Ricardo Chacón, Faustino Palmero, Pedro J. Martínez, Somnath Roy
An energy-based theory of autoresonance in driven dissipative chains of coupled generic oscillators is discussed on the basis of a variational principle concerning the energy functional. The theory is applied to chains of delayed Duffing-Ueda oscillators and the equations that together govern the autoresonance forces and solutions are derived and solved analytically for generic values of parameters and initial conditions, including the case of quenched time-delay disorder. Remarkably, the presence of retarded potentials with time-delayed feedback drastically modify the autoresonance scenario preventing the growth of the energy oscillation over specific regions of the parameter space. Additionally, effective harmonic forces with a slowly varying frequency are derived from the exact autoresonant excitations and the effectiveness of the theory is demonstrated at suppressing the chaos induced by homogeneous periodic excitations in such oscillator chains. Numerical experiments confirmed all the theoretical predictions.
{"title":"Energy-based theory of autoresonance in chains of coupled damped-driven generic oscillators","authors":"Ricardo Chacón, Faustino Palmero, Pedro J. Martínez, Somnath Roy","doi":"arxiv-2405.04556","DOIUrl":"https://doi.org/arxiv-2405.04556","url":null,"abstract":"An energy-based theory of autoresonance in driven dissipative chains of\u0000coupled generic oscillators is discussed on the basis of a variational\u0000principle concerning the energy functional. The theory is applied to chains of\u0000delayed Duffing-Ueda oscillators and the equations that together govern the\u0000autoresonance forces and solutions are derived and solved analytically for\u0000generic values of parameters and initial conditions, including the case of\u0000quenched time-delay disorder. Remarkably, the presence of retarded potentials\u0000with time-delayed feedback drastically modify the autoresonance scenario\u0000preventing the growth of the energy oscillation over specific regions of the\u0000parameter space. Additionally, effective harmonic forces with a slowly varying\u0000frequency are derived from the exact autoresonant excitations and the\u0000effectiveness of the theory is demonstrated at suppressing the chaos induced by\u0000homogeneous periodic excitations in such oscillator chains. Numerical\u0000experiments confirmed all the theoretical predictions.","PeriodicalId":501370,"journal":{"name":"arXiv - PHYS - Pattern Formation and Solitons","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938347","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sherzod R. Otajonov, Bakhram A. Umarov, Fatkhulla Kh. Abdullaev
In this work, we investigate the modulational instability of plane wave solutions within a modified Gross-Pitaevskii equation framework. The equation features cubic and quartic nonlinearity. It models the behaviour of quasi-one-dimensional Bose-Einstein condensates in symmetric Bose-Bose mixtures of ultra-dilute cold atoms. Our study demonstrates the pivotal role of the competition between mean-field attractions and quantum fluctuation-induced repulsions. This competition significantly affects the emergence and evolution of modulational instability. By employing linear stability analysis, we identify the essential conditions that lead to modulational instability. We find that the stability of plane wave solutions significantly depends on the interaction among system parameters. Further development of the instability leads to the fragmentation of the BEC into a chain of quantum droplets. We calculated the quantity of quantum droplets generated during the nonlinear phase of the instability. Our analytical results are corroborated by numerical simulations of the modified quasi-1D Gross-Pitaevskii equation. These simulations vividly depict the formation, interaction, and coalescence of droplets during the nonlinear phase of modulational instability. The investigation shows that linear stability analysis of the modified Gross-Pitaevskii equation, considering quantum fluctuations, precisely forecasts modulational instability phenomena across different domains of parameter spaces.
{"title":"Modulational instability in a quasi-one-dimensional Bose-Einstein condensates","authors":"Sherzod R. Otajonov, Bakhram A. Umarov, Fatkhulla Kh. Abdullaev","doi":"arxiv-2405.02282","DOIUrl":"https://doi.org/arxiv-2405.02282","url":null,"abstract":"In this work, we investigate the modulational instability of plane wave\u0000solutions within a modified Gross-Pitaevskii equation framework. The equation\u0000features cubic and quartic nonlinearity. It models the behaviour of\u0000quasi-one-dimensional Bose-Einstein condensates in symmetric Bose-Bose mixtures\u0000of ultra-dilute cold atoms. Our study demonstrates the pivotal role of the\u0000competition between mean-field attractions and quantum fluctuation-induced\u0000repulsions. This competition significantly affects the emergence and evolution\u0000of modulational instability. By employing linear stability analysis, we\u0000identify the essential conditions that lead to modulational instability. We\u0000find that the stability of plane wave solutions significantly depends on the\u0000interaction among system parameters. Further development of the instability\u0000leads to the fragmentation of the BEC into a chain of quantum droplets. We\u0000calculated the quantity of quantum droplets generated during the nonlinear\u0000phase of the instability. Our analytical results are corroborated by numerical\u0000simulations of the modified quasi-1D Gross-Pitaevskii equation. These\u0000simulations vividly depict the formation, interaction, and coalescence of\u0000droplets during the nonlinear phase of modulational instability. The\u0000investigation shows that linear stability analysis of the modified\u0000Gross-Pitaevskii equation, considering quantum fluctuations, precisely\u0000forecasts modulational instability phenomena across different domains of\u0000parameter spaces.","PeriodicalId":501370,"journal":{"name":"arXiv - PHYS - Pattern Formation and Solitons","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140885289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a tunable magnetoelastic lattice with a multistable onsite potential, focusing on a tristable potential. Through experimental and numerical analysis, we verify the existence of three types of transition waves with distinct amplitudes and velocities. Additionally, we establish the presence of a scaling law that elucidates various characteristics of these transition waves. By manipulating the onsite potential, we investigate the collision dynamics of two transition waves within the system. In chains featuring an asymmetric potential well, the collision of similar transition waves leads to the remote nucleation of a new phase. In chains with a symmetric potential well, the collision of dissimilar transition waves results in the formation of a stationary domain wall
{"title":"Remote Nucleation and Stationary Domain Walls via Transition Waves in Tristable Magnetoelastic Lattices","authors":"Anusree Ray, Samanvay Anand, Vivekanand Dabade, Rajesh Chaunsali","doi":"arxiv-2405.01168","DOIUrl":"https://doi.org/arxiv-2405.01168","url":null,"abstract":"We present a tunable magnetoelastic lattice with a multistable onsite\u0000potential, focusing on a tristable potential. Through experimental and\u0000numerical analysis, we verify the existence of three types of transition waves\u0000with distinct amplitudes and velocities. Additionally, we establish the\u0000presence of a scaling law that elucidates various characteristics of these\u0000transition waves. By manipulating the onsite potential, we investigate the\u0000collision dynamics of two transition waves within the system. In chains\u0000featuring an asymmetric potential well, the collision of similar transition\u0000waves leads to the remote nucleation of a new phase. In chains with a symmetric\u0000potential well, the collision of dissimilar transition waves results in the\u0000formation of a stationary domain wall","PeriodicalId":501370,"journal":{"name":"arXiv - PHYS - Pattern Formation and Solitons","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140828973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We examine the scattering of Ostrovsky wave packets, generated from an incident solitary wave, in a two layered waveguide with a delamination in the centre and soft (imperfect) bonding either side of the centre. The layers of the waveguide are assumed to consist of different materials, and the strains are described by a system of coupled Boussinesq equations. A semi-analytical approach consisting of matched asymptotic multiple-scale expansions is applied, leading to Ostrovsky equations in soft bonded regions and Korteweg-de Vries equations in the delaminated region. This semi-analytical method has good agreement with direct numerical simulations, validating the approach. In the delaminated regions, Ostrovsky wave packets evolve into a train of solitary waves, which subsequently evolve into Ostrovsky wave packets in the second bonded region. Analysis of the phase shift in the wave packet, introduced from the delaminated region, allows us to predict both the position and the length of the delamination; the first time this has been achieved using nonlinear waves. These results motivate experiments to validate the theoretical results, with the aim of creating a tool to monitor the integrity of layered structures.
{"title":"Delamination Detection in Layered Waveguides using Ostrovsky Wave Packets","authors":"J. S. Tamber, D. J. Chappell, M. R. Tranter","doi":"arxiv-2405.00388","DOIUrl":"https://doi.org/arxiv-2405.00388","url":null,"abstract":"We examine the scattering of Ostrovsky wave packets, generated from an\u0000incident solitary wave, in a two layered waveguide with a delamination in the\u0000centre and soft (imperfect) bonding either side of the centre. The layers of\u0000the waveguide are assumed to consist of different materials, and the strains\u0000are described by a system of coupled Boussinesq equations. A semi-analytical\u0000approach consisting of matched asymptotic multiple-scale expansions is applied,\u0000leading to Ostrovsky equations in soft bonded regions and Korteweg-de Vries\u0000equations in the delaminated region. This semi-analytical method has good\u0000agreement with direct numerical simulations, validating the approach. In the delaminated regions, Ostrovsky wave packets evolve into a train of\u0000solitary waves, which subsequently evolve into Ostrovsky wave packets in the\u0000second bonded region. Analysis of the phase shift in the wave packet,\u0000introduced from the delaminated region, allows us to predict both the position\u0000and the length of the delamination; the first time this has been achieved using\u0000nonlinear waves. These results motivate experiments to validate the theoretical\u0000results, with the aim of creating a tool to monitor the integrity of layered\u0000structures.","PeriodicalId":501370,"journal":{"name":"arXiv - PHYS - Pattern Formation and Solitons","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
I. Ioannou Sougleridis, O. Richoux, V. Achilleos G. Theocharis, D. J. Frantzeskakis
We study the propagation of both low- and high-amplitude ring-shaped sound waves in a 2D square lattice of acoustic waveguides with Helmholtz resonators. We show that the inclusion of the Helmholtz resonators suppresses the inherent anisotropy of the system in the low frequency regime allowing for radially symmetric solutions. By employing the electroacoustic analogue approach and asymptotic methods we derive an effective cylindrical Korteweg de Vries (cKdV) equation. Low-amplitude waveforms are self-similar structures of the Airy function profile, while high-amplitude ones are of the form of cylindrical solitons. Our analytical predictions are corroborated by results of direct numerical simulations, with a very good agreement between the two.
我们研究了低振幅和高振幅环形声波在带有亥姆霍兹谐振器的二维方形声波导管晶格中的传播。通过采用电声模拟方法和渐近方法，我们得出了一个有效的圆柱 Korteweg de Vries (cKdV) 方程。低振幅波形是艾里函数剖面的自相似结构，而高振幅波形则是圆柱孤子形式。我们的分析预测得到了直接数值模拟结果的证实，两者之间的吻合度非常高。
{"title":"Ring-Shaped Linear Waves and Solitons in a Square Lattice of Acoustic Waveguides","authors":"I. Ioannou Sougleridis, O. Richoux, V. Achilleos G. Theocharis, D. J. Frantzeskakis","doi":"arxiv-2404.18966","DOIUrl":"https://doi.org/arxiv-2404.18966","url":null,"abstract":"We study the propagation of both low- and high-amplitude ring-shaped sound\u0000waves in a 2D square lattice of acoustic waveguides with Helmholtz resonators.\u0000We show that the inclusion of the Helmholtz resonators suppresses the inherent\u0000anisotropy of the system in the low frequency regime allowing for radially\u0000symmetric solutions. By employing the electroacoustic analogue approach and\u0000asymptotic methods we derive an effective cylindrical Korteweg de Vries (cKdV)\u0000equation. Low-amplitude waveforms are self-similar structures of the Airy\u0000function profile, while high-amplitude ones are of the form of cylindrical\u0000solitons. Our analytical predictions are corroborated by results of direct\u0000numerical simulations, with a very good agreement between the two.","PeriodicalId":501370,"journal":{"name":"arXiv - PHYS - Pattern Formation and Solitons","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829148","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We examine the oscillon decay in a model with a single scalar field and two asymmetric fundamental states. Specifically, we verify how the evolution of an initially excited oscillon leads to the formation of antikink-kink pairs in the presence of a centered long-lived pulse. In such a context, the magnitude of the original perturbation stands for the most important factor influencing the process. For intermediary values of this magnitude, we find a transitory behavior which suggests that the energy exchange between translational and vibrational modes begins to play an important role in the overall evolution. As the magnitude increases, the centered pulse intermediates an interaction between the inner structures which compose the pairs whose behavior mimics that of a standard antikink-kink collision and therefore reinforces the role played by the energy transfer mechanism. For different values of width, an almost pure antikink-kink pair escapes to the infinity after multiple bounces.
{"title":"Generation of kink-antikink pairs through the excitation of an oscillon in the $φ^4$ model","authors":"Fabiano C. Simas, E. da Hora","doi":"arxiv-2404.17848","DOIUrl":"https://doi.org/arxiv-2404.17848","url":null,"abstract":"We examine the oscillon decay in a model with a single scalar field and two\u0000asymmetric fundamental states. Specifically, we verify how the evolution of an\u0000initially excited oscillon leads to the formation of antikink-kink pairs in the\u0000presence of a centered long-lived pulse. In such a context, the magnitude of\u0000the original perturbation stands for the most important factor influencing the\u0000process. For intermediary values of this magnitude, we find a transitory\u0000behavior which suggests that the energy exchange between translational and\u0000vibrational modes begins to play an important role in the overall evolution. As\u0000the magnitude increases, the centered pulse intermediates an interaction\u0000between the inner structures which compose the pairs whose behavior mimics that\u0000of a standard antikink-kink collision and therefore reinforces the role played\u0000by the energy transfer mechanism. For different values of width, an almost pure\u0000antikink-kink pair escapes to the infinity after multiple bounces.","PeriodicalId":501370,"journal":{"name":"arXiv - PHYS - Pattern Formation and Solitons","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829100","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Patrick Sprenger, Christopher Chong, Emmanuel Okyere, Michael Herrmann, P. G. Kevrekidis, Mark A. Hoefer
The Riemann problem for the discrete conservation law $2 dot{u}_n + u^2_{n+1} - u^2_{n-1} = 0$ is classified using Whitham modulation theory, a quasi-continuum approximation, and numerical simulations. A surprisingly elaborate set of solutions to this simple discrete regularization of the inviscid Burgers' equation is obtained. In addition to discrete analogues of well-known dispersive hydrodynamic solutions -- rarefaction waves (RWs) and dispersive shock waves (DSWs) -- additional unsteady solution families and finite time blow-up are observed. Two solution types exhibit no known conservative continuum correlates: (i) a counterpropagating DSW and RW solution separated by a symmetric, stationary shock and (ii) an unsteady shock emitting two counter-propagating periodic wavetrains with the same frequency connected to a partial DSW or a RW. Another class of solutions called traveling DSWs, (iii), consists of a partial DSW connected to a traveling wave comprised of a periodic wavetrain with a rapid transition to a constant. Portions of solutions (ii) and (iii) are interpreted as shock solutions of the Whitham modulation equations.
{"title":"Hydrodynamics of a Discrete Conservation Law","authors":"Patrick Sprenger, Christopher Chong, Emmanuel Okyere, Michael Herrmann, P. G. Kevrekidis, Mark A. Hoefer","doi":"arxiv-2404.16750","DOIUrl":"https://doi.org/arxiv-2404.16750","url":null,"abstract":"The Riemann problem for the discrete conservation law $2 dot{u}_n +\u0000u^2_{n+1} - u^2_{n-1} = 0$ is classified using Whitham modulation theory, a\u0000quasi-continuum approximation, and numerical simulations. A surprisingly\u0000elaborate set of solutions to this simple discrete regularization of the\u0000inviscid Burgers' equation is obtained. In addition to discrete analogues of\u0000well-known dispersive hydrodynamic solutions -- rarefaction waves (RWs) and\u0000dispersive shock waves (DSWs) -- additional unsteady solution families and\u0000finite time blow-up are observed. Two solution types exhibit no known\u0000conservative continuum correlates: (i) a counterpropagating DSW and RW solution\u0000separated by a symmetric, stationary shock and (ii) an unsteady shock emitting\u0000two counter-propagating periodic wavetrains with the same frequency connected\u0000to a partial DSW or a RW. Another class of solutions called traveling DSWs,\u0000(iii), consists of a partial DSW connected to a traveling wave comprised of a\u0000periodic wavetrain with a rapid transition to a constant. Portions of solutions\u0000(ii) and (iii) are interpreted as shock solutions of the Whitham modulation\u0000equations.","PeriodicalId":501370,"journal":{"name":"arXiv - PHYS - Pattern Formation and Solitons","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140804163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The study of higher order interactions in the dynamics of Kuramoto oscillators has been a topic of intense recent research. Arguments based on dimensional reduction using the Ott-Antonsen ansatz show that such interactions usually facilitate synchronization, giving rise to bi-stability and hysteresis. Here we show that three body interactions shift the critical coupling for synchronization towards higher values in all dimensions, except D=2, where a cancellation occurs. After the transition, three and four body interactions combine to facilitate synchronization. We show simulations in D=3 and 4 to illustrate the dynamics.
{"title":"Third order interactions shift the critical coupling in multidimensional Kuramoto models","authors":"Ricardo Fariello, Marcus A. M. de Aguiar","doi":"arxiv-2404.16715","DOIUrl":"https://doi.org/arxiv-2404.16715","url":null,"abstract":"The study of higher order interactions in the dynamics of Kuramoto\u0000oscillators has been a topic of intense recent research. Arguments based on\u0000dimensional reduction using the Ott-Antonsen ansatz show that such interactions\u0000usually facilitate synchronization, giving rise to bi-stability and hysteresis.\u0000Here we show that three body interactions shift the critical coupling for\u0000synchronization towards higher values in all dimensions, except D=2, where a\u0000cancellation occurs. After the transition, three and four body interactions\u0000combine to facilitate synchronization. We show simulations in D=3 and 4 to\u0000illustrate the dynamics.","PeriodicalId":501370,"journal":{"name":"arXiv - PHYS - Pattern Formation and Solitons","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140804155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mikheil Kharbedia, Niccolò Caselli, Macarena Calero, Lara H. Moleiro, Jesús F. Castillo, José A. Santiago, Diego Herráez-Aguilar, Francisco Monroy
Surface stiffnesses engender steady patterns of Faraday waves (FWs), so called hydrodynamic crystals as correspond to ordered wave lattices made of discrete subharmonics under monochromatic driving. Mastering rules are both inertia-imposed parametric resonance for frequency-halving together with rigidity-driven nonlinearity for wavefield self-focusing. They harness the discretization needed for coherent FW-packets to localize in space and time. Collective lattice excitations are observed as dispersionless propagating dislocations that lead periodic modulations arising from explicit symmetry breaking. In a field theory perspective, a halving genesis for the collective distorting modes is revealed as the natural pathway for hydrodynamic crystal melting.
{"title":"Collective lattice excitations in the dynamic route for melting hydrodynamic 2D-crystals","authors":"Mikheil Kharbedia, Niccolò Caselli, Macarena Calero, Lara H. Moleiro, Jesús F. Castillo, José A. Santiago, Diego Herráez-Aguilar, Francisco Monroy","doi":"arxiv-2404.15912","DOIUrl":"https://doi.org/arxiv-2404.15912","url":null,"abstract":"Surface stiffnesses engender steady patterns of Faraday waves (FWs), so\u0000called hydrodynamic crystals as correspond to ordered wave lattices made of\u0000discrete subharmonics under monochromatic driving. Mastering rules are both\u0000inertia-imposed parametric resonance for frequency-halving together with\u0000rigidity-driven nonlinearity for wavefield self-focusing. They harness the\u0000discretization needed for coherent FW-packets to localize in space and time.\u0000Collective lattice excitations are observed as dispersionless propagating\u0000dislocations that lead periodic modulations arising from explicit symmetry\u0000breaking. In a field theory perspective, a halving genesis for the collective\u0000distorting modes is revealed as the natural pathway for hydrodynamic crystal\u0000melting.","PeriodicalId":501370,"journal":{"name":"arXiv - PHYS - Pattern Formation and Solitons","volume":"92 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140804098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The computation of the amplitude, $alpha$, of asymptotic standing wave tails of weakly delocalized, stationary solutions in a fifth-order Korteweg-de Vries equation is revisited. Assuming the coefficient of the fifth order derivative term, $epsilon^2ll1$, a new derivation of the ``beyond all orders in $epsilon$'' amplitude, $alpha$, is presented. It is shown by asymptotic matching techniques, extended to higher orders in $epsilon$, that the value of $alpha$ can be obtained from the asymmetry at the center of the unique solution exponentially decaying in one direction. This observation, complemented by some fundamental results of Hammersley and Mazzarino [Proc. R. Soc. Lond. A 424, 19 (1989)], not only sheds new light on the computation of $alpha$, but also greatly facilitates its numerical determination to a remarkable precision for so small values of $epsilon$, which are beyond the capabilities of standard numerical methods.
{"title":"A new derivation of the amplitude of asymptotic oscillatory tails of weakly delocalized solitons","authors":"Gyula Fodor, Péter Forgács, Muneeb Mushtaq","doi":"arxiv-2404.15020","DOIUrl":"https://doi.org/arxiv-2404.15020","url":null,"abstract":"The computation of the amplitude, $alpha$, of asymptotic standing wave tails\u0000of weakly delocalized, stationary solutions in a fifth-order Korteweg-de Vries\u0000equation is revisited. Assuming the coefficient of the fifth order derivative\u0000term, $epsilon^2ll1$, a new derivation of the ``beyond all orders in\u0000$epsilon$'' amplitude, $alpha$, is presented. It is shown by asymptotic\u0000matching techniques, extended to higher orders in $epsilon$, that the value of\u0000$alpha$ can be obtained from the asymmetry at the center of the unique\u0000solution exponentially decaying in one direction. This observation,\u0000complemented by some fundamental results of Hammersley and Mazzarino [Proc. R.\u0000Soc. Lond. A 424, 19 (1989)], not only sheds new light on the computation of\u0000$alpha$, but also greatly facilitates its numerical determination to a\u0000remarkable precision for so small values of $epsilon$, which are beyond the\u0000capabilities of standard numerical methods.","PeriodicalId":501370,"journal":{"name":"arXiv - PHYS - Pattern Formation and Solitons","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140806618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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