Searcharxiv⌕ Search

arXiv subjects

Dirk Hennig

Publications and source records attributed to Dirk Hennig.

At least 19 recordsLinked to original sources
arXiv2025

Discrete Nonlinear Schr\"odinger versus Ablowitz-Ladik: Existence and dynamics of generalized focusing NLS-type lattices over a nonzero background

The question of well-posedness of the generalized focusing Ablowitz-Ladik and Discrete Nonlinear Schr\"{o}dinger equations with \textit{nonzero} boundary conditions on the infinite lattice is far less understood than in the case where the models are supplemented with vanishing boundary conditions. This question remains largely unexplored even in the standard case of cubic nonlinearities in which, in particular, the Ablowitz-Ladik equation is completely integrable while the Discrete Nonlinear Schr\"{o}dinger equation is not (in contrast with its continuous counterpart). We establish local well-posedness for both of these generalized nonlinear systems supplemented with a broad class of nonzero boundary conditions and, in addition, derive analytical upper bounds for the minimal guaranteed lifespan of their solutions. These bounds depend explicitly on the norm of the initial data, the background, and the nonlinearity exponents. In particular, they suggest the possibility of finite-time collapse (blow-up) of solutions. Furthermore, by comparing models with different nonlinearity exponents, we prove estimates for the distance between their respective solutions (measured in suitable metrics), valid up to their common minimal guaranteed lifespan. Highly accurate numerical studies illustrate that solutions of the generalized Ablowitz-Ladik equation may collapse in finite time. Importantly, the numerically observed blow-up time is in excellent agreement with the theoretically predicted order of the minimal guaranteed lifespan. Furthermore, in the case of the Discrete Nonlinear Schr\"{o}dinger equation on a finite lattice we prove global existence of solutions; this is consistent with our numerical observations of the phenomenon of \textit{quasi-collapse}, manifested by narrow oscillatory spikes that nevertheless persist throughout time ... continued in pdf

math.AP↗
arXiv2024

Nonlinear discrete Schr\"odinger equations with a point defect

We study the $d$-dimensional discrete nonlinear Schr\"odinger equation with general power nonlinearity and a delta potential. Our interest lies in the interplay between two localization mechanisms. On the one hand, the attractive (repulsive) delta potential acting as a point defect breaks the translational invariance of the lattice so that a linear staggering (non-staggering) bound state is formed with negative (positive) energy. On the other hand, focusing nonlinearity may lead to self-trapping of excitation energy. For focusing nonlinearity we prove the existence of a spatially exponentially localized and time-periodic ground state and investigate the impact of an attractive respectively repulsive delta potential on the existence of an excitation threshold, i.e. supercritical $l^2$ norm, for the creation of such a ground states. Explicit expressions for the lower excitation thresholds are given. Reciprocally, we discuss the influence of defocusing nonlinearity on the durability of the linear bound states and provide upper thresholds of the $l^2-$norm for their preservation. Regarding the asymptotic behavior of the solutions we establish that for a $l^2-$norm below the excitation threshold the solutions scatter to a solution of the linear problem in $l^{p>2}$.

math.AP↗
arXiv2023

On the proximity between the wave dynamics of the integrable focusing nonlinear Schrödinger equation and its non-integrable generalizations

The question of whether features and behaviors that are characteristic to completely integrable systems persist in the transition to non-integrable settings is a central one in the field of nonlinear dispersive equations. In this work, we investigate this topic in the context of focusing nonlinear Schrödinger (NLS) equations. In particular, we consider non-integrable counterparts of the (integrable) focusing cubic NLS equation, which are distinct generalizations of cubic NLS and involve a broad class of nonlinearities, with the cases of power and saturable nonlinearities serving as illustrative examples. This is a notably different direction from the one explored in other works, where the non-integrable models considered are only small perturbations of the integrable one. We study the Cauchy problem on the real line for both vanishing and non-vanishing boundary conditions at infinity and quantify the proximity of solutions between the integrable and non-integrable models via estimates in appropriate metrics as well as pointwise. These results establish that the distance of solutions grows at most linearly with respect to time, while the growth rate of each solution is chiefly controlled by the size of the initial data and the nonlinearity parameters. A major implication of these closeness estimates is that integrable dynamics emerging from small initial conditions may persist in the non-integrable setting for significantly long times. In the case of zero boundary conditions at infinity, this persistence includes soliton and soliton collision dynamics, while in the case of nonzero boundary conditions at infinity, it establishes the nonlinear behavior of the non-integrable models at the early stages of the ubiquitous phenomenon of modulational instability. For this latter and more challenging type of boundary conditions, ... (full abstract in article)

nlin.PS↗
arXiv2023

Periodic travelling wave solutions of discrete nonlinear Klein-Gordon lattices

We prove the existence of periodic travelling wave solutions for general discrete nonlinear Klein-Gordon systems, considering both cases of hard and soft on-site potentials. In the case of hard on-site potentials we implement a fixed point theory approach, combining Schauder's fixed point theorem and the contraction mapping principle. This approach enables us to identify a ring in the energy space for non-trivial solutions to exist, energy (norm) thresholds for their existence and upper bounds on their velocity. In the case of soft on-site potentials, the proof of existence of periodic travelling wave solutions is facilitated by a variational approach based on the Mountain Pass Theorem. Thresholds on the averaged kinetic energy for these solutions to exist are also derived.

math.AP↗
arXiv2023

Exponentially stable breather solutions in nonautonomous dissipative nonlinear Schrödinger lattices

We consider damped and forced discrete nonlinear Schrödinger equations on the lattice $\mathbb{Z}$. First we establish the existence of periodic and quasiperiodic breather solutions for periodic and quasiperiodic driving, respectively. Notably, quasiperiodic breathers cannot exist in the system without damping and driving. Afterwards the existence of a global uniform attractor for the dissipative dynamics of the system is shown. For strong dissipation we prove that the global uniform attractor has finite fractal dimension and consists of a single trajectory that is confined to a finite dimensional subspace of the infinite dimensional phase space, attracting any bounded set in phase space exponentially fast. Conclusively, for strong damping and periodic (quasiperiodic) forcing the single periodic (quasiperiodic) breather solution possesses a finite number of modes and is exponentially stable.

math-ph↗
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

Existence of breathers in nonlinear Klein-Gordon lattices

We prove the existence of time-periodic solutions and spatially localised solutions (breathers), in general nonlinear Klein-Gordon infinite lattices. The existence problem is converted into a fixed point problem for an operator on some appropriate function space which is solved by means of Schauder's Fixed Point Theorem.

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

Existence of exponentially spatially localised breather solutions for lattices of nonlinearly coupled particles: Schauder's fixed point theorem approach

The problem of showing the existence of localised modes in nonlinear lattices has attracted considerable efforts from the physical but also from the mathematical viewpoint where a rich variety of methods has been employed. In this paper we prove that a fixed point theory approach based on the celebrated Schauder's Fixed Point Theorem may provide a general method to establish concisely not only the existence of localised structures but also a required rate of spatial localisation. As a case study we consider lattices of coupled particles with nonlinear nearest neighbour interaction and prove the existence of exponentially spatially localised breathers exhibiting either even-parity or odd-parity symmetry under necessary non-resonant conditions accompanied with the proof of energy bounds of the solutions.

math.AP↗
arXiv2021

Dynamics of nonlocal and local discrete Ginzburg-Landau equations: global attractors and their congruence

Discrete Ginzburg-Landau (DGL) equations with non-local nonlinearities have been established as significant inherently discrete models in numerous physical contexts, similar to their counterparts with local nonlinear terms. We study two prototypical examples of non-local and local DGLs on the one-dimensional infinite lattice. For the non-local DGL, we identify distinct scenarios for the asymptotic behavior of the globally existing in time solutions depending on certain parametric regimes. One of these scenarios is associated with a restricted compact attractor according to J. K. Hale's definition. We also prove the closeness of the solutions of the two models in the sense of a "continuous dependence on their initial data" in the $l^2$ metric under general conditions on the intrinsic linear gain or loss incorporated in the model. As a consequence of the closeness results, in the dissipative regime we establish the congruence of the attractors possessed by the semiflows of the non-local and of the local model respectively, for initial conditions in a suitable domain of attraction defined by the non-local system.

math.AP↗
arXiv2021

The closeness of localised structures between the Ablowitz-Ladik lattice and Discrete Nonlinear Schrödinger equations II: Generalised AL and DNLS systems

The Ablowitz-Ladik system, being one of the few integrable nonlinear lattices, admits a wide class of analytical solutions, ranging from exact spatially localised solitons to rational solutions in the form of the spatiotemporally localised discrete Peregrine soliton. Proving a closeness result between the solutions of the Ablowitz-Ladik and a wide class of Discrete Nonlinear Schrödinger systems in a sense of a continuous dependence on their initial data, we establish that such small amplitude waveforms may be supported in the nonintegrable lattices, for significant large times. The nonintegrable systems exhibiting such behavior include a generalisation of the Ablowitz-Ladik system with a power-law nonlinearity and the Discrete Nonlinear Schrödinger with power-law and saturable nonlinearities. The outcome of numerical simulations illustrates in an excellent agreement with the analytical results the persistence of small amplitude Ablowitz-Ladik analytical solutions in all the nonintegrable systems considered in this work, with the most striking example being that of the Peregine soliton.

nlin.PS↗
arXiv2021

Existence and congruence of global attractors for damped and forced integrable and nonintegrable discrete nonlinear Schrödinger equations

We study two damped and forced discrete nonlinear Schrödinger equations on the one-dimensional infinite lattice. Without damping and forcing they are represented by the integrable Ablowitz-Ladik equation (AL) featuring non-local cubic nonlinear terms, and its standard (nonintegrable) counterpart with local cubic nonlinear terms (DNLS). The global existence of a unique solution to the initial value problem for both, the damped and forced AL and DNLS, is proven. It is further shown that for sufficiently close initial data, their corresponding solutions stay close for all times. Concerning the asymptotic behaviour of the solutions to the damped and forced AL and DNLS, for the former a sufficient condition for the existence of a restricted global attractor is established while it is shown that the latter possesses a global attractor. Finally, we prove the congruence of the restricted global AL attractor and the DNLS attractor for dynamics ensuing from initial data contained in an appropriate bounded subset in a Banach space.

math.AP↗
arXiv2017

Periodic travelling wave solutions of discrete nonlinear Schrödinger equations

The existence of nonzero periodic travelling wave solutions for a general discrete nonlinear Schrödinger equation (DNLS) on finite one-dimensional lattices is proved. The DNLS features a general nonlinear term and variable range of interactions going beyond the usual nearest-neighbour interaction. The problem of the existence of travelling wave solutions is converted into a fixed point problem for an operator on some appropriate function space which is solved by means of Schauder's Fixed Point Theorem.

math.DS↗
arXiv2016

Existence of localised normal modes in nonlinear lattices

We prove the existence of exponentially localised and time-periodic solutions in general nonlinear Hamiltonian lattice systems. Like normal modes, these localised solutions are characterised by collective oscillations at the lattice sites with a uniform time-dependence. The proof of existence uses the comparison principle for differential equations to demonstrate that at each lattice site every half of the fundamental period of oscillations, contributing to localised solutions of the nonlinear lattice, is sandwiched between two oscillatory states of auxiliary linear equations. For soft (hard) on-site potentials the allowed frequencies of the in-phase (out-of-phase) localised periodic solutions lie below (above) the lower (upper) value of the linear spectrum of phonon frequencies. By varying a control parameter the exponential decay of the localised states can be tuned continuously.

nlin.PS↗
arXiv2015

Existence of breathing patterns in globally coupled finite-size nonlinear lattices

We prove the existence of time-periodic solutions consisting of patterns built up from two states, one with small amplitude and the other one with large amplitude, in general nonlinear Hamiltonian finite-size lattices with global coupling. Utilising the comparison principle for differential equations it is demonstrated that for a two site segment of the nonlinear lattice one can construct solutions that are sandwiched between two oscillatory localised lattice states. Subsequently, it is proven that such a localised state can be embedded in the extended nonlinear lattice forming a breathing pattern with a single site of large amplitude against a background of uniform small-amplitude states. Furthermore, it is demonstrated that spatial patterns are possible that are built up from any combination of the small-amplitude state and the large-amplitude state. It is shown that for soft (hard) on-site potentials the range of allowed frequencies of the in-phase (out-of-phase) breathing patterns extends to values below (above) the lower (upper) value of the bivalued degenerate linear spectrum of phonon frequencies.

nlin.PS↗
arXiv2015

Existence of nonlinear normal modes for coupled nonlinear oscillators

We prove the existence of nonlinear normal modes for general systems of two coupled nonlinear oscillators. Facilitating the comparison principle for ordinary differential equations it is shown that there exist exact solutions representing a vibration in unison of the system. The associated spatially localised time-periodic solutions feature out-of-phase and in-phase motion of the oscillators.

nlin.PS↗
arXiv2014

Modulational instability and resonant wave modes act on the metastability of oscillator chains

We describe the emergence and interactions of breather modes and resonant wave modes within a two-dimensional ring-like oscillator chain in a microcanonical situation. Our analytical results identify different dynamical regimes characterized by the potential dominance of either type of mode. The chain is initially placed in a meta-stable state which it can leave by passing over the brim of the applied Mexican-hat-like potential. We elucidate the influence of the different wave modes on the mean-first passage time. A central finding is that also in this complex potential landscape a fast noise-free escape scenario solely relying on nonlinear cooperative effects is accomplishable even in a low energy setting.

nlin.CD↗