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Soliton, breather, and rogue wave solutions for solving the nonlinear Schrodinger equation using a deep learning method with physical constraints 被引量:5
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作者 Jun-Cai Pu Jun Li Yong Chen 《Chinese Physics B》 SCIE EI CAS CSCD 2021年第6期77-87,共11页
The nonlinear Schrodinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particu... The nonlinear Schrodinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schrodinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schrodinger equation can be well reconstructed by utilizing this physically-constrained deep learning method. 展开更多
关键词 deep learning method neural network soliton solutions breather solution rogue wave solutions
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A more general form of lump solution,lumpoff,and instanton/rogue wave solutions of a reduced (3+1)-dimensional nonlinear evolution equation 被引量:2
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作者 Panfeng Zheng Man Jia 《Chinese Physics B》 SCIE EI CAS CSCD 2018年第12期147-156,共10页
In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution ... In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution to the special case for z=x.Furthermore,a more general form of lump solution of the equation is found which possesses seven arbitrary parameters and four constraint conditions.By cutting the lump by the induced soliton(s),lumpoff and instanton/rogue wave solutions are also constructed by the more general form of lump solution. 展开更多
关键词 a reduced(3 + 1)-dimensional nonlinear evolution equation more general form of lump solution soliton induced by lump lumpoff and instanton/rogue wave solutions
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Soliton, Breather and Rogue Wave Solutions for the Nonlinear Schrdinger Equation Coupled to a Multiple Self-Induced Transparency System 被引量:1
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作者 王鑫 王雷 《Chinese Physics Letters》 SCIE CAS CSCD 2018年第3期1-4,共4页
We derive an N-fold Darboux transformation for the nonlinear Schrdinger equation coupled to a multiple selfinduced transparency system, which is applicable to optical fiber communications in the erbium-doped medium.Th... We derive an N-fold Darboux transformation for the nonlinear Schrdinger equation coupled to a multiple selfinduced transparency system, which is applicable to optical fiber communications in the erbium-doped medium.The N-soliton, N-breather and N th-order rogue wave solutions in the compact determinant representations are derived using the Darboux transformation and limit technique. Dynamics of such solutions from the first-to second-order ones are shown. 展开更多
关键词 LIM SOLITON dinger Equation Coupled to a Multiple Self-Induced Transparency System Breather and rogue wave solutions for the Nonlinear Schr
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Soliton and rogue wave solutions of two-component nonlinear Schr?dinger equation coupled to the Boussinesq equation 被引量:1
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作者 宋彩芹 肖冬梅 朱佐农 《Chinese Physics B》 SCIE EI CAS CSCD 2017年第10期28-37,共10页
The nonlinear Schrodinger (NLS) equation and Boussinesq equation are two very important integrable equations. They have widely physical applications. In this paper, we investigate a nonlinear system, which is the tw... The nonlinear Schrodinger (NLS) equation and Boussinesq equation are two very important integrable equations. They have widely physical applications. In this paper, we investigate a nonlinear system, which is the two-component NLS equation coupled to the Boussinesq equation. We obtain the bright-bright, bright-dark, and dark-dark soliton solutions to the nonlinear system. We discuss the collision between two solitons. We observe that the collision of bright-bright soliton is inelastic and two solitons oscillating periodically can happen in the two parallel-traveling bright-bright or bright-dark soliton solution. The general breather and rogue wave solutions are also given. Our results show again that there are more abundant dynamical properties for multi-component nonlinear systems. 展开更多
关键词 multi-component NLS-Boussinesq equation soliton solution rogue wave solution
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Rogue wave solutions and rogue-breather solutions to the focusing nonlinear Schr?dinger equation
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作者 Si-Jia Chen Xing Lü 《Communications in Theoretical Physics》 SCIE CAS CSCD 2024年第3期33-41,共9页
Based on the long wave limit method,the general form of the second-order and third-order rogue wave solutions to the focusing nonlinear Schr?dinger equation are given by introducing some arbitrary parameters.The inter... Based on the long wave limit method,the general form of the second-order and third-order rogue wave solutions to the focusing nonlinear Schr?dinger equation are given by introducing some arbitrary parameters.The interaction solutions between the first-order rogue wave and one-breather wave are constructed by taking a long wave limit on the two-breather solutions.By applying the same method to the three-breather solutions,two types of interaction solutions are obtained,namely the first-order rogue wave and two breather waves,the second-order rogue wave and one-breather wave,respectively.The influence of the parameters related to the phase on the interaction phenomena is graphically demonstrated.Collisions occur among the rogue waves and breather waves.After the collisions,the shape of them remains unchanged.The abundant interaction phenomena in this paper will contribute to a better understanding of the propagation and control of nonlinear waves. 展开更多
关键词 nonlinear Schrodinger equation rogue wave solutions interaction solutions
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Localized wave solutions and interactions of the (2+1)-dimensional Hirota-Satsuma-Ito equation
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作者 巩乾坤 王惠 王云虎 《Chinese Physics B》 SCIE EI CAS CSCD 2024年第4期409-416,共8页
This paper studies the(2+1)-dimensional Hirota-Satsuma-Ito equation.Based on an associated Hirota bilinear form,lump-type solution,two types of interaction solutions,and breather wave solution of the(2+1)-dimensional ... This paper studies the(2+1)-dimensional Hirota-Satsuma-Ito equation.Based on an associated Hirota bilinear form,lump-type solution,two types of interaction solutions,and breather wave solution of the(2+1)-dimensional Hirota-Satsuma-Ito equation are obtained,which are all related to the seed solution of the equation.It is interesting that the rogue wave is aroused by the interaction between one-lump soliton and a pair of resonance stripe solitons,and the fusion and fission phenomena are also found in the interaction between lump solitons and one-stripe soliton.Furthermore,the breather wave solution is also obtained by reducing the two-soliton solutions.The trajectory and period of the one-order breather wave are analyzed.The corresponding dynamical characteristics are demonstrated by the graphs. 展开更多
关键词 lump solution rogue wave solution breather wave solution (2+1)-dimensional Hirota-Satsuma-Ito equation
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Controllable Discrete Rogue Wave Solutions of the Ablowitz–Ladik Equation in Optics
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作者 闻小永 《Communications in Theoretical Physics》 SCIE CAS CSCD 2016年第7期29-34,共6页
With the aid of symbolic computation Maple, the discrete Ablowitz–Ladik equation is studied via an algebra method, some new rational solutions with four arbitrary parameters are constructed. By analyzing related para... With the aid of symbolic computation Maple, the discrete Ablowitz–Ladik equation is studied via an algebra method, some new rational solutions with four arbitrary parameters are constructed. By analyzing related parameters, the discrete rogue wave solutions with alterable positions and amplitude for the focusing Ablowitz–Ladik equations are derived. Some properties are discussed by graphical analysis, which might be helpful for understanding physical phenomena in optics. 展开更多
关键词 symbolic computation Maple Ablowitz–Ladik equation rational solutions discrete rogue wave solutions
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Darboux Transformations, Higher-Order Rational Solitons and Rogue Wave Solutions for a(2+1)-Dimensional Nonlinear Schrdinger Equation 被引量:1
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作者 Mi Chen Biao Li Ya-Xuan Yu 《Communications in Theoretical Physics》 SCIE CAS CSCD 2019年第1期27-36,共10页
By Taylor expansion of Darboux matrix, a new generalized Darboux transformations(DTs) for a(2 + 1)-dimensional nonlinear Schrdinger(NLS) equation is derived, which can be reduced to two(1 + 1)-dimensional equation:a m... By Taylor expansion of Darboux matrix, a new generalized Darboux transformations(DTs) for a(2 + 1)-dimensional nonlinear Schrdinger(NLS) equation is derived, which can be reduced to two(1 + 1)-dimensional equation:a modified KdV equation and an NLS equation. With the help of symbolic computation, some higher-order rational solutions and rogue wave(RW) solutions are constructed by its(1, N-1)-fold DTs according to determinants. From the dynamic behavior of these rogue waves discussed under some selected parameters, we find that the RWs and solitons are demonstrated some interesting structures including the triangle, pentagon, heptagon profiles, etc. Furthermore, we find that the wave structure can be changed from the higher-order RWs into higher-order rational solitons by modulating the main free parameter. These results may give an explanation and prediction for the corresponding dynamical phenomena in some physically relevant systems. 展开更多
关键词 Darboux transformations nonlinear Schrdinger equation higher-order rational solution rogue wave solution
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Multiple rational rogue waves for higher dimensional nonlinear evolution equations via symbolic computation approach 被引量:1
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作者 Saima Arshed Nauman Raza +2 位作者 Asma Rashid Butt Ahmad Javid J.F.Gómez-Aguilar 《Journal of Ocean Engineering and Science》 SCIE 2023年第1期33-41,共9页
The paper investigates the multiple rogue wave solutions associated with the generalized Hirota-Satsuma-Ito(HSI)equation and the newly proposed extended(3+1)-dimensional Jimbo-Miwa(JM)equation with the help of a symbo... The paper investigates the multiple rogue wave solutions associated with the generalized Hirota-Satsuma-Ito(HSI)equation and the newly proposed extended(3+1)-dimensional Jimbo-Miwa(JM)equation with the help of a symbolic computation technique.By incorporating a direct variable trans-formation and utilizing Hirota’s bilinear form,multiple rogue wave structures of different orders are ob-tained for both generalized HSI and JM equation.The obtained bilinear forms of the proposed equations successfully investigate the 1st,2nd and 3rd-order rogue waves.The constructed solutions are verified by inserting them into original equations.The computations are assisted with 3D graphs to analyze the propagation dynamics of these rogue waves.Physical properties of these waves are governed by different parameters that are discussed in details. 展开更多
关键词 Generalized Hirota-Satsuma-Ito(HSI) equation The new extended(3+1)-dimensional Jimbo-Miwaequation Symbolic computation approach Bilinear form rogue wave solutions
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Multiple rogue wave and solitary solutions for the generalized BK equation via Hirota bilinear and SIVP schemes arising in fluid mechanics
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作者 Jalil Manafian Onur Alp Ilhan +1 位作者 As’ad Alizadeh Sizar Abid Mohammed 《Communications in Theoretical Physics》 SCIE CAS CSCD 2020年第7期8-20,共13页
The multiple lump solutions method is employed for the purpose of obtaining multiple soliton solutions for the generalized Bogoyavlensky-Konopelchenko(BK)equation.The solutions obtained contain first-order,second-orde... The multiple lump solutions method is employed for the purpose of obtaining multiple soliton solutions for the generalized Bogoyavlensky-Konopelchenko(BK)equation.The solutions obtained contain first-order,second-order,and third-order wave solutions.At the critical point,the second-order derivative and Hessian matrix for only one point is investigated,and the lump solution has one maximum value.He’s semi-inverse variational principle(SIVP)is also used for the generalized BK equation.Three major cases are studied,based on two different ansatzes using the SIVP.The physical phenomena of the multiple soliton solutions thus obtained are then analyzed and demonstrated in the figures below,using a selection of suitable parameter values.This method should prove extremely useful for further studies of attractive physical phenomena in the fields of heat transfer,fluid dynamics,etc. 展开更多
关键词 multiple rogue wave solutions multiple soliton solutions generalized Bogoyavlensky–Konopelchenko equation semi-inverse variational principle
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Multiple-order line rogue wave,lump and its interaction,periodic,and cross-kink solutions for the generalized CHKP equation
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作者 Yufeng Qian Jalil Manafian +3 位作者 Sherin Youns Mohyaldeen Liqaa S.Esmail Sergey Alekseevich Gorovoy Gurpreet Singh 《Propulsion and Power Research》 SCIE 2021年第3期277-293,共17页
The multiple-order line rogue wave solutions method is emp loyed for searching the multiple soliton solutions for the generalized(2+1)-dimensional Camassa-HolmKadomtsev-Petviashvili(CHKP)equation,which contains first-... The multiple-order line rogue wave solutions method is emp loyed for searching the multiple soliton solutions for the generalized(2+1)-dimensional Camassa-HolmKadomtsev-Petviashvili(CHKP)equation,which contains first-order,second-order,and third-order waves solutions.At the critical point,the second-order derivative and Hessian matrix for only one point will be investigated and the lump solution has one minimum value.For the case,the lump solution will be shown the bright-dark lump structure and for another case can be present the dark lump structure-two small peaks and one deep hole.Also,the interaction of lump with periodic waves and the interaction between lump and soliton can be obtained by introducing the Hirota forms.In the meanwhile,the cross-kink wave and periodic wave solutions can be gained by the Hirota operator.The physical phenomena of these gained multiple soliton solutions are analyzed and indicated in figures by selecting suitable values.We alternative offer that the determining method is general,impressive,outspoken,and powerful and can be exerted to create exact solutions of various kinds of nonlinear models originated in mathematical physics and engineering. 展开更多
关键词 Multiple rogue wave solutions Multiple soliton solutions Generalized CamassaHolm-KadomtsevPetviashvili equation Lump solution Hirota operator
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A Novel Method for Solving Nonlinear Schrödinger Equation with a Potential by Deep Learning
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作者 Chaojun Zhang Yuexing Bai 《Journal of Applied Mathematics and Physics》 2022年第10期3175-3190,共16页
The improved physical information neural network algorithm has been proven to be used to study integrable systems. In this paper, the improved physical information neural network algorithm is used to study the defocus... The improved physical information neural network algorithm has been proven to be used to study integrable systems. In this paper, the improved physical information neural network algorithm is used to study the defocusing nonlinear Schr&#246;dinger (NLS) equation with time-varying potential, and the rogue wave solution of the equation is obtained. At the same time, the influence of the number of network layers, neurons and the number of sampling points on the network performance is studied. Experiments show that the number of hidden layers and the number of neurons in each hidden layer affect the relative L<sub>2</sub>-norm error. With fixed configuration points, the relative norm error does not decrease with the increase in the number of boundary data points, which indicates that in this case, the number of boundary data points has no obvious influence on the error. Through the experiment, the rogue wave solution of the defocusing NLS equation is successfully captured by IPINN method for the first time. The experimental results of this paper are also compared with the results obtained by the physical information neural network method and show that the improved algorithm has higher accuracy. The results of this paper will be contributed to the generalization of deep learning algorithms for solving defocusing NLS equations with time-varying potential. 展开更多
关键词 Physics-Informed Neural Networks Improved Physics-Informed Neural Net-works Defocusing NLS Equation rogue wave Solution
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