We present three families of soliton solutions to the generalized (3+l)-dimensional nonlinear Schrodinger equation with distributed coefficients. We investigate the dynamics of these solitons in nonlinear optics wi...We present three families of soliton solutions to the generalized (3+l)-dimensional nonlinear Schrodinger equation with distributed coefficients. We investigate the dynamics of these solitons in nonlinear optics with some selected parameters. Different shapes of bright solitons, a train of bright solitons and dark solitons are observed. The obtained results may raise the possibilities of relevant experiments and potential applications.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
In this paper, the extended symmetry transformation of (3+1)-dimensional (3D) generalized nonlinear Schrodinger (NLS) equations with variable coefficients is investigated by using the extended symmetry approach...In this paper, the extended symmetry transformation of (3+1)-dimensional (3D) generalized nonlinear Schrodinger (NLS) equations with variable coefficients is investigated by using the extended symmetry approach and symbolic computation. Then based on the extended symmetry, some 3D variable coefficient NLS equations are reduced to other variable coefficient NLS equations or the constant coefficient 3D NLS equation. By using these symmetry transformations, abundant exact solutions of some 3D NLS equations with distributed dispersion, nonlinearity, and gain or loss are obtained from the constant coefficient 3D NLS equation.展开更多
The symmetries and the exact solutions of the (3+l)-dimensional nonlinear incompressible non-hydrostatic Boussi- nesq (INHB) equations, which describe atmospheric gravity waves, are studied in this paper. The cal...The symmetries and the exact solutions of the (3+l)-dimensional nonlinear incompressible non-hydrostatic Boussi- nesq (INHB) equations, which describe atmospheric gravity waves, are studied in this paper. The calculation on symmetry shows that the equations are invariant under the Galilean transformations, the scaling transformations, and the space-time translations. Three types of symmetry reduction equations and similar solutions for the (3+ 1)-dimensional INHB equations are proposed. Traveling and non-traveling wave solutions of the INHB equations are demonstrated. The evolutions of the wind velocities in latitudinal, longitudinal, and vertical directions with space-time are demonstrated. The periodicity and the atmosphere viscosity are displayed in the (3+1)-dimensional INHB system.展开更多
This paper investigates an important high-dimensional model in the atmospheric and oceanic dynamics-(3+1)- dimensional nonlinear baroclinic potential vorticity equation by the classical Lie group method. Its symmet...This paper investigates an important high-dimensional model in the atmospheric and oceanic dynamics-(3+1)- dimensional nonlinear baroclinic potential vorticity equation by the classical Lie group method. Its symmetry algebra, symmetry group and group-invariant solutions are analysed. Otherwise, some exact explicit solutions are obtained from the corresponding (2+1)-dimensional equation, the inviscid barotropic nondivergent vorticy equation. To show the properties and characters of these solutions, some plots as well as their possible physical meanings of the atmospheric circulation are given out.展开更多
In order to get the exact traveling wave solutions to nonlinear partial differential equation, the complete discrimination system for polynomial and direct integral method are applied to the considered equation. All s...In order to get the exact traveling wave solutions to nonlinear partial differential equation, the complete discrimination system for polynomial and direct integral method are applied to the considered equation. All single traveling wave solutions to the equation can be obtained. As an example, we give the solutions to (3 + 1)-dimensional breaking soliton equation.展开更多
In this paper,we mainly focus on proving the existence of lump solutions to a generalized(3+1)-dimensional nonlinear differential equation.Hirota’s bilinear method and a quadratic function method are employed to deri...In this paper,we mainly focus on proving the existence of lump solutions to a generalized(3+1)-dimensional nonlinear differential equation.Hirota’s bilinear method and a quadratic function method are employed to derive the lump solutions localized in the whole plane for a(3+1)-dimensional nonlinear differential equation.Three examples of such a nonlinear equation are presented to investigate the exact expressions of the lump solutions.Moreover,the 3d plots and corresponding density plots of the solutions are given to show the space structures of the lump waves.In addition,the breath-wave solutions and several interaction solutions of the(3+1)-dimensional nonlinear differential equation are obtained and their dynamics are analyzed.展开更多
We applied the multiple exp-function scheme to the(2+1)-dimensional Sawada-Kotera(SK) equation and(3+1)-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analyti...We applied the multiple exp-function scheme to the(2+1)-dimensional Sawada-Kotera(SK) equation and(3+1)-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analytic particular solutions contain one-soliton, two-soliton, and three-soliton type solutions. With the assistance of Maple, we demonstrated the efficiency and advantages of the procedure that generalizes Hirota's perturbation scheme. The obtained solutions can be used as a benchmark for numerical solutions and describe the physical phenomena behind the model.展开更多
In this paper, we first obtain a bilinear form with small perturbation u_0 for a generalized(3+1)-dimensional nonlinear wave equation in liquid with gas bubbles. Based on that, a new bilinear B?cklund transformation w...In this paper, we first obtain a bilinear form with small perturbation u_0 for a generalized(3+1)-dimensional nonlinear wave equation in liquid with gas bubbles. Based on that, a new bilinear B?cklund transformation which consists of four bilinear equations and involves seven arbitrary parameters is constructed. After that, by applying a new symbolic computation method, we construct the higher order rogue waves with controllable center to the generalized(3+1)-dimensional nonlinear wave equation. The rogue waves present new structure, which contain two free parametersα and β. The dynamic properties of the higher order rogue waves are demonstrated graphically. The graphs tell that the parameters α and β can control the center of the rogue waves.展开更多
We investigate a generalized (3 + 1)-dimensional nonlinear wave equation, which can be used to depict many nonlinear phenomena in liquid containing gas bubbles. By employing the Hirota bilinear method, we derive its b...We investigate a generalized (3 + 1)-dimensional nonlinear wave equation, which can be used to depict many nonlinear phenomena in liquid containing gas bubbles. By employing the Hirota bilinear method, we derive its bilinear formalism and soliton solutions succinctly. Meanwhile, the first-order lump wave solution and second-order lump wave solution are well presented based on the corresponding two-soliton solution and four-soliton solution. Furthermore, two types of hybrid solutions are systematically established by using the long wave limit method. Finally, the graphical analyses of the obtained solutions are represented in order to better understand their dynamical behaviors.展开更多
In this paper,a new(3+1)-dimensional nonlinear evolution equation is introduced,through the generalized bilinear operators based on prime number p=3.By Maple symbolic calculation,one-,two-lump,and breather-type period...In this paper,a new(3+1)-dimensional nonlinear evolution equation is introduced,through the generalized bilinear operators based on prime number p=3.By Maple symbolic calculation,one-,two-lump,and breather-type periodic soliton solutions are obtained,where the condition of positiveness and analyticity of the lump solution are considered.The interaction solutions between the lump and multi-kink soliton,and the interaction between the lump and breather-type periodic soliton are derived,by combining multi-exponential function or trigonometric sine and cosine functions with a quadratic one.In addition,new interaction solutions between a lump,periodic-solitary waves,and one-,two-or even three-kink solitons are constructed by using the ansatz technique.Finally,the characteristics of these various solutions are exhibited and illustrated graphically.展开更多
In this paper, the(3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation for water waves is investigated. Through the Hirota method and Kadomtsev–Petviashvili hierarchy reduction, we obtain the first-o...In this paper, the(3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation for water waves is investigated. Through the Hirota method and Kadomtsev–Petviashvili hierarchy reduction, we obtain the first-order,higher-order, multiple rogue waves and lump solitons based on the solutions in terms of the Gramian. The first-order rogue waves are the line rogue waves which arise from the constant background and then disappear into the constant background again, while the first-order lump solitons propagate stably. Interactions among several first-order rogue waves which are described by the multiple rogue waves are presented. Elastic interactions of several first-order lump solitons are also presented. We find that the higher-order rogue waves and lump solitons can be treated as the superpositions of several first-order ones, while the interaction between the second-order lump solitons is inelastic.展开更多
In this paper,the third model of four(3+1)-dimensional nonlinear evolution equations,generated by the Jaulent-Miodek hierarchy,is investigated by the bifurcation method of planar dynamical systems.The 2-parameters dif...In this paper,the third model of four(3+1)-dimensional nonlinear evolution equations,generated by the Jaulent-Miodek hierarchy,is investigated by the bifurcation method of planar dynamical systems.The 2-parameters different bifurcation regions are obtained.According to the different phase portraits in 2-parameters different bifurcation regions,we obtain kink(anti-kink)wave solutions,solitary wave solutions and periodic wave solutions for the third of these models in the different subsets of 4-parameters space by dynamical system method.Furthermore,the explicit exact expressions of these bounded traveling waves are obtained.All these wave solutions are characterized by distinct physical structures.展开更多
基金supported by the Zhejiang Provincial Natural Science Foundations,China(Grant No.Y6090592)the National Natural Science Foundation of China(Grant Nos.11041003 and 10735030)+1 种基金the Ningbo Natural Science Foundation,China(Grant Nos.2010A610095,2010A610103,and 2009B21003)K.C.Wong Magna Fund in Ningbo University,China
文摘We present three families of soliton solutions to the generalized (3+l)-dimensional nonlinear Schrodinger equation with distributed coefficients. We investigate the dynamics of these solitons in nonlinear optics with some selected parameters. Different shapes of bright solitons, a train of bright solitons and dark solitons are observed. The obtained results may raise the possibilities of relevant experiments and potential applications.
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Project supported by the National Natural Science Foundation of China (Grant No. 11041003)the Ningbo Natural Science Foundation, China (Grant No. 2009B21003)K.C. Wong Magna Fund in Ningbo University, China
文摘In this paper, the extended symmetry transformation of (3+1)-dimensional (3D) generalized nonlinear Schrodinger (NLS) equations with variable coefficients is investigated by using the extended symmetry approach and symbolic computation. Then based on the extended symmetry, some 3D variable coefficient NLS equations are reduced to other variable coefficient NLS equations or the constant coefficient 3D NLS equation. By using these symmetry transformations, abundant exact solutions of some 3D NLS equations with distributed dispersion, nonlinearity, and gain or loss are obtained from the constant coefficient 3D NLS equation.
基金Project supported by the Natural Science Foundation of Guangdong Province, China (Grant Nos. 10452840301004616 and S2011040000403)the National Natural Science Foundation of China (Grant No. 41176005)the Science and Technology Project Foundation of Zhongshan, China (Grnat No. 20123A326)
文摘The symmetries and the exact solutions of the (3+l)-dimensional nonlinear incompressible non-hydrostatic Boussi- nesq (INHB) equations, which describe atmospheric gravity waves, are studied in this paper. The calculation on symmetry shows that the equations are invariant under the Galilean transformations, the scaling transformations, and the space-time translations. Three types of symmetry reduction equations and similar solutions for the (3+ 1)-dimensional INHB equations are proposed. Traveling and non-traveling wave solutions of the INHB equations are demonstrated. The evolutions of the wind velocities in latitudinal, longitudinal, and vertical directions with space-time are demonstrated. The periodicity and the atmosphere viscosity are displayed in the (3+1)-dimensional INHB system.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10735030,90718041 and 40975038)Shanghai Leading Academic Discipline Project(Grant No.B412)Program for Changjiang Scholars and Innovative Research Team in University(Grant No.IRT0734)
文摘This paper investigates an important high-dimensional model in the atmospheric and oceanic dynamics-(3+1)- dimensional nonlinear baroclinic potential vorticity equation by the classical Lie group method. Its symmetry algebra, symmetry group and group-invariant solutions are analysed. Otherwise, some exact explicit solutions are obtained from the corresponding (2+1)-dimensional equation, the inviscid barotropic nondivergent vorticy equation. To show the properties and characters of these solutions, some plots as well as their possible physical meanings of the atmospheric circulation are given out.
文摘In order to get the exact traveling wave solutions to nonlinear partial differential equation, the complete discrimination system for polynomial and direct integral method are applied to the considered equation. All single traveling wave solutions to the equation can be obtained. As an example, we give the solutions to (3 + 1)-dimensional breaking soliton equation.
基金supported by the National Natural Science Foundation of China(Nos.12101572,12371256)2023 Shanxi Province Graduate Innovation Project(No.2023KY614)the 19th Graduate Science and Technology Project of North University of China(No.20231943)。
文摘In this paper,we mainly focus on proving the existence of lump solutions to a generalized(3+1)-dimensional nonlinear differential equation.Hirota’s bilinear method and a quadratic function method are employed to derive the lump solutions localized in the whole plane for a(3+1)-dimensional nonlinear differential equation.Three examples of such a nonlinear equation are presented to investigate the exact expressions of the lump solutions.Moreover,the 3d plots and corresponding density plots of the solutions are given to show the space structures of the lump waves.In addition,the breath-wave solutions and several interaction solutions of the(3+1)-dimensional nonlinear differential equation are obtained and their dynamics are analyzed.
基金Supported by the National Natural Science Foundation of China under Grant No.10974177by the Ministry of Science and Technology of China under Grant No.2010DFA04690
文摘We applied the multiple exp-function scheme to the(2+1)-dimensional Sawada-Kotera(SK) equation and(3+1)-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analytic particular solutions contain one-soliton, two-soliton, and three-soliton type solutions. With the assistance of Maple, we demonstrated the efficiency and advantages of the procedure that generalizes Hirota's perturbation scheme. The obtained solutions can be used as a benchmark for numerical solutions and describe the physical phenomena behind the model.
基金Supported by the National Natural Science Foundation of China(11471004,11501498)Shaanxi Key Research and Development Programs(2018SF-251)the Research Project at Yuncheng University [XK2012007]
文摘In this paper, we first obtain a bilinear form with small perturbation u_0 for a generalized(3+1)-dimensional nonlinear wave equation in liquid with gas bubbles. Based on that, a new bilinear B?cklund transformation which consists of four bilinear equations and involves seven arbitrary parameters is constructed. After that, by applying a new symbolic computation method, we construct the higher order rogue waves with controllable center to the generalized(3+1)-dimensional nonlinear wave equation. The rogue waves present new structure, which contain two free parametersα and β. The dynamic properties of the higher order rogue waves are demonstrated graphically. The graphs tell that the parameters α and β can control the center of the rogue waves.
基金the Natural Science Foundation of Jiangsu Province (Grant No. BK20181351)the ‘Qinglan Engineering project’ of Jiangsu Universities, the National Natural Science Foundation of China (Grant No. 11301527)+1 种基金the Fundamental Research Fund for the Central Universities (Grant No. 2019QNA35)the General Financial Grant from the China Postdoctoral Science Foundation (Grant Nos. 2015M570498, 2017T100413).
文摘We investigate a generalized (3 + 1)-dimensional nonlinear wave equation, which can be used to depict many nonlinear phenomena in liquid containing gas bubbles. By employing the Hirota bilinear method, we derive its bilinear formalism and soliton solutions succinctly. Meanwhile, the first-order lump wave solution and second-order lump wave solution are well presented based on the corresponding two-soliton solution and four-soliton solution. Furthermore, two types of hybrid solutions are systematically established by using the long wave limit method. Finally, the graphical analyses of the obtained solutions are represented in order to better understand their dynamical behaviors.
基金supported by the National Natural Science Foundation of China No.11835011 and No.11675146。
文摘In this paper,a new(3+1)-dimensional nonlinear evolution equation is introduced,through the generalized bilinear operators based on prime number p=3.By Maple symbolic calculation,one-,two-lump,and breather-type periodic soliton solutions are obtained,where the condition of positiveness and analyticity of the lump solution are considered.The interaction solutions between the lump and multi-kink soliton,and the interaction between the lump and breather-type periodic soliton are derived,by combining multi-exponential function or trigonometric sine and cosine functions with a quadratic one.In addition,new interaction solutions between a lump,periodic-solitary waves,and one-,two-or even three-kink solitons are constructed by using the ansatz technique.Finally,the characteristics of these various solutions are exhibited and illustrated graphically.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11772017,11272023,and 11471050by the Open Fund of State Key Laboratory of Information Photonics and Optical Communications(Beijing University of Posts and Telecommunications),China(IPOC:2017ZZ05)by the Fundamental Research Funds for the Central Universities of China under Grant No.2011BUPTYB02
文摘In this paper, the(3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation for water waves is investigated. Through the Hirota method and Kadomtsev–Petviashvili hierarchy reduction, we obtain the first-order,higher-order, multiple rogue waves and lump solitons based on the solutions in terms of the Gramian. The first-order rogue waves are the line rogue waves which arise from the constant background and then disappear into the constant background again, while the first-order lump solitons propagate stably. Interactions among several first-order rogue waves which are described by the multiple rogue waves are presented. Elastic interactions of several first-order lump solitons are also presented. We find that the higher-order rogue waves and lump solitons can be treated as the superpositions of several first-order ones, while the interaction between the second-order lump solitons is inelastic.
基金the National Natural Science Foundation of China through Grant No.11772007the Natural Science Foundation of Beijing through Grant No.1172002+2 种基金the International Science and Technology Cooperation Program of China through Grant No.2014DFR61080the Research Fund for Higher Education of Gansu No.2018B-48the Research Fund for 13th Five-Year Plan for Educational Science of Gansu No.GS[2018]GHBBK072.
文摘In this paper,the third model of four(3+1)-dimensional nonlinear evolution equations,generated by the Jaulent-Miodek hierarchy,is investigated by the bifurcation method of planar dynamical systems.The 2-parameters different bifurcation regions are obtained.According to the different phase portraits in 2-parameters different bifurcation regions,we obtain kink(anti-kink)wave solutions,solitary wave solutions and periodic wave solutions for the third of these models in the different subsets of 4-parameters space by dynamical system method.Furthermore,the explicit exact expressions of these bounded traveling waves are obtained.All these wave solutions are characterized by distinct physical structures.