrable limit has not been studied as of yet. It is thus the purpose of this work to shed light on the existence and stability of time-periodic solutions of the Salerno model. In particular, we vary the homotopy parameter of the model by employing a pseudo-arclength continuation algorithm where time-periodic solutions are identified via fixed-point iterations. We show that the solutions transform into time-periodic patterns featuring small, yet non-decaying far-field oscillations. Remarkably, our numerical results support the existence of previously unknown time-periodic solutions {\it even} at the integrable case whose stability is explored by using Floquet theory. A continuation of these patterns towards the DNLS limit is also discussed.Comment: 9 pages, 4 figure

Rogue Waves in Nonlinear Schrodinger Models with Variable Coefficients : Application to Bose Einstein Condensates

We explore the form of rogue waves solution sin a select set of case examples of non linear Schrodinger equations with variable coefficients. We focus on systems with constant dispersion, and present three different models that describe atomic Bose Einstein condensates in different experimentally relevant settings. For these models, we identify exact rogue waves solutions. Our analytical findings are corroborated by direct numerical integration of the original equations, performed by two different schemes. Very good agreement between numerical results and analytical predictions for the emergence of the rogue waves is identified. Additionally, the nontrivial fate of small numerically induced perturbations to the exact rogue waves solutions is also discussed

Dark bright solitons in coupled nonlinear Schrodinger equations with unequal dispersion coefficients

We study a two component nonlinear Schrodinger system with equal, repulsive cubic interactions and different dispersion coefficients in the two components. We consider states that have a dark solitary wave in one component. Treating it as a frozen one, we explore the possibility of the formation of bright solitonic structures in the other component. We identify bifurcation points at which such states emerge in the bright component in the linear limit and explore their continuation into the nonlinear regime. An additional analytically tractable limit is found to be that of vanishing dispersion of the bright component. We numerically identify regimes of potential stability, not only of the single peak ground state (the dark bright soliton), but also of excited states with one or more zero crossings in the bright component. When the states are identifies as unstable, direct numerical simulations are used to investigate the outcome of the instability development. Although out principal focus is n the homogeneous setting, we also briefly touch upon the counter intuitive impact of the potential presence of a parabolic trap on the states of interest