arXiv subjects

Jesús Cuevas-Maraver

Publications and source records attributed to Jesús Cuevas-Maraver.

15 recordsLinked to original sources

arXiv2026

Dark solitons in the fractional NLS equation

In the present work we consider the subject of dark fractional solitary waves in the realm of generalized (fractional) forms of the nonlinear Schrödinger (NLS) equation. While earlier studies have examined such states in the realm of real field theories, we showcase the existence and stability of individual dark solitary waves in such NLS settings and subsequently turn to two-soliton solutions. We find different branches of such two-soliton solution equilibria and contrary to the real field-theoretic setting all possible branches of two-soliton equilibria are found to be potentially unstable, although with different types of instabilities. Odd branches are potentially subject to oscillatory instabilities, while even branches are always exponentially unstable. The dynamics that results from the instabilities is also examined and is found to potentially feature breathing characteristics. This prompts us to seek and find associated periodic (breathing) orbits that are also unprecedented in this context, to the best of our knowledge. The effective particle-like dynamics of the solitary waves also prompts us to seek ordinary differential equation (ODE) descriptions to the dark soliton interaction dynamics. These are shown to hold promise toward providing us with effective reduced order models, indicating some potential directions for further investigation.

nlin.PS↗

arXiv2025

Stable bi-frequency spinor modes as Dark Matter candidates

We show that spinor systems with scalar self-interaction, such as the Dirac--Klein--Gordon system with Yukawa coupling or the Soler model, generically have bi-frequency solitary wave solutions. We develop the approach to stability properties of such waves and use the radial reduction to show that indeed the (linear) stability is available for a wide range of parameters. We show that only bi-frequency modes can be dynamically stable and suggest that stable bi-frequency modes can serve as storages of the Dark Matter. The approach is based on linear stability results of one-frequency solitary waves in (3+1)D Soler model, which we obtain as a by-product.

math.AP↗

arXiv2024

Standing and Traveling Waves in a Nonlinearly Dispersive Lattice Model

In the work of Colliander et al. (2010), a minimal lattice model was constructed describing the transfer of energy to high frequencies in the defocusing nonlinear Schrödinger equation. In the present work, we present a systematic study of the coherent structures, both standing and traveling, that arise in the context of this model. We find that the nonlinearly dispersive nature of the model is responsible for standing waves in the form of discrete compactons. On the other hand, analysis of the dynamical features of the simplest nontrivial variant of the model, namely the dimer case, yields both solutions where the intensity is trapped in a single site and solutions where the intensity moves between the two sites, which suggests the possibility of moving excitations in larger lattices. Such excitations are also suggested by the dynamical evolution associated with modulational instability. Our numerical computations confirm this expectation, and we systematically construct such traveling states as exact solutions in lattices of varying size, as well as explore their stability. A remarkable feature of these traveling lattice waves is that they are of "antidark" type, i.e., they are mounted on top of a non-vanishing background. These studies shed light on the existence, stability and dynamics of such standing and traveling states in $1+1$ dimensions, and pave the way for exploration of corresponding configurations in higher dimensions.

nlin.PS↗

arXiv2024

Stability of Breathers for a Periodic Klein-Gordon Equation

The existence of breather type solutions, i.e., periodic in time, exponentially localized in space solutions, is a very unusual feature for continuum, nonlinear wave type equations. Following an earlier work [Comm. Math. Phys. {\bf 302}, 815-841 (2011)], establishing a theorem for the existence of such structures, we bring to bear a combination of analysis-inspired numerical tools that permit the construction of such wave forms to a desired numerical accuracy. In addition, this enables us to explore their numerical stability. Our computations show that for the spatially heterogeneous form of the $ϕ^4$ model considered herein, the breather solutions are generically found to be unstable. Their instability seems to generically favor the motion of the relevant structures. We expect that these results may inspire further studies towards the identification of stable continuous breathers in spatially-heterogeneous, continuum nonlinear wave equation models.

nlin.PS↗

arXiv2024

The Dissipative Effect of Caputo--Time-Fractional Derivatives and its Implications for the Solutions of Nonlinear Wave Equations

In honor of the great Russian mathematician A. N. Kolmogorov, we would like to draw attention in the present paper to a curious mathematical observation concerning fractional differential equations describing physical systems, whose time evolution for integer derivatives has a time-honored conservative form. This observation, although known to the general mathematical community, has not, in our view, been satisfactorily addressed. More specifically, we follow the recent exploration of Caputo-Riesz time-space-fractional nonlinear wave equation, in which two of the present authors introduced an energy-type functional and proposed a finite-difference scheme to approximate the solutions of the continuous model. The relevant Klein-Gordon equation considered here has the form: \begin{equation} \frac {\partial ^βϕ(x , t)} {\partial t ^β} - Δ^αϕ(x , t) + F ^\prime (ϕ(x , t)) = 0, \quad \forall (x , t) \in (-\infty,\infty) \end{equation} where we explore the sine-Gordon nonlinearity $F(ϕ)=1-\cos(ϕ)$ with smooth initial data. For $α=β=2$, we naturally retrieve the exact, analytical form of breather waves expected from the literature. Focusing on the Caputo temporal derivative variation within $1< β< 2$ values for $α=2$, however, we observe artificial dissipative effects, which lead to complete breather disappearance, over a time scale depending on the value of $β$. We compare such findings to single degree-of-freedom linear and nonlinear oscillators in the presence of Caputo temporal derivatives and also consider anti-damping mechanisms to counter the relevant effect. These findings also motivate some interesting directions for further study, e.g., regarding the consideration of topological solitary waves, such as kinks/antikinks and their dynamical evolution in this model.

nlin.PS↗

arXiv2023

Standing and Traveling Waves in a Model of Periodically Modulated One-dimensional Waveguide Arrays

In the present work, we study coherent structures in a one-dimensional discrete nonlinear Schrödinger lattice in which the coupling between waveguides is periodically modulated. Numerical experiments with single-site initial conditions show that, depending on the power, the system exhibits two fundamentally different behaviors. At low power, initial conditions with intensity concentrated in a single site give rise to transport, with the energy moving unidirectionally along the lattice, whereas high power initial conditions yield stationary solutions. We explain these two behaviors, as well as the nature of the transition between the two regimes, by analyzing a simpler model where the couplings between waveguides are given by step functions. In this case, we numerically construct both stationary and moving coherent structures, which are solutions reproducing themselves exactly after an integer multiple of the coupling period. For the stationary solutions, which are true periodic orbits, we use Floquet analysis to determine the parameter regime for which they are spectrally stable. Typically, the traveling solutions are characterized by having small-amplitude, oscillatory tails, although we identify a set of parameters for which these tails disappear. These parameters turn out to be independent of the lattice size, and our simulations suggest that for these parameters, numerically exact traveling solutions are stable.

nlin.PS↗

arXiv2023

Dissipative localised structures for the complex Discrete Ginzburg-Landau equation

The discrete complex Ginzburg-Landau equation is a fundamental model for the dynamics of nonlinear lattices incorporating competitive dissipation and energy gain effects. Such mechanisms are of particular importance for the study of survival/destruction of localised structures in many physical situations. In this work, we prove that in the discrete complex Ginzburg-Landau equation dissipative solitonic waveforms persist for significant times by introducing a dynamical transitivity argument. This argument is based on a combination of the notions of ``inviscid limits'' and of the ``continuous dependence of solutions on their initial data'', between the dissipative system and its Hamiltonian counterparts. Thereby, it establishes closeness of the solutions of the Ginzburg-Landau lattice to those of the conservative ideals described by the Discrete Nonlinear Schrödinger and Ablowitz-Ladik lattices. Such a closeness holds when the initial conditions of the systems are chosen to be sufficiently small in the suitable metrics and for small values of the dissipation or gain strengths. Our numerical findings are in excellent agreement with the analytical predictions for the dynamics of the dissipative bright, dark or even Peregrine-type solitonic waveforms.

nlin.PS↗

arXiv2022

Revisiting Multi-breathers in the discrete Klein-Gordon equation: A Spatial Dynamics Approach

We consider the existence and spectral stability of multi-breather structures in the discrete Klein-Gordon equation, both for soft and hard symmetric potentials. To obtain analytical results, we project the system onto a finite-dimensional Hilbert space consisting of the first $M$ Fourier modes, for arbitrary $M$. On this approximate system, we then take a spatial dynamics approach and use Lin's method to construct multi-breathers from a sequence of well-separated copies of the primary, single-site breather. We then locate the eigenmodes in the Floquet spectrum associated with the interaction between the individual breathers of such multi-breather states by reducing the spectral problem to a matrix equation. Expressions for these eigenmodes for the approximate, finite-dimensional system are obtained in terms of the primary breather and its kernel eigenfunctions, and these are found to be in very good agreement with the numerical Floquet spectrum results. This is supplemented with results from numerical timestepping experiments, which are interpreted using the spectral computations.

nlin.PS↗

arXiv2022

Floquet solitons in square lattices: Existence, Stability and Dynamics

In the present work, we revisit a recently proposed and experimentally realized topological 2D lattice with periodically time-dependent interactions. We identify the fundamental solitons, previously observed in experiments and direct numerical simulations, as exact, exponentially localized, periodic in time solutions. This is done for a variety of phase-shift angles of the central nodes upon a period oscillation of the coupling strength. Subsequently, we perform a systematic Floquet stability analysis of the relevant structures. We analyze both their point and their continuous spectrum and find that the solutions are generically stable, aside from the possible emergence of complex quartets due to the collision of bands of continuous spectrum. The relevant instabilities become weaker as the lattice size gets larger. Finally, we also consider multi-soliton analogues of these Floquet states, inspired by the corresponding discrete nonlinear Schrödinger (DNLS) lattice. When exciting initially multiple sites in phase, we find that the solutions reflect the instability of their DNLS multi-soliton counterparts, while for configurations with multiple excited sites in alternating phases, the Floquet states are spectrally stable, again in analogy to their DNLS counterparts.

nlin.PS↗

arXiv2021

The closeness of the Ablowitz-Ladik lattice to the Discrete Nonlinear Schrödinger equation

While the Ablowitz-Ladik lattice is integrable, the Discrete Nonlinear Schrödinger equation, which is more significant for physical applications, is not. We prove closeness of the solutions of both systems in the sense of a "continuous dependence" on their initial data in the $l^2$ and $l^{\infty}$ metrics. The most striking relevance of the analytical results is that small amplitude solutions of the Ablowitz-Ladik system persist in the Discrete Nonlinear Schrödinger one. It is shown that the closeness results are also valid in higher dimensional lattices as well as for generalised nonlinearities. For illustration of the applicability of the approach, a brief numerical study is included, showing that when the 1-soliton solution of the Ablowitz-Ladik system is initiated in the Discrete Nonlinear Schrödinger system with cubic and saturable nonlinearity, it persists for long-times. Thereby excellent agreement of the numerical findings with the theoretical predicti ions is obtained.

nlin.PS↗

arXiv2021

Discrete embedded solitary waves and breathers in one-dimensional nonlinear lattices

For a one-dimensional linear lattice, earlier work has shown how to systematically construct a slowly-decaying linear potential bearing a localized eigenmode embedded in the continuous spectrum. Here, we extend this idea in two directions: The first one is in the realm of the discrete nonlinear Schrodinger equation, where the linear operator of the Schrodinger type is considered in the presence of a Kerr focusing or defocusing nonlinearity and the embedded linear mode is continued into the nonlinear regime as a discrete solitary wave. The second case is the Klein-Gordon setting, where the presence of a cubic nonlinearity leads to the emergence of embedded-in-the-continuum discrete breathers. In both settings, it is seen that the stability of the modes near the linear limit turns into instability as nonlinearity is increased past a critical value, leading to a dynamical delocalization of the solitary wave (or breathing) state. Finally, we suggest a concrete experiment to observe these embedded modes using a bi-inductive electrical lattice.

nlin.PS↗

arXiv2018

Experimental and numerical observation of dark and bright discrete solitons in the band-gap of a diatomic-like electrical lattice

We observe dark and bright intrinsic localized modes (ILMs) or discrete breathers (DB) experimentally and numerically in a diatomic-like electrical lattice. The generation of dark ILMs by driving a dissipative lattice with spatially-homogenous amplitude is, to our knowledge, unprecedented. In addition, the experimental manifestation of bright breathers within the bandgap is also novel in this system. In experimental measurements the dark modes appear just below the bottom of the top branch in frequency. As the frequency is then lowered further into the band-gap, the dark DBs persist, until the nonlinear localization pattern reverses and bright DBs appear on top of the finite background. Deep into the bandgap, only a single bright structure survives in a lattice of 32 nodes. The vicinity of the bottom band also features bright and dark self-localized excitations. These results pave the way for a more systematic study of dark breathers and their bifurcations in diatomic-like chains.

nlin.PS↗

arXiv2016

Energy criterion for the spectral stability of discrete breathers

Discrete breathers are ubiquitous structures in nonlinear anharmonic models ranging from the prototypical example of the Fermi-Pasta-Ulam model to Klein-Gordon nonlinear lattices, among many others. We propose a general criterion for the emergence of instabilities of discrete breathers analogous to the well-established Vakhitov-Kolokolov criterion for solitary waves. The criterion involves the change of monotonicity of the discrete breather's energy as a function of the breather frequency. Our analysis suggests and numerical results corroborate that breathers with increasing (decreasing) energy-frequency dependence are generically unstable in soft (hard) nonlinear potentials.

nlin.PS↗

arXiv2016

Impulse-induced localized control of chaos in starlike networks

Locally decreasing the impulse transmitted by periodic pulses is shown to be a reliable method of taming chaos in starlike networks of dissipative nonlinear oscillators, leading to both synchronous periodic states and equilibria (oscillation death). Specifically, the paradigmatic model of damped kicked rotators is studied in which it is assumed that when the rotators are driven synchronously, i.e., all driving pulses transmit the same impulse, the networks display chaotic dynamics. It is found that the taming effect of decreasing the impulse transmitted by the pulses acting on particular nodes strongly depends on their number and degree of connectivity. A theoretical analysis is given explaining the basic physical mechanism as well as the main features of the chaos-control scenario.

nlin.CD↗

arXiv2015

Nonlinear instabilities of multi-site breathers in Klein-Gordon lattices

In the present work, we explore the possibility of excited breather states in a nonlinear Klein--Gordon lattice to become nonlinearly unstable, even if they are found to be spectrally stable. The mechanism for this fundamentally nonlinear instability is through the resonance with the wave continuum of a multiple of an internal mode eigenfrequency in the linearization of excited breather states. For the nonlinear instability, the internal mode must have its Krein signature opposite to that of the wave continuum. This mechanism is not only theoretically proposed, but also numerically corroborated through two concrete examples of the Klein--Gordon lattice with a soft (Morse) and a hard ($ϕ^4$) potential. Compared to the case of the nonlinear Schr{ö}dinger lattice, the Krein signature of the internal mode relative to that of the wave continuum may change depending on the period of the excited breather state. For the periods for which the Krein signatures of the internal mode and the wave continuum coincide, excited breather states are observed to be nonlinearly stable.

nlin.PS↗