We uncover the virtual monopoles underlying the nontrivial phases of the one-dimensional nonlinear excitations of rogue waves by extending the Dirac magnetic monopole theory to a complex plane. We find that the densit...We uncover the virtual monopoles underlying the nontrivial phases of the one-dimensional nonlinear excitations of rogue waves by extending the Dirac magnetic monopole theory to a complex plane. We find that the density zeros of the nonlinear waves on the extended complex plane constitute the virtual monopole fields with a quantized flux of elementary π. We then explain the exotic properties of rogue waves by means of a virtual monopole collision mechanism and find that the maximum amplitude amplification ratio and multiple phase steps of the high-order rogue waves are closely related to the number of their contained monopoles. These results open a new avenue for studying topological properties of nonlinear waves and provide an alternative way to understand their dynamics.展开更多
To the nonlinear Schrodinger–Boussinesq system,with the aid of Adler–Moser polynomials we predict the patterns of higher-order rogue wave solutions containing multiple large parameters.The new interesting rogue wave...To the nonlinear Schrodinger–Boussinesq system,with the aid of Adler–Moser polynomials we predict the patterns of higher-order rogue wave solutions containing multiple large parameters.The new interesting rogue wave patterns of a number of true and predicted solutions are graphically illustrated,including fan-,heart-shaped structures and their skewed versions.The results are significant for both experimental and theoretical studies of rogue wave patterns of integrable systems.展开更多
In this paper,the nonlinearization of the Lax pair and the Darboux transformation method are used to construct the rogue wave on the elliptic function background in the reduced Maxwell–Bloch system,which is described...In this paper,the nonlinearization of the Lax pair and the Darboux transformation method are used to construct the rogue wave on the elliptic function background in the reduced Maxwell–Bloch system,which is described by four component nonlinear evolution equations(NLEEs).On the background of the Jacobian elliptic function,we obtain the admissible eigenvalues and the corresponding non-periodic eigenfunctions of the model spectrum problem.Then,with the help of the one-fold Darboux transformation and two-fold Darboux transformation,rogue waves on a dn-periodic background and cn-periodic background are derived,respectively.Finally,the corresponding complex dynamical properties and evolutions of the four components are illustrated graphically by choosing suitable parameters.展开更多
A compact Grammian form for N-breather solution to the complex m Kd V equation is derived using the bilinear Kadomtsev–Petviashvili hierarchy reduction method.The propagation trajectory,period,maximum points,and peak...A compact Grammian form for N-breather solution to the complex m Kd V equation is derived using the bilinear Kadomtsev–Petviashvili hierarchy reduction method.The propagation trajectory,period,maximum points,and peak value of the 1-breather solution are calculated.Additionally,through the asymptotic analysis of 2-breather solution,we show that two breathers undergo an elastic collision.By applying the generalized long-wave limit method,the fundamental and second-order rogue wave solutions for the complex m Kd V equation are obtained from the 1-breather and 2-breather solutions,respectively.We also construct the hybrid solution of a breather and a fundamental rogue wave for the complex m Kd V equation from the 2-breather solution.Furthermore,the hybrid solution of two breathers and a fundamental rogue wave as well as the hybrid solution of a breather and a second-order rogue wave for the complex m Kd V equation are derived from the 3-breather solution via the generalized long-wave limit method.By controlling the phase parameters of breathers,the diverse phenomena of interaction between the breathers and the rogue waves are demonstrated.展开更多
We analytically give the financial rogue waves in the nonlinear option pricing model due to Ivancevic,which is nonlinear wave alternative of the Black-Scholes model.These rogue wave solutions may be used to describe t...We analytically give the financial rogue waves in the nonlinear option pricing model due to Ivancevic,which is nonlinear wave alternative of the Black-Scholes model.These rogue wave solutions may be used to describe thepossible physical mechanisms for rogue wave phenomenon in financial markets and related fields.展开更多
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
Analytical solutions in terms of rational-like functions are presented for a （3＋1）-dimensional nonlinear Schrodinger equation with time-varying coefficients and a harmonica potential using the similarity transforma...Analytical solutions in terms of rational-like functions are presented for a （3＋1）-dimensional nonlinear Schrodinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz. Several free functions of time t are involved to generate abundant wave structures. Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations.展开更多
In this paper,we construct the rogue wave solutions of the sixth-order nonlinear Schrodinger equation on a background of Jacobian elliptic functions dn and cn by means of the nonlinearization of a spectral problem and...In this paper,we construct the rogue wave solutions of the sixth-order nonlinear Schrodinger equation on a background of Jacobian elliptic functions dn and cn by means of the nonlinearization of a spectral problem and Darboux transformation approach.The solutions we find present the dynamic phenomena of higher-order nonlinear wave equations.展开更多
The derivative nonlinear Schrodinger equation, which is extensively applied in plasma physics and nonlinear optics, is analytically studied by Hirota method. Space periodic solutions are determined by means of Hirota...The derivative nonlinear Schrodinger equation, which is extensively applied in plasma physics and nonlinear optics, is analytically studied by Hirota method. Space periodic solutions are determined by means of Hirota＇s bilinear formalism, and the rogue wave solution is derived as a long-wave limit of the space periodic solution.展开更多
In this paper, a new type （or the second type） of transformation which is used to map the variable coefficient nonlinear Schr6dinger （VCNLS） equation to the usual nonlinear Schrodinger （NLS） equation is given. A...In this paper, a new type （or the second type） of transformation which is used to map the variable coefficient nonlinear Schr6dinger （VCNLS） equation to the usual nonlinear Schrodinger （NLS） equation is given. As a special case, a new kind of nonautonomous NLS equation with a t-dependent potential is introduced. Further, by using the new transformation and making full use of the known soliton and rogue wave solutions of the usual NLS equation, the corresponding kinds of solutions of a special model of the new nonautonomous NLS equation are discussed respectively. Additionally, through using the new transformation, a new expression, i.e., the non-rational formula, of the rogue wave of a special VCNLS equation is given analytically. The main differences between the two types of transformation mentioned above are listed by three items.展开更多
In this paper, we provide determinant representation of the n-th order rogue wave solutions for a higherorder nonlinear Schr6dinger equation （HONLS） by the Darboux transformation and confirm the decomposition rule o...In this paper, we provide determinant representation of the n-th order rogue wave solutions for a higherorder nonlinear Schr6dinger equation （HONLS） by the Darboux transformation and confirm the decomposition rule of the rogue wave solutions up to fourth-order. These solutions have two parameters a and ;3 which denote the contribution of the higher-order terms （dispersions and nonlinear effects） included in the HONLS equation. Two localized properties, i.e., length and width of the first-order rogue wave solution are expressed by above two parameters, which show analytically a remarkable influence of higher-order terms on the rogue wave. Moreover, profiles of the higher-order rogue wave solutions demonstrate graphically a strong compression effect along t-direction given by higher-order terms.展开更多
The generalized binary Darboux transformation for the （1 ＋2）-dimensional non-isospectral KP-H equation is presented. Moreover, as a direct application, the new rogue wave solutions for the （1＋2）-dimensional non-...The generalized binary Darboux transformation for the （1 ＋2）-dimensional non-isospectral KP-H equation is presented. Moreover, as a direct application, the new rogue wave solutions for the （1＋2）-dimensional non-isospectral KP-II equation are constructed by the generalized binary Darboux transformation.展开更多
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
Studied in this paper is a(2+1)-dimensional coupled nonlinear Schr?dinger system with variable coefficients,which describes the propagation of an optical beam inside the two-dimensional graded-index waveguide amplifie...Studied in this paper is a(2+1)-dimensional coupled nonlinear Schr?dinger system with variable coefficients,which describes the propagation of an optical beam inside the two-dimensional graded-index waveguide amplifier with the polarization effects. According to the similarity transformation, we derive the type-Ⅰ and type-Ⅱ rogue-wave solutions. We graphically present two types of the rouge wave and discuss the influence of the diffraction parameter on the rogue waves.When the diffraction parameters are exponentially-growing-periodic, exponential, linear and quadratic parameters, we obtain the periodic rogue wave and composite rogue waves respectively.展开更多
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
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
The (2+1)-dimension nonlocal nonlinear Schrödinger (NLS) equation with the self-induced parity-time symmetric potential is introduced, which provides spatially two-dimensional analogues of the nonlocal NLS equati...The (2+1)-dimension nonlocal nonlinear Schrödinger (NLS) equation with the self-induced parity-time symmetric potential is introduced, which provides spatially two-dimensional analogues of the nonlocal NLS equation introduced by Ablowitz et al. [Phys. Rev. Lett. 110 (2013) 064105]. General periodic solutions are derived by the bilinear method. These periodic solutions behave as growing and decaying periodic line waves arising from the constant background and decaying back to the constant background again. By taking long wave limits of the obtained periodic solutions, rogue waves are obtained. It is also shown that these line rogue waves arise from the constant background with a line profile and disappear into the constant background again in the plane.展开更多
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
In this paper,the fractional-order model is used to study dust acoustic rogue waves in dusty plasma.Firstly,based on control equations,the multi-scale analysis and reduced perturbation method are used to derive the(3+...In this paper,the fractional-order model is used to study dust acoustic rogue waves in dusty plasma.Firstly,based on control equations,the multi-scale analysis and reduced perturbation method are used to derive the(3+1)-dimensional modified Kadomtsev–Petviashvili(MKP)equation.Secondly,using the semi-inverse method and the fractional variation principle,the(3+1)-dimensional time-fractional modified Kadomtsev–Petviashvili(TF-MKP)equation is derived.Then,the Riemann–Liouville fractional derivative is used to study the symmetric property and conservation laws of the(3+1)-dimensional TF-MKP equation.Finally,the exact solution of the(3+1)-dimensional TF-MKP equation is obtained by using fractional order transformations and the definition and properties of Bell polynomials.Based on the obtained solution,we analyze and discuss dust acoustic rogue waves in dusty plasma.展开更多
It is proved that rogue waves can be found in Korteweg de-Vries（KdV） systems if real nonintegrable effects, higher order nonlinearity and nonlinear diffusion are considered. Rogue waves can also be formed without mo...It is proved that rogue waves can be found in Korteweg de-Vries（KdV） systems if real nonintegrable effects, higher order nonlinearity and nonlinear diffusion are considered. Rogue waves can also be formed without modulation instability which is considered as the main formation mechanism of the rogue waves.展开更多
基金supported by the National Natural Science Foundation of China (Grant Nos.12375005,12022513,and12235007)the National Safety Academic Fund(Grant No.U2330401)。
文摘We uncover the virtual monopoles underlying the nontrivial phases of the one-dimensional nonlinear excitations of rogue waves by extending the Dirac magnetic monopole theory to a complex plane. We find that the density zeros of the nonlinear waves on the extended complex plane constitute the virtual monopole fields with a quantized flux of elementary π. We then explain the exotic properties of rogue waves by means of a virtual monopole collision mechanism and find that the maximum amplitude amplification ratio and multiple phase steps of the high-order rogue waves are closely related to the number of their contained monopoles. These results open a new avenue for studying topological properties of nonlinear waves and provide an alternative way to understand their dynamics.
基金supported by the National Natural Science Foundation of China(Grant Nos.11871396,12271433)Shaanxi Fundamental Science Research Project for Mathematics and Physics(Grant No.23JSY036)partly supported by Graduate Student Innovation Project of Northwest University(Grant No.CX2024129)。
文摘To the nonlinear Schrodinger–Boussinesq system,with the aid of Adler–Moser polynomials we predict the patterns of higher-order rogue wave solutions containing multiple large parameters.The new interesting rogue wave patterns of a number of true and predicted solutions are graphically illustrated,including fan-,heart-shaped structures and their skewed versions.The results are significant for both experimental and theoretical studies of rogue wave patterns of integrable systems.
基金supported by the National Natural Science Foundation of China(Grant No.12361052)the Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region(Grant No.NMGIRT2414)+3 种基金the Fundamental Research Funds for the Inner Mongolia Normal University,China(Grant Nos.2022JBTD007,2022JBXC013)Graduate Students'Research and Innovation Fund of Inner Mongolia Autonomous Region(Grant No.B20231053Z)the Key Laboratory of Infinite-Dimensional Hamiltonian System and Its Algorithm Application(Inner Mongolia Normal University),the Ministry of Education(Grant Nos.2023KFZR01,2023KFZR02)the First-Class Disciplines Project,Inner Mongolia Autonomous Region,China(Grant No.YLXKZX-NSD-001)。
文摘In this paper,the nonlinearization of the Lax pair and the Darboux transformation method are used to construct the rogue wave on the elliptic function background in the reduced Maxwell–Bloch system,which is described by four component nonlinear evolution equations(NLEEs).On the background of the Jacobian elliptic function,we obtain the admissible eigenvalues and the corresponding non-periodic eigenfunctions of the model spectrum problem.Then,with the help of the one-fold Darboux transformation and two-fold Darboux transformation,rogue waves on a dn-periodic background and cn-periodic background are derived,respectively.Finally,the corresponding complex dynamical properties and evolutions of the four components are illustrated graphically by choosing suitable parameters.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.12061051 and 12461048)。
文摘A compact Grammian form for N-breather solution to the complex m Kd V equation is derived using the bilinear Kadomtsev–Petviashvili hierarchy reduction method.The propagation trajectory,period,maximum points,and peak value of the 1-breather solution are calculated.Additionally,through the asymptotic analysis of 2-breather solution,we show that two breathers undergo an elastic collision.By applying the generalized long-wave limit method,the fundamental and second-order rogue wave solutions for the complex m Kd V equation are obtained from the 1-breather and 2-breather solutions,respectively.We also construct the hybrid solution of a breather and a fundamental rogue wave for the complex m Kd V equation from the 2-breather solution.Furthermore,the hybrid solution of two breathers and a fundamental rogue wave as well as the hybrid solution of a breather and a second-order rogue wave for the complex m Kd V equation are derived from the 3-breather solution via the generalized long-wave limit method.By controlling the phase parameters of breathers,the diverse phenomena of interaction between the breathers and the rogue waves are demonstrated.
基金Supported by National Natural Science Foundation of China under Grant No.60821002/F02
文摘We analytically give the financial rogue waves in the nonlinear option pricing model due to Ivancevic,which is nonlinear wave alternative of the Black-Scholes model.These rogue wave solutions may be used to describe thepossible physical mechanisms for rogue wave phenomenon in financial markets and related fields.
基金supported by the National Natural Science Foundation of China (Grant No. 11675054)the Fund from Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213)the Project of Science and Technology Commission of Shanghai Municipality (Grant No. 18dz2271000)。
文摘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.
基金Project supported by the National Natural Science Foundation of China （Grant No. 10772110） and the Natural Science Foundation of Zhejiang Province, China （Grant Nos. Y606049, Y6090681, and Y6100257）.
文摘Analytical solutions in terms of rational-like functions are presented for a （3＋1）-dimensional nonlinear Schrodinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz. Several free functions of time t are involved to generate abundant wave structures. Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations.
基金supported by the National Natural Science Foundation of China(Grant Nos.11861050,11261037)the Natural Science Foundation of Inner Mongolia Autonomous Region,China(Grant No.2020LH01010)the Inner Mongolia Normal University Graduate Students Research and Innovation Fund(Grant No.CXJJS21119)。
文摘In this paper,we construct the rogue wave solutions of the sixth-order nonlinear Schrodinger equation on a background of Jacobian elliptic functions dn and cn by means of the nonlinearization of a spectral problem and Darboux transformation approach.The solutions we find present the dynamic phenomena of higher-order nonlinear wave equations.
基金Supported by the Teaching Steering Committee Research Project of Higher-Learning Institutions of Ministry of Education(JZW-16-DD-15)
文摘The derivative nonlinear Schrodinger equation, which is extensively applied in plasma physics and nonlinear optics, is analytically studied by Hirota method. Space periodic solutions are determined by means of Hirota＇s bilinear formalism, and the rogue wave solution is derived as a long-wave limit of the space periodic solution.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10971109 and 10971211supported by Program for New Century Excellent Talents in University under Grant No.NCET-08-0515
文摘In this paper, a new type （or the second type） of transformation which is used to map the variable coefficient nonlinear Schr6dinger （VCNLS） equation to the usual nonlinear Schrodinger （NLS） equation is given. As a special case, a new kind of nonautonomous NLS equation with a t-dependent potential is introduced. Further, by using the new transformation and making full use of the known soliton and rogue wave solutions of the usual NLS equation, the corresponding kinds of solutions of a special model of the new nonautonomous NLS equation are discussed respectively. Additionally, through using the new transformation, a new expression, i.e., the non-rational formula, of the rogue wave of a special VCNLS equation is given analytically. The main differences between the two types of transformation mentioned above are listed by three items.
基金Supported by the National Natural Science Foundation of China under Grant No.11271210the K.C.Wong Magna Fund in Ningbo University
文摘In this paper, we provide determinant representation of the n-th order rogue wave solutions for a higherorder nonlinear Schr6dinger equation （HONLS） by the Darboux transformation and confirm the decomposition rule of the rogue wave solutions up to fourth-order. These solutions have two parameters a and ;3 which denote the contribution of the higher-order terms （dispersions and nonlinear effects） included in the HONLS equation. Two localized properties, i.e., length and width of the first-order rogue wave solution are expressed by above two parameters, which show analytically a remarkable influence of higher-order terms on the rogue wave. Moreover, profiles of the higher-order rogue wave solutions demonstrate graphically a strong compression effect along t-direction given by higher-order terms.
基金Supported by the National Natural Science Foundation of China under Grant No. 11061003 and Guangxi Natural Science Foundation under Grant No. 2013GXNSFAA019001
文摘The generalized binary Darboux transformation for the （1 ＋2）-dimensional non-isospectral KP-H equation is presented. Moreover, as a direct application, the new rogue wave solutions for the （1＋2）-dimensional non-isospectral KP-II equation are constructed by the generalized binary Darboux transformation.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11675084 and 11435005)the Fund from the Educational Commission of Zhejiang Province,China(Grant No.Y201737177)+1 种基金Ningbo Natural Science Foundation(Grant No.2015A610159)the K C Wong Magna Fund in Ningbo University
文摘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.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11772017,11272023,and 11471050the Fund of State Key Laboratory of Information Photonics and Optical Communications(Beijing University of Posts and Telecommunications),China(IPOC:2017ZZ05)the Fundamental Research Funds for the Central Universities of China under Grant No.2011BUPTYB02
文摘Studied in this paper is a(2+1)-dimensional coupled nonlinear Schr?dinger system with variable coefficients,which describes the propagation of an optical beam inside the two-dimensional graded-index waveguide amplifier with the polarization effects. According to the similarity transformation, we derive the type-Ⅰ and type-Ⅱ rogue-wave solutions. We graphically present two types of the rouge wave and discuss the influence of the diffraction parameter on the rogue waves.When the diffraction parameters are exponentially-growing-periodic, exponential, linear and quadratic parameters, we obtain the periodic rogue wave and composite rogue waves respectively.
基金Supported by National Natural Science Foundation of China under Grant Nos.11775121 and 11435005K.C.Wong Magna Fund in Ningbo University
文摘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.
基金supported by the National Natural Science Foundation of China(Grant Nos.11371248,11431008,11271254,11428102,and 11671255)the Fund from the Ministry of Economy and Competitiveness of Spain(Grant Nos.MTM2012-37070 and MTM2016-80276-P(AEI/FEDER,EU))
文摘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.
基金Supported by the National Natural Science Foundation of China under Grant Nos 11271211,11275072 and 11435005the Ningbo Natural Science Foundation under Grant No 2015A610159+1 种基金the Opening Project of Zhejiang Provincial Top Key Discipline of Physics Sciences in Ningbo University under Grant No xkzw11502the K.C.Wong Magna Fund in Ningbo University
文摘The (2+1)-dimension nonlocal nonlinear Schrödinger (NLS) equation with the self-induced parity-time symmetric potential is introduced, which provides spatially two-dimensional analogues of the nonlocal NLS equation introduced by Ablowitz et al. [Phys. Rev. Lett. 110 (2013) 064105]. General periodic solutions are derived by the bilinear method. These periodic solutions behave as growing and decaying periodic line waves arising from the constant background and decaying back to the constant background again. By taking long wave limits of the obtained periodic solutions, rogue waves are obtained. It is also shown that these line rogue waves arise from the constant background with a line profile and disappear into the constant background again in the plane.
基金Supported by the National Natural Science Foundation of China under Grant Nos 11705290 and 11305060the China Postdoctoral Science Foundation under Grant No 2016M602252
文摘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.
基金supported by the National Natural Science Foundation of China(Grants 11975143)。
文摘In this paper,the fractional-order model is used to study dust acoustic rogue waves in dusty plasma.Firstly,based on control equations,the multi-scale analysis and reduced perturbation method are used to derive the(3+1)-dimensional modified Kadomtsev–Petviashvili(MKP)equation.Secondly,using the semi-inverse method and the fractional variation principle,the(3+1)-dimensional time-fractional modified Kadomtsev–Petviashvili(TF-MKP)equation is derived.Then,the Riemann–Liouville fractional derivative is used to study the symmetric property and conservation laws of the(3+1)-dimensional TF-MKP equation.Finally,the exact solution of the(3+1)-dimensional TF-MKP equation is obtained by using fractional order transformations and the definition and properties of Bell polynomials.Based on the obtained solution,we analyze and discuss dust acoustic rogue waves in dusty plasma.
基金Supported by the National Natural Science Foundation of China under Grant Nos 11675084 and 11435005the K.C.Wong Magna Fund in Ningbo University
文摘It is proved that rogue waves can be found in Korteweg de-Vries（KdV） systems if real nonintegrable effects, higher order nonlinearity and nonlinear diffusion are considered. Rogue waves can also be formed without modulation instability which is considered as the main formation mechanism of the rogue waves.
