Based on the method of Hirota＇s bilinear derivative transform, the derivative nonlinear Schrodinger equation with vanishing boundary condition has been directly solved. The oneand two-soliton solutions are given as t...Based on the method of Hirota＇s bilinear derivative transform, the derivative nonlinear Schrodinger equation with vanishing boundary condition has been directly solved. The oneand two-soliton solutions are given as two typical examples in the illustration of the general procedures and the concrete cut-off technique of the series-form solution, and the n-soliton solution is also attained by induction method. Our study shows their equivalence to the existing soliton solutions by a simple parameter transformation. The methodological importance of bilinear derivative transform in dealing with an integrable nonlinear equation has also been emphasized. The evolution of one and two-soliton solution with respect to time and space has been discussed in detail. The collision among the solitons has been manifested through an example of two-soliton case, revealing the elastic essence of the collision and the invariance of the soliton form and characteristics.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
In this paper,the(3+1)-dimensional nonlinear evolution equation is studied analytically.The bilinear form of given model is achieved by using the Hirota bilinear method.As a result,the lump waves and col-lisions betwe...In this paper,the(3+1)-dimensional nonlinear evolution equation is studied analytically.The bilinear form of given model is achieved by using the Hirota bilinear method.As a result,the lump waves and col-lisions between lumps and periodic waves,the collision among lump wave and single,double-kink soliton solutions as well as the collision between lump,periodic,and single,double-kink soliton solutions for the given model are constructed.Furthermore,some new traveling wave solutions are developed by applying the exp(−φ(ξ))expansion method.The 3D,2D and contours plots are drawn to demonstrate the nature of the nonlinear model for setting appropriate set of parameters.As a result,a collection of bright,dark,periodic,rational function and elliptic function solutions are established.The applied strategies appear to be more powerful and efficient approaches to construct some new traveling wave structures for various contemporary models of recent era.展开更多
In this paper,we describe the nonlinear behavior of a generalized fourth-order Hietarinta-type equa-tion for dispersive waves in(2+1)dimension.The various wave formations are retrieved by using Hirota’s bilinear meth...In this paper,we describe the nonlinear behavior of a generalized fourth-order Hietarinta-type equa-tion for dispersive waves in(2+1)dimension.The various wave formations are retrieved by using Hirota’s bilinear method(HBM)and various test function perspectives.The Hirota method is a widely used and robust mathematical tool for finding soliton solutions of nonlinear partial differential equa-tions(NLPDEs)in a variety of disciplines like mathematical physics,nonlinear dynamics,oceanography,engineering sciences,and others requires bilinearization of nonlinear PDEs.The different wave structures in the forms of new breather,lump-periodic,rogue waves,and two-wave solutions are recovered.In addi-tion,the physical behavior of the acquired solutions is illustrated in three-dimensional,two-dimensional,density,and contour profiles by the assistance of suitable parameters.Based on the obtained results,we can assert that the employed methodology is straightforward,dynamic,highly efficient,and will serve as a valuable tool for discussing complex issues in a diversity of domains specifically ocean and coastal engineering.We have also made an important first step in understanding the structure and physical be-havior of complex structures with our findings here.We believe this research is timely and relevant to a wide range of engineering modelers.The results obtained are useful for comprehending the fundamental scenarios of nonlinear sciences.展开更多
Starting from the multi-soliton solutions obtained by the Hirota bilinear method,the soli ton molecule structures for the combined mKdV-type bilinear equation(Dt+∑n=1NαnDx2n+1)f*·f=0 are investigated using the ...Starting from the multi-soliton solutions obtained by the Hirota bilinear method,the soli ton molecule structures for the combined mKdV-type bilinear equation(Dt+∑n=1NαnDx2n+1)f*·f=0 are investigated using the velocity resonance mechanism.The two-soliton molecules of the mKdV-35 equation and the three-soliton molecules of the mKdV-357 equation are specifically demonstrated in this paper.With particular selections of the involved arbitrary parameters,especially the wave numbers,it is confirmed that,besides the usual multi-bright soliton molecules,the multi-dark soliton molecules and the mixed multibright-dark soliton molecules can also be obtained.In addition,we discuss the existence of the multi-soliton molecules for the combined mKdV-type bilinear equation with more higher order nonlinear terms and dispersions.The results demonstrate that when N≥4,the combined mKdVtype bilinear equation no longer admits soliton molecules comprising more than four solitons.展开更多
2N line-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equation can be presented by resorting tothe Hirota bilinear method.In this paper,N periodic-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equation...2N line-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equation can be presented by resorting tothe Hirota bilinear method.In this paper,N periodic-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equationare obtained from the 2N line-soliton solutions by selecting the parameters into conjugated complex parameters in pairs.展开更多
A class of lump solutions of(2+1)-dimensional Boussinesq equation are obtained with the help of Maple by using Hirota bilinear method.Some contour plots with different determinant values are sequentially made to show ...A class of lump solutions of(2+1)-dimensional Boussinesq equation are obtained with the help of Maple by using Hirota bilinear method.Some contour plots with different determinant values are sequentially made to show that the corresponding lump solution tends to zero when the determinant approaches zero.The particular lump solutions with specific values of the involved parameters are plotted,as illustrative examples.展开更多
Lump solutions are one of the most common solutions for nonlinear evolution equations.This study aspires to investigate the generalized Hietarintatype equation.We auspiciously provide multiple M-lump waves.On the othe...Lump solutions are one of the most common solutions for nonlinear evolution equations.This study aspires to investigate the generalized Hietarintatype equation.We auspiciously provide multiple M-lump waves.On the other hand,collision phenomena to multiple M-lump waves with soliton wave solutions are also provided.During the collision,the amplitude of the lump will change significantly over the processes,whereas the amplitude of the soliton will just minimally alter.As it is of paramount importance,we use suitable values of parameter to put out the physical features of the reported results through three dimensional and contour graphics.The results presented express physical features of lump and lump interaction phenomena of different kinds of nonlinear physical processes.Further,this study serves to enrich nonlinear dynamics and provide insight into how nonlinear waves propagate.展开更多
This article investigates the Hirota-Satsuma-Ito equation with variable coefficient using the Hirota bilinear method and the long wave limit method.The equation is proved to be Painlevé integrable by Painlevé...This article investigates the Hirota-Satsuma-Ito equation with variable coefficient using the Hirota bilinear method and the long wave limit method.The equation is proved to be Painlevé integrable by Painlevé analysis.On the basis of the bilinear form,the forms of two-soliton solutions,three-soliton solutions,and four-soliton solutions are studied specifically.The appropriate parameter values are chosen and the corresponding figures are presented.The breather waves solutions,lump solutions,periodic solutions and the interaction of breather waves solutions and soliton solutions,etc.are given.In addition,we also analyze the different effects of the parameters on the figures.The figures of the same set of parameters in different planes are presented to describe the dynamical behavior of solutions.These are important for describing water waves in nature.展开更多
In this work,we study a new(2+1)-dimensional generalized breaking soliton equation which admits the Painleve property for one special set of parameters.We derive multiple soliton solutions,traveling wave solutions,and...In this work,we study a new(2+1)-dimensional generalized breaking soliton equation which admits the Painleve property for one special set of parameters.We derive multiple soliton solutions,traveling wave solutions,and periodic solutions as well.We use the simplified Hirotas method and a variety of ansatze to achieve our goal.展开更多
基金Supported by the National Natural Science Foundation of China (10775105)
文摘Based on the method of Hirota＇s bilinear derivative transform, the derivative nonlinear Schrodinger equation with vanishing boundary condition has been directly solved. The oneand two-soliton solutions are given as two typical examples in the illustration of the general procedures and the concrete cut-off technique of the series-form solution, and the n-soliton solution is also attained by induction method. Our study shows their equivalence to the existing soliton solutions by a simple parameter transformation. The methodological importance of bilinear derivative transform in dealing with an integrable nonlinear equation has also been emphasized. The evolution of one and two-soliton solution with respect to time and space has been discussed in detail. The collision among the solitons has been manifested through an example of two-soliton case, revealing the elastic essence of the collision and the invariance of the soliton form and characteristics.
基金supported in part by the National Basic Research Program (2007CB814906)the National Natural Science Foundation of China (10471103 and 10771158)+2 种基金Social Science Foundation of the Ministry of Education of China (Numerical methods for convertible bonds, 06JA630047)Tianjin Natural Science Foundation (07JCYBJC14300)the National Science Foundation under Grant No. EAR-0934747
文摘This article summarizes our recent work on uniform error estimates for various finite elementmethods for time-dependent advection-diffusion equations.
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.
文摘In this paper,the(3+1)-dimensional nonlinear evolution equation is studied analytically.The bilinear form of given model is achieved by using the Hirota bilinear method.As a result,the lump waves and col-lisions between lumps and periodic waves,the collision among lump wave and single,double-kink soliton solutions as well as the collision between lump,periodic,and single,double-kink soliton solutions for the given model are constructed.Furthermore,some new traveling wave solutions are developed by applying the exp(−φ(ξ))expansion method.The 3D,2D and contours plots are drawn to demonstrate the nature of the nonlinear model for setting appropriate set of parameters.As a result,a collection of bright,dark,periodic,rational function and elliptic function solutions are established.The applied strategies appear to be more powerful and efficient approaches to construct some new traveling wave structures for various contemporary models of recent era.
基金support provided for this research via Open Fund of State Key Laboratory of Power Grid Environmental Protection (No.GYW51202101374).
文摘In this paper,we describe the nonlinear behavior of a generalized fourth-order Hietarinta-type equa-tion for dispersive waves in(2+1)dimension.The various wave formations are retrieved by using Hirota’s bilinear method(HBM)and various test function perspectives.The Hirota method is a widely used and robust mathematical tool for finding soliton solutions of nonlinear partial differential equa-tions(NLPDEs)in a variety of disciplines like mathematical physics,nonlinear dynamics,oceanography,engineering sciences,and others requires bilinearization of nonlinear PDEs.The different wave structures in the forms of new breather,lump-periodic,rogue waves,and two-wave solutions are recovered.In addi-tion,the physical behavior of the acquired solutions is illustrated in three-dimensional,two-dimensional,density,and contour profiles by the assistance of suitable parameters.Based on the obtained results,we can assert that the employed methodology is straightforward,dynamic,highly efficient,and will serve as a valuable tool for discussing complex issues in a diversity of domains specifically ocean and coastal engineering.We have also made an important first step in understanding the structure and physical be-havior of complex structures with our findings here.We believe this research is timely and relevant to a wide range of engineering modelers.The results obtained are useful for comprehending the fundamental scenarios of nonlinear sciences.
基金the National Natural Science Foundation of China(Grant Nos.11975204 and 12075208)the Project of Zhoushan City Science and Technology Bureau(Grant No.2021C21015)the Training Program for Leading Talents in Universities of Zhejiang Province。
文摘Starting from the multi-soliton solutions obtained by the Hirota bilinear method,the soli ton molecule structures for the combined mKdV-type bilinear equation(Dt+∑n=1NαnDx2n+1)f*·f=0 are investigated using the velocity resonance mechanism.The two-soliton molecules of the mKdV-35 equation and the three-soliton molecules of the mKdV-357 equation are specifically demonstrated in this paper.With particular selections of the involved arbitrary parameters,especially the wave numbers,it is confirmed that,besides the usual multi-bright soliton molecules,the multi-dark soliton molecules and the mixed multibright-dark soliton molecules can also be obtained.In addition,we discuss the existence of the multi-soliton molecules for the combined mKdV-type bilinear equation with more higher order nonlinear terms and dispersions.The results demonstrate that when N≥4,the combined mKdVtype bilinear equation no longer admits soliton molecules comprising more than four solitons.
基金supported by the State Key Basic Research Program of China under Grant No.2004CB318000National Natural Science Foundation of China under Grant No.10771072
文摘2N line-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equation can be presented by resorting tothe Hirota bilinear method.In this paper,N periodic-soliton solutions of the (3+1)-dimensional Jimbo-Miwa equationare obtained from the 2N line-soliton solutions by selecting the parameters into conjugated complex parameters in pairs.
基金Supported by the National Natural Science Foundation of China under Grant No.10647112the Fund of Science and Technology Commission of Shanghai Municipality under Grant No.ZX201307000014
文摘A class of lump solutions of(2+1)-dimensional Boussinesq equation are obtained with the help of Maple by using Hirota bilinear method.Some contour plots with different determinant values are sequentially made to show that the corresponding lump solution tends to zero when the determinant approaches zero.The particular lump solutions with specific values of the involved parameters are plotted,as illustrative examples.
文摘Lump solutions are one of the most common solutions for nonlinear evolution equations.This study aspires to investigate the generalized Hietarintatype equation.We auspiciously provide multiple M-lump waves.On the other hand,collision phenomena to multiple M-lump waves with soliton wave solutions are also provided.During the collision,the amplitude of the lump will change significantly over the processes,whereas the amplitude of the soliton will just minimally alter.As it is of paramount importance,we use suitable values of parameter to put out the physical features of the reported results through three dimensional and contour graphics.The results presented express physical features of lump and lump interaction phenomena of different kinds of nonlinear physical processes.Further,this study serves to enrich nonlinear dynamics and provide insight into how nonlinear waves propagate.
基金This work was supported by the National Natural Science Foundation of China(Grant No.11505090)Research Award Foundation for Outstanding Young Scientists of Shandong Province(Grant No.BS2015SF009)+2 种基金the Doctoral Foundation of Liaocheng University(Grant No.318051413)Liaocheng University Level Science and Technology Research Fund(Grant No.318012018)Discipline with Strong Characteristics of Liaocheng University–Intelligent Science and Technology(Grant No.319462208).
文摘This article investigates the Hirota-Satsuma-Ito equation with variable coefficient using the Hirota bilinear method and the long wave limit method.The equation is proved to be Painlevé integrable by Painlevé analysis.On the basis of the bilinear form,the forms of two-soliton solutions,three-soliton solutions,and four-soliton solutions are studied specifically.The appropriate parameter values are chosen and the corresponding figures are presented.The breather waves solutions,lump solutions,periodic solutions and the interaction of breather waves solutions and soliton solutions,etc.are given.In addition,we also analyze the different effects of the parameters on the figures.The figures of the same set of parameters in different planes are presented to describe the dynamical behavior of solutions.These are important for describing water waves in nature.
文摘In this work,we study a new(2+1)-dimensional generalized breaking soliton equation which admits the Painleve property for one special set of parameters.We derive multiple soliton solutions,traveling wave solutions,and periodic solutions as well.We use the simplified Hirotas method and a variety of ansatze to achieve our goal.
