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Localised nonlinear modes in the PT-symmetric double-delta well Gross-Pitaevskii equation pt对称双delta井Gross-Pitaevskii方程的局部非线性模态
arXiv: Pattern Formation and Solitons Pub Date : 2015-11-20 DOI: 10.1007/978-3-319-31356-6_8
I. V. Barashenkov, D. Zezyulin
{"title":"Localised nonlinear modes in the PT-symmetric double-delta well Gross-Pitaevskii equation","authors":"I. V. Barashenkov, D. Zezyulin","doi":"10.1007/978-3-319-31356-6_8","DOIUrl":"https://doi.org/10.1007/978-3-319-31356-6_8","url":null,"abstract":"","PeriodicalId":154515,"journal":{"name":"arXiv: Pattern Formation and Solitons","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115202380","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}
引用次数: 8
A nonlinear generalization of the Camassa-Holm equation with peakon solutions 带峰值解的Camassa-Holm方程的非线性推广
arXiv: Pattern Formation and Solitons Pub Date : 2015-11-01 DOI: 10.3934/PROC.2015.0029
S. Anco, E. Recio, M. Gandarias, M. Bruzón
{"title":"A nonlinear generalization of the Camassa-Holm equation with peakon solutions","authors":"S. Anco, E. Recio, M. Gandarias, M. Bruzón","doi":"10.3934/PROC.2015.0029","DOIUrl":"https://doi.org/10.3934/PROC.2015.0029","url":null,"abstract":"A nonlinearly generalized Camassa-Holm equation, depending an arbitrary nonlinearity power $p neq 0$, is considered. This equation reduces to the Camassa-Holm equation when $p=1$ and shares one of the Hamiltonian structures of the Camassa-Holm equation. Two main results are obtained. A classification of point symmetries is presented and a peakon solution is derived, for all powers $p neq 0$.","PeriodicalId":154515,"journal":{"name":"arXiv: Pattern Formation and Solitons","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134251645","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}
引用次数: 14
Analytical, Optimal, and Sparse Optimal Control of Traveling Wave Solutions to Reaction-Diffusion Systems 反应扩散系统行波解的解析、最优和稀疏最优控制
arXiv: Pattern Formation and Solitons Pub Date : 2015-07-29 DOI: 10.1007/978-3-319-28028-8_10
C. Ryll, Jakob Lober, S. Martens, H. Engel, F. Troltzsch
{"title":"Analytical, Optimal, and Sparse Optimal Control of Traveling Wave Solutions to Reaction-Diffusion Systems","authors":"C. Ryll, Jakob Lober, S. Martens, H. Engel, F. Troltzsch","doi":"10.1007/978-3-319-28028-8_10","DOIUrl":"https://doi.org/10.1007/978-3-319-28028-8_10","url":null,"abstract":"","PeriodicalId":154515,"journal":{"name":"arXiv: Pattern Formation and Solitons","volume":"111 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128506698","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}
引用次数: 23
Ghosts of bump attractors in stochastic neural fields: Bottlenecks and extinction 随机神经场中凹凸吸引子的幽灵:瓶颈和消去
arXiv: Pattern Formation and Solitons Pub Date : 2015-05-23 DOI: 10.3934/DCDSB.2016044
Z. Kilpatrick
{"title":"Ghosts of bump attractors in stochastic neural fields: Bottlenecks and extinction","authors":"Z. Kilpatrick","doi":"10.3934/DCDSB.2016044","DOIUrl":"https://doi.org/10.3934/DCDSB.2016044","url":null,"abstract":"We study the effects of additive noise on stationary bump solutions to spatially extended neural fields near a saddle-node bifurcation. The integral terms of these evolution equations have a weight kernel describing synaptic interactions between neurons at different locations of the network. Excited regions of the neural field correspond to parts of the domain whose fraction of active neurons exceeds a sharp threshold of a firing rate nonlinearity. For sufficiently low firing threshold, a stable bump coexists with an unstable bump and a homogeneous quiescent state. As the threshold is increased, the stable and unstable branch of bump solutions annihilate in a saddle node bifurcation. Near this criticality, we derive a quadratic amplitude equation that describes the slow evolution of the even mode (bump contractions) as it depends on the distance from the bifurcation. Beyond the bifurcation, bumps eventually become extinct, and the time it takes for this to occur increases for systems nearer the bifurcation. When noise is incorporated, a stochastic amplitude equation for the even mode can be derived, which can be analyzed to reveal bump extinction time both below and above the saddle-node.","PeriodicalId":154515,"journal":{"name":"arXiv: Pattern Formation and Solitons","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132354774","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}
引用次数: 5
A Supersonic Crowdion in Mica 云母的超音速人群
arXiv: Pattern Formation and Solitons Pub Date : 2015-05-05 DOI: 10.1007/978-3-319-21045-2_3
J. Archilla, Y. Kosevich, N. Jim'enez, V. S'anchez-Morcillo, L. M. Garc'ia-Raffi
{"title":"A Supersonic Crowdion in Mica","authors":"J. Archilla, Y. Kosevich, N. Jim'enez, V. S'anchez-Morcillo, L. M. Garc'ia-Raffi","doi":"10.1007/978-3-319-21045-2_3","DOIUrl":"https://doi.org/10.1007/978-3-319-21045-2_3","url":null,"abstract":"","PeriodicalId":154515,"journal":{"name":"arXiv: Pattern Formation and Solitons","volume":"137 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129798803","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}
引用次数: 18
Localized States and Dynamics in the Nonlinear Schrödinger/Gross-Pitaevskii Equation 非线性Schrödinger/Gross-Pitaevskii方程的局域状态和动力学
arXiv: Pattern Formation and Solitons Pub Date : 2015-04-21 DOI: 10.1007/978-3-319-19935-1_2
M. Weinstein
{"title":"Localized States and Dynamics in the Nonlinear Schrödinger/Gross-Pitaevskii Equation","authors":"M. Weinstein","doi":"10.1007/978-3-319-19935-1_2","DOIUrl":"https://doi.org/10.1007/978-3-319-19935-1_2","url":null,"abstract":"","PeriodicalId":154515,"journal":{"name":"arXiv: Pattern Formation and Solitons","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123511063","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}
引用次数: 14
Non-holonomic constraints and their impact on discretizations of Klein-Gordon lattice dynamical models 非完整约束及其对Klein-Gordon晶格动力学模型离散化的影响
arXiv: Pattern Formation and Solitons Pub Date : 2015-03-18 DOI: 10.3934/PROC.2015.0696
P. Kevrekidis, V. Putkaradze, Z. Rapti
{"title":"Non-holonomic constraints and their impact on discretizations of Klein-Gordon lattice dynamical models","authors":"P. Kevrekidis, V. Putkaradze, Z. Rapti","doi":"10.3934/PROC.2015.0696","DOIUrl":"https://doi.org/10.3934/PROC.2015.0696","url":null,"abstract":"We explore a new type of discretizations of lattice dynamical models of the Klein-Gordon type relevant to the existence and long-term mobility of nonlinear waves. The discretization is based on non-holonomic constraints and is shown to retrieve the \"proper\" continuum limit of the model. Such discretizations are useful in exactly preserving a discrete analogue of the momentum. It is also shown that for generic initial data, the momentum and energy conservation laws cannot be achieved concurrently. Finally, direct numerical simulations illustrate that our models yield considerably higher mobility of strongly nonlinear solutions than the well-known \"standard\" discretizations, even in the case of highly discrete systems when the coupling between the adjacent nodes is weak. Thus, our approach is better suited for cases where an accurate description of mobility for nonlinear traveling waves is important.","PeriodicalId":154515,"journal":{"name":"arXiv: Pattern Formation and Solitons","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133173032","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}
引用次数: 2
Modelling Pattern Formation in Dip-Coating Experiments 浸渍实验中模型模式的形成
arXiv: Pattern Formation and Solitons Pub Date : 2015-02-12 DOI: 10.1051/mmnp/201510402
M. Wilczek, Walter Tewes, S. Gurevich, M. Kopf, L. Chi, U. Thiele
{"title":"Modelling Pattern Formation in Dip-Coating Experiments","authors":"M. Wilczek, Walter Tewes, S. Gurevich, M. Kopf, L. Chi, U. Thiele","doi":"10.1051/mmnp/201510402","DOIUrl":"https://doi.org/10.1051/mmnp/201510402","url":null,"abstract":"We briefly review selected mathematical models that describe the dynamics of pattern formation phenomena in dip-coating and Langmuir-Blodgett transfer experiments, where solutions or suspensions are transferred onto a substrate producing patterned deposit layers with structure length from hundreds of nanometres to tens of micrometres. The models are presented with a focus on their gradient dynamics formulations that clearly shows how the dynamics is governed by particular free energy functionals and facilitates the comparison of the models. In particular, we include a discussion of models based on long-wave hydrodynamics as well as of more phenomenological models that focus on the pattern formation processes in such systems. The models and their relations are elucidated and examples of resulting patterns are discussed before we conclude with a discussion of implications of the gradient dynamics formulation and of some related open issues.","PeriodicalId":154515,"journal":{"name":"arXiv: Pattern Formation and Solitons","volume":"320 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132224717","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}
引用次数: 28
High-order nonlinear Schr"odinger equation for the envelope of slowly modulated gravity waves on the surface of finite-depth fluid and its quasi-soliton solutions 有限深度流体表面慢调制重力波包络的高阶非线性Schr odinger方程及其拟孤子解
arXiv: Pattern Formation and Solitons Pub Date : 2014-12-17 DOI: 10.15407/UJPE59.12.1201
I. Gandzha, Y. Sedletsky, D. Dutykh
{"title":"High-order nonlinear Schr\"odinger equation for the envelope of slowly modulated gravity waves on the surface of finite-depth fluid and its quasi-soliton solutions","authors":"I. Gandzha, Y. Sedletsky, D. Dutykh","doi":"10.15407/UJPE59.12.1201","DOIUrl":"https://doi.org/10.15407/UJPE59.12.1201","url":null,"abstract":"We consider the high-order nonlinear Schrodinger equation derived earlier by Sedletsky [Ukr. J. Phys. 48(1), 82 (2003)] for the first-harmonic envelope of slowly modulated gravity waves on the surface of finite-depth irrotational, inviscid, and incompressible fluid with flat bottom. This equation takes into account the third-order dispersion and cubic nonlinear dispersive terms. We rewrite this equation in dimensionless form featuring only one dimensionless parameter $kh$, where $k$ is the carrier wavenumber and $h$ is the undisturbed fluid depth. We show that one-soliton solutions of the classical nonlinear Schrodinger equation are transformed into quasi-soliton solutions with slowly varying amplitude when the high-order terms are taken into consideration. These quasi-soliton solutions represent the secondary modulations of gravity waves.","PeriodicalId":154515,"journal":{"name":"arXiv: Pattern Formation and Solitons","volume":"115 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129687361","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}
引用次数: 11
Bi-$cal{PT}$ symmetry in nonlinearly damped dynamical systems and tailoring $cal{PT}$ regions with position dependent loss-gain profiles 非线性阻尼动力系统的双-$cal{PT}$对称性及位置相关损益曲线裁剪$cal{PT}$区域
arXiv: Pattern Formation and Solitons Pub Date : 2014-12-08 DOI: 10.1103/PHYSREVA.93.012102
S. Karthiga, V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan
{"title":"Bi-$cal{PT}$ symmetry in nonlinearly damped dynamical systems and tailoring $cal{PT}$ regions with position dependent loss-gain profiles","authors":"S. Karthiga, V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan","doi":"10.1103/PHYSREVA.93.012102","DOIUrl":"https://doi.org/10.1103/PHYSREVA.93.012102","url":null,"abstract":"We investigate the remarkable role of position dependent damping in determining the parametric regions of symmetry breaking in nonlinear $cal{PT}$-symmetric systems. We illustrate the nature of $cal{PT}$-symmetry preservation and breaking with reference to a remarkable integrable scalar nonlinear system. In the two dimensional cases of such position dependent damped systems, we unveil the existence of a class of novel bi-$cal{PT}$-symmetric systems which have two fold $cal{PT}$ symmetries. We analyze the dynamics of these systems and show how symmetry breaking occurs, that is whether the symmetry breaking of the two $cal{PT}$ symmetries occurs in pair or occurs one by one. The addition of linear damping in these nonlinearly damped systems induces competition between the two types of damping. This competition results in a $cal{PT}$ phase transition in which the $cal{PT}$ symmetry is broken for lower loss/gain strength and is restored by increasing the loss/gain strength. We also show that by properly designing the form of the position dependent damping, we can tailor the $cal{PT}$-symmetric regions of the system.","PeriodicalId":154515,"journal":{"name":"arXiv: Pattern Formation and Solitons","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121884967","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}
引用次数: 10
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