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Articles 1 - 28 of 28
Full-Text Articles in Dynamical Systems
Symmetry-Breaking Bifurcation In The Nonlinear Schrödinger Equation With Symmetric Potentials, E. Kirr, Panos Kevrekidis, D. E. Pelinovsky
Symmetry-Breaking Bifurcation In The Nonlinear Schrödinger Equation With Symmetric Potentials, E. Kirr, Panos Kevrekidis, D. E. Pelinovsky
Panos Kevrekidis
We consider the focusing (attractive) nonlinear Schr\"odinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the …
Multiple Dark-Bright Solitons In Atomic Bose-Einstein Condensates, D. Yan, J. J. Chang, C. Hamner, Panos Kevrekidis, P. Engels, V. Achilleos, D. J. Frantzeskakis, R. Carretero-Gonz´Alez, P. Schmelcher
Multiple Dark-Bright Solitons In Atomic Bose-Einstein Condensates, D. Yan, J. J. Chang, C. Hamner, Panos Kevrekidis, P. Engels, V. Achilleos, D. J. Frantzeskakis, R. Carretero-Gonz´Alez, P. Schmelcher
Panos Kevrekidis
Motivated by recent experimental results, we present a systematic theoretical analysis of dark-bright-soliton interactions and multiple-dark-bright-soliton complexes in atomic two-component Bose-Einstein condensates. We study analytically the interactions between two dark-bright solitons in a homogeneous condensate and then extend our considerations to the presence of the trap. We illustrate the existence of robust stationary dark-bright-soliton “molecules,” composed of two or more solitons, which are formed due to the competition of the interaction forces between the dark- and bright-soliton components and the trap force. Our analysis is based on an effective equation of motion, derived for the distance between two dark-bright solitons. …
Breathers In Oscillator Chains With Hertzian Interactions, Guillaume James, Panos Kevrekidis, Jesus Cuevas
Breathers In Oscillator Chains With Hertzian Interactions, Guillaume James, Panos Kevrekidis, Jesus Cuevas
Panos Kevrekidis
We prove nonexistence of breathers (spatially localized and time-periodic oscillations) for a class of Fermi-Pasta-Ulam lattices representing an uncompressed chain of beads interacting via Hertz's contact forces. We then consider the setting in which an additional on-site potential is present, motivated by the Newton's cradle under the effect of gravity. Using both direct numerical computations and a simplified asymptotic model of the oscillator chain, the so-called discrete p-Schr\"odinger (DpS) equation, we show the existence of discrete breathers and study their spectral properties and mobility. Due to the fully nonlinear character of Hertzian interactions, breathers are found to be much more …
Rabi Flopping Induces Spatial Demixing Dynamics, E. Nicklas, H. Strobel, T. Zibold, C. Gross, B. A. Malomed, Panos Kevrekidis, M. K. Oberthaler
Rabi Flopping Induces Spatial Demixing Dynamics, E. Nicklas, H. Strobel, T. Zibold, C. Gross, B. A. Malomed, Panos Kevrekidis, M. K. Oberthaler
Panos Kevrekidis
We experimentally investigate the mixing and demixing dynamics of Bose-Einstein condensates in the presence of a linear coupling between two internal states. The observed amplitude reduction of the Rabi oscillations can be understood as a result of demixing dynamics of dressed states as experimentally confirmed by reconstructing the spatial profile of dressed state amplitudes. The observations are in quantitative agreement with numerical integration of coupled Gross-Pitaevskii equations without free parameters, which also reveals the criticality of the dynamics on the symmetry of the system. Our observations demonstrate new possibilities for changing effective atomic interactions and studying critical phenomena.
Fluctuating And Dissipative Dynamics Of Dark Solitons In Quasicondensates S., S. P. Cockburn, H. E. Nistazakis, T. P. Horikis, Panos Kevrekidis, N. P. Proukakis, D. J. Frantzeskakis
Fluctuating And Dissipative Dynamics Of Dark Solitons In Quasicondensates S., S. P. Cockburn, H. E. Nistazakis, T. P. Horikis, Panos Kevrekidis, N. P. Proukakis, D. J. Frantzeskakis
Panos Kevrekidis
The fluctuating and dissipative dynamics of matter-wave dark solitons within harmonically trapped, partially condensed Bose gases is studied both numerically and analytically. A study of the stochastic Gross-Pitaevskii equation, which correctly accounts for density and phase fluctuations at finite temperatures, reveals dark-soliton decay times to be lognormally distributed at each temperature, thereby characterizing the previously predicted long-lived soliton trajectories within each ensemble of numerical realizations [ S. P. Cockburn et al. Phys. Rev. Lett. 104 174101 (2010)]. Expectation values for the average soliton lifetimes extracted from these distributions are found to agree well with both numerical and analytic predictions based …
Dark–Bright Ring Solitons In Bose–Einstein Condensates, J. Stockhofe, Panos Kevrekidis, D. J. Frantzeskakis, P. Schmelcher
Dark–Bright Ring Solitons In Bose–Einstein Condensates, J. Stockhofe, Panos Kevrekidis, D. J. Frantzeskakis, P. Schmelcher
Panos Kevrekidis
We study dark–bright (DB) ring solitons in two-component Bose–Einstein condensates. In the limit of large densities of the dark component, we describe the soliton dynamics by means of an equation of motion for the ring radius. The presence of the bright, 'filling' species is demonstrated to have a stabilizing effect on the ring dark soliton. Near the linear limit, we discuss the symmetry-breaking bifurcations of DB soliton stripes and vortex-bright soliton clusters from the DB ring and relate the stabilizing effect of filling to changes in the bifurcation diagram. Finally, we show that the stabilization by means of a second …
Stationary States Of A Nonlinear Schrödinger Lattice With A Harmonic Trap, V. Achilleos, G. Theocharis, Panos Kevrekidis, N. I. Karachalios, F. K. Diakonos, D. J. Frantzeskakis
Stationary States Of A Nonlinear Schrödinger Lattice With A Harmonic Trap, V. Achilleos, G. Theocharis, Panos Kevrekidis, N. I. Karachalios, F. K. Diakonos, D. J. Frantzeskakis
Panos Kevrekidis
We study a discrete nonlinear Schrödinger lattice with a parabolic trapping potential. The model, describing, e.g., an array of repulsive Bose-Einstein condensate droplets confined in the wells of an optical lattice, is analytically and numerically investigated. Starting from the linear limit of the problem, we use global bifurcation theory to rigorously prove that – in the discrete regime – all linear states lead to nonlinear generalizations thereof, which assume the form of a chain of discrete dark solitons (as the density increases). The stability of the ensuing nonlinear states is studied and it is found that the ground state is …
Emergence And Stability Of Vortex Clusters In Bose-Einstein Condensates: A Bifurcation Approach Near The Linear Limit, S. Middelkamp, Panos Kevrekidis, D. J. Frantzeskakis, R. Carretero-Gonzalez, P. Schmelcher
Emergence And Stability Of Vortex Clusters In Bose-Einstein Condensates: A Bifurcation Approach Near The Linear Limit, S. Middelkamp, Panos Kevrekidis, D. J. Frantzeskakis, R. Carretero-Gonzalez, P. Schmelcher
Panos Kevrekidis
We study the existence and stability properties of clusters of alternating charge vortices in Bose-Einstein condensates. It is illustrated that such states emerge from cascades of symmetry-breaking bifurcations that can be analytically tracked near the linear limit of the system via weakly nonlinear few-mode expansions. We present the resulting states that emerge near the first few eigenvalues of the linear limit, and illustrate how the nature of the bifurcations can be used to understand their stability. Rectilinear, polygonal and diagonal vortex clusters are only some of the obtained states while mixed states, consisting of dark solitons and vortex clusters, are …
Discrete Breathers In A Nonlinear Electric Line: Modeling, Computation, And Experiment, F. Palmero, L. Q. English, J. Cuevas, R. Carretero-Gonz´Alez, Panos Kevrekidis
Discrete Breathers In A Nonlinear Electric Line: Modeling, Computation, And Experiment, F. Palmero, L. Q. English, J. Cuevas, R. Carretero-Gonz´Alez, Panos Kevrekidis
Panos Kevrekidis
We study experimentally and numerically the existence and stability properties of discrete breathers in a periodic nonlinear electric line. The electric line is composed of single cell nodes, containing a varactor diode and an inductor, coupled together in a periodic ring configuration through inductors and driven uniformly by a harmonic external voltage source. A simple model for each cell is proposed by using a nonlinear form for the varactor characteristics through the current and capacitance dependence on the voltage. For an electrical line composed of 32 elements, we find the regions, in driver voltage and frequency, where n-peaked breather solutions …
Dynamics Of Vortex Dipoles In Confined Bose-Einstein Condensates, P. J. Torres, Panos Kevrekidis, D. J. Frantzeskakis, R. Carretero-Gonzalez, P. Schmelcher, D. S. Hall
Dynamics Of Vortex Dipoles In Confined Bose-Einstein Condensates, P. J. Torres, Panos Kevrekidis, D. J. Frantzeskakis, R. Carretero-Gonzalez, P. Schmelcher, D. S. Hall
Panos Kevrekidis
We present a systematic theoretical analysis of the motion of a pair of straight counter-rotating vortex lines within a trapped Bose-Einstein condensate. We introduce the dynamical equations of motion, identify the associated conserved quantities, and illustrate the integrability of the ensuing dynamics. The system possesses a stationary equilibrium as a special case in a class of exact solutions that consist of rotating guiding-center equilibria about which the vortex lines execute periodic motion; thus, the generic two-vortex motion can be classified as quasi-periodic. We conclude with an analysis of the linear and nonlinear stability of these stationary and rotating equilibria.
Guiding-Center Dynamics Of Vortex Dipoles In Bose-Einstein Condensates, S. Middelkamp, P. J. Torres, Panos Kevrekidis, D. J. Frantzeskakis, R. Carretero-Gonz´Alez, P. Schmelcher, D. V. Freilich, D. S. Hall
Guiding-Center Dynamics Of Vortex Dipoles In Bose-Einstein Condensates, S. Middelkamp, P. J. Torres, Panos Kevrekidis, D. J. Frantzeskakis, R. Carretero-Gonz´Alez, P. Schmelcher, D. V. Freilich, D. S. Hall
Panos Kevrekidis
A quantized vortex dipole is the simplest vortex molecule, comprising two countercirculating vortex lines in a superfluid. Although vortex dipoles are endemic in two-dimensional superfluids, the precise details of their dynamics have remained largely unexplored. We present here several striking observations of vortex dipoles in dilute-gas Bose-Einstein condensates, and develop a vortex-particle model that generates vortex line trajectories that are in good agreement with the experimental data. Interestingly, these diverse trajectories exhibit essentially identical quasiperiodic behavior, in which the vortex lines undergo stable epicyclic orbits.
Variational Approximations In Discrete Nonlinear Schrödinger Equations With Next-Nearest-Neighbor Couplings, Panos Kevrekidis, C. Chong, R. Carretero-González, B. A. Malomed
Variational Approximations In Discrete Nonlinear Schrödinger Equations With Next-Nearest-Neighbor Couplings, Panos Kevrekidis, C. Chong, R. Carretero-González, B. A. Malomed
Panos Kevrekidis
Solitons of a discrete nonlinear Schr\"{o}dinger equation which includes the next-nearest-neighbor interactions are studied by means of a variational approximation and numerical computations. A large family of multi-humped solutions, including those with a nontrivial phase structure which are a feature particular to the next-nearest-neighbor interaction model, are accurately predicted by the variational approximation. Bifurcations linking solutions with the trivial and nontrivial phase structures are also captured remarkably well, including a prediction of critical parameter values.
Nonlinear Excitations, Stability Inversions, And Dissipative Dynamics In Quasi-One-Dimensional Polariton Condensates, J. Cuevas, A. S. Rodrigues, R. Carretero-Gonz´Alez, Panos Kevrekidis, D. J. Frantzeskakis
Nonlinear Excitations, Stability Inversions, And Dissipative Dynamics In Quasi-One-Dimensional Polariton Condensates, J. Cuevas, A. S. Rodrigues, R. Carretero-Gonz´Alez, Panos Kevrekidis, D. J. Frantzeskakis
Panos Kevrekidis
We study the existence, stability, and dynamics of the ground state and nonlinear excitations, in the form of dark solitons, for a quasi-one-dimensional polariton condensate in the presence of nonresonant pumping and nonlinear damping. We find a series of remarkable features that can be directly contrasted to the case of the typically energy-conserving ultracold alkali-atom Bose-Einstein condensates. For some sizable parameter ranges, the nodeless (“ground”) state becomes unstabletoward the formation of stable nonlinear single- or multi-dark-soliton excitations. It is also observed that for suitable parametric choices, the instability of single dark solitons can nucleate multi-dark-soliton states. Also, for other parametric …
Pt-Symmetric Oligomers: Analytical Solutions, Linear Stability, And Nonlinear Dynamics, K. Li, Panos Kevrekidis
Pt-Symmetric Oligomers: Analytical Solutions, Linear Stability, And Nonlinear Dynamics, K. Li, Panos Kevrekidis
Panos Kevrekidis
In the present work we focus on the case of (few-site) configurations respecting the parity-time (PT) symmetry, i.e., with a spatially odd gain-loss profile. We examine the case of such “oligomers” with not only two sites, as in earlier works, but also the cases of three and four sites. While in the former case of recent experimental interest the picture of existing stationary solutions and their stability is fairly straightforward, the latter cases reveal a considerable additional complexity of solutions, including ones that exist past the linear PT-symmetry breaking point in the case of the trimer, and symmetry-breaking bifurcations, as …
Dark-Bright Gap Solitons In Coupled-Mode One-Dimensional Saturable Waveguide Arrays, Rong Dong, Christian E. Ruter, Detlef Kip, Jesus Cuevas, Panos Kevrekidis, Daohong Song, Jingjun Xu
Dark-Bright Gap Solitons In Coupled-Mode One-Dimensional Saturable Waveguide Arrays, Rong Dong, Christian E. Ruter, Detlef Kip, Jesus Cuevas, Panos Kevrekidis, Daohong Song, Jingjun Xu
Panos Kevrekidis
In the present work, we consider the dynamics of dark solitons as one mode of a defocusing photorefractive lattice coupled with bright solitons as a second mode of the lattice. Our investigation is motivated by an experiment that illustrates that such coupled states can exist with both components in the first gap of the linear band spectrum. This finding is further extended by the examination of different possibilities from a theoretical perspective, such as symbiotic ones where the bright component is supported by states of the dark component in the first or second gap, or nonsymbiotic ones where the bright …
Nonlinear Waves In Lattices: Past, Present, Future, Panos Kevrekidis
Nonlinear Waves In Lattices: Past, Present, Future, Panos Kevrekidis
Panos Kevrekidis
In the present work, we attempt a brief summary of various areas where non-linear waves have been emerging in the phenomenology of lattice dynamical systems. These areas include non-linear optics, atomic physics, mechanical systems, electrical lattices, non-linear metamaterials, plasma dynamics and granular crystals. We give some of the recent developments in each one of these areas and speculate on some of the potentially interesting directions for future study.
Symmetry Breaking, Coupling Management, And Localized Modes In Dual-Core Discrete Nonlinear-Schrödinger Lattices, H. Susanto, Panos Kevrekidis, F. Kh. Abdullaev, Boris A. Malomed
Symmetry Breaking, Coupling Management, And Localized Modes In Dual-Core Discrete Nonlinear-Schrödinger Lattices, H. Susanto, Panos Kevrekidis, F. Kh. Abdullaev, Boris A. Malomed
Panos Kevrekidis
We introduce a system of two linearly coupled discrete nonlinear Schr\"{o}dinger equations (DNLSEs), with the coupling constant subject to a rapid temporal modulation. The model can be realized in bimodal Bose-Einstein condensates (BEC). Using an averaging procedure based on the multiscale method, we derive a system of averaged (autonomous) equations, which take the form of coupled DNLSEs with additional nonlinear coupling terms of the four-wave-mixing type. We identify stability regions for fundamental onsite discrete symmetric solitons (single-site modes with equal norms in both components), as well as for two-site in-phase and twisted modes, the in-phase ones being completely unstable. The …
Quasidiscrete Microwave Solitons In A Split-Ring-Resonator-Based Left-Handed Coplanar Waveguide, G. P. Veldes, J. Cuevas, Panos Kevrekidis, D. J. Frantzeskakis
Quasidiscrete Microwave Solitons In A Split-Ring-Resonator-Based Left-Handed Coplanar Waveguide, G. P. Veldes, J. Cuevas, Panos Kevrekidis, D. J. Frantzeskakis
Panos Kevrekidis
We study the propagation of quasidiscrete microwave solitons in a nonlinear left-handed coplanar waveguide coupled with split-ring resonators. By considering the relevant transmission line analog, we derive a nonlinear lattice model which is studied analytically by means of a quasidiscrete approximation. We derive a nonlinear Schrödinger equation, and find that the system supports bright envelope soliton solutions in a relatively wide subinterval of the left-handed frequency band. We perform systematic numerical simulations, in the framework of the nonlinear lattice model, to study the propagation properties of the quasidiscrete microwave solitons. Our numerical findings are in good agreement with the analytical …
Tunable Vibrational Band Gaps In One-Dimensional Diatomic Granular Crystals With Three-Particle Unit Cells, N. Boechler, J. Yang, G. Theocharis, Panos Kevrekidis, C. Daraio
Tunable Vibrational Band Gaps In One-Dimensional Diatomic Granular Crystals With Three-Particle Unit Cells, N. Boechler, J. Yang, G. Theocharis, Panos Kevrekidis, C. Daraio
Panos Kevrekidis
We investigate the tunable vibration filtering properties of statically compressed one-dimensional diatomic granular crystals composed of arrays of stainless steel spheres and cylinders interacting via Hertzian contact. The arrays consist of periodically repeated three-particle unit cells (sphere-cylinder-sphere) in which the length of the cylinder is varied systematically. We investigate the response of these granular crystals, given small amplitude dynamic displacements relative to those due to the static compression, and characterize their linear frequency spectrum. We find good agreement between theoretical dispersion relation analysis (for infinite systems), state-space analysis (for finite systems), and experiments. We report the observation of three distinct …
Dark-Bright Discrete Solitons: A Numerical Study Of Existence, Stability And Dynamics, A. Alvarez, J. Cuevas, F. R. Romero, Panos Kevrekidis
Dark-Bright Discrete Solitons: A Numerical Study Of Existence, Stability And Dynamics, A. Alvarez, J. Cuevas, F. R. Romero, Panos Kevrekidis
Panos Kevrekidis
In the present work, we numerically explore the existence and stability properties of different types of configurations of dark-bright solitons, dark-bright soliton pairs and pairs of dark-bright and dark solitons in discrete settings, starting from the anti-continuum limit. We find that while single discrete dark-bright solitons have similar stability properties to discrete dark solitons, their pairs may only be stable if the bright components are in phase and are always unstable if the bright components are out of phase. Pairs of dark-bright solitons with dark ones have similar stability properties as individual dark or dark-bright ones. Lastly, we consider collisions …
Radial Standing And Self-Similar Waves For The Hyperbolic Cubic Nls In 2d, Panos Kevrekidis, Andrea R. Nahmod, Chongchun Zeng
Radial Standing And Self-Similar Waves For The Hyperbolic Cubic Nls In 2d, Panos Kevrekidis, Andrea R. Nahmod, Chongchun Zeng
Panos Kevrekidis
In this note we propose a new set of coordinates to study the hyperbolic or non-elliptic cubic nonlinear Schrodinger equation in two dimensions. Based on these coordinates, we study the existence of bounded and continuous hyperbolically radial standing waves, as well as hyperbolically radial self-similar solutions. Many of the arguments can easily be adapted to more general nonlinearities.
Control Of The Symmetry Breaking In Double-Well Potentials By The Resonant Nonlinearity Management, H. E. Nistazakis, B. A. Malomed, Panos Kevrekidis, D. J. Frantzeskakis
Control Of The Symmetry Breaking In Double-Well Potentials By The Resonant Nonlinearity Management, H. E. Nistazakis, B. A. Malomed, Panos Kevrekidis, D. J. Frantzeskakis
Panos Kevrekidis
We introduce a one-dimensional model of Bose–Einstein condensates (BECs), combining the double-well potential, which is a usual setting for the onset of spontaneous-symmetry-breaking (SSB) effects, and time-periodic modulation of the nonlinearity, which may be implemented by means of the Feshbach-resonance-management (FRM) technique. Both cases of the nonlinearity that is repulsive or attractive on the average are considered. In the former case, the main effect produced by the application of the FRM is spontaneous self-trapping of the condensate in either of the two potential wells in parameter regimes where it would remain untrapped in the absence of the management. In the …
Impact Of Anisotropy On Vortex Clusters And Their Dynamics, J. Stockhofe, S. Middelkamp, Panos Kevrekidis, P. Schmelcher
Impact Of Anisotropy On Vortex Clusters And Their Dynamics, J. Stockhofe, S. Middelkamp, Panos Kevrekidis, P. Schmelcher
Panos Kevrekidis
We investigate the effects of anisotropy on the stability and dynamics of vortex cluster states which arise in Bose-Einstein condensates. Sufficiently strong anisotropies are shown to stabilize states with arbitrary numbers of vortices that are highly unstable in the isotropic limit. Conversely, anisotropy can be used to destabilize states which are stable in the isotropic limit. Near the linear limit, we identify the bifurcations of vortex states including their emergence from linear eigenstates, while in the strongly nonlinear limit, a particle-like description of the dynamics of the vortices in the anisotropic trap is developed. Both are in very good agreement …
Termodynamika Procesowa (Dla Me Aparatura Procesowa) Ćw., Wojciech M. Budzianowski
Termodynamika Procesowa (Dla Me Aparatura Procesowa) Ćw., Wojciech M. Budzianowski
Wojciech Budzianowski
No abstract provided.
The Analysis Of Heat Transfer In A Gas-Gas Heat Exchanger Operated Under A Heat-Recirculating Mode, Mariusz Salaniec, Wojciech M. Budzianowski
The Analysis Of Heat Transfer In A Gas-Gas Heat Exchanger Operated Under A Heat-Recirculating Mode, Mariusz Salaniec, Wojciech M. Budzianowski
Wojciech Budzianowski
The present paper presents the analysis of heat transfer in a gas-gas heat exchanger operated in a heat-recirculating mode.
An Overview Of Technologies For Upgrading Of Biogas To Biomethane, Wojciech M. Budzianowski
An Overview Of Technologies For Upgrading Of Biogas To Biomethane, Wojciech M. Budzianowski
Wojciech Budzianowski
The present contribution presents an overview of technologies available for upgrading of biogas to biomethane. Technologies under study include pressure swing adsorption (PSA), high-pressure water wash (HPWW), reactive absorption (RA), physical absorption (PA), membrane separation (MS) and cryogenic separation (CS).
Influence Of Energy Policy On The Rate Of Implementation Of Biogas Power Plants In Germany During The 2001-2010 Decade, Izabela Chasiak, Wojciech M. Budzianowski
Influence Of Energy Policy On The Rate Of Implementation Of Biogas Power Plants In Germany During The 2001-2010 Decade, Izabela Chasiak, Wojciech M. Budzianowski
Wojciech Budzianowski
The current article describes energy policy tools, which caused intensive development of biogas-based power generation in Germany during the 2001-2010 decade. The German system of financial support to biogas power plants is presented in details. It is shown that in Germany, i.e. in a country characterised by similar climate and potentials to renewable energy to Poland, biogas power plants cover 10,7% of electricity demands in 2010, while all renewable energy sources cover only 5,4% of electricity demands. It is emphasised that under favourable Polish energy policy, the development of biogas energy can be very rapid.
On The Existence Of Solitary Traveling Waves For Generalized Hertzian Chains, Panos Kevrekidis, Atanas Stefanov
On The Existence Of Solitary Traveling Waves For Generalized Hertzian Chains, Panos Kevrekidis, Atanas Stefanov
Panos Kevrekidis
We consider the question of existence of “bell-shaped” (i.e. non-increasing for x > 0 and non-decreasing for x < 0) traveling waves for the strain variable of the generalized Hertzian model describing, in the special case of a p = 3/2 exponent, the dynamics of a granular chain. The proof of existence of such waves is based on the English and Pego [Proceedings of the AMS 133, 1763 (2005)] formulation of the problem. More specifically, we construct an appropriate energy functional, for which we show that the constrained minimization problem over bell-shaped entries has a solution. We also provide an alternative proof of the Friesecke-Wattis result [Comm. Math. Phys 161, 394 (1994)], by using the same approach (but where the minimization is not constrained over bell-shaped curves). We briefly discuss and illustrate numerically the implications on the doubly exponential decay properties of the waves, as well as touch upon the modifications of these properties in the presence of a finite precompression force in the model.