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
Through the Hirota bilinear formulation and the symbolic computation software Maple, we construct lump-type solutions for a generalized(3+1)-dimensional Kadomtsev-Petviashvili(KP) equation in three cases of the coeffi...Through the Hirota bilinear formulation and the symbolic computation software Maple, we construct lump-type solutions for a generalized(3+1)-dimensional Kadomtsev-Petviashvili(KP) equation in three cases of the coefficients in the equation. Then the sufficient and necessary conditions to guarantee the analyticity of the resulting lump-type solutions(or the positivity of the corresponding quadratic solutions to the associated bilinear equation) are discussed. To illustrate the generality of the obtained solutions, two concrete lump-type solutions are explicitly presented, and to analyze the dynamic behaviors of the solutions specifically, the three-dimensional plots and contour profiles of these two lump-type solutions with particular choices of the involved free parameters are well displayed.展开更多
We study a simplified(3+1)-dimensional model equation and construct a lump solution for the special case of z=y using the Hirota bilinear method.Then,a more general form of lump solution is constructed,which contains ...We study a simplified(3+1)-dimensional model equation and construct a lump solution for the special case of z=y using the Hirota bilinear method.Then,a more general form of lump solution is constructed,which contains more arbitrary autocephalous parameters.In addition,a lumpoff solution is also derived based on the general lump solutions and a stripe soliton.Furthermore,we figure out instanton/rogue wave solutions via introducing two stripe solitons.Finally,one can better illustrate these propagation phenomena of these solutions by analyzing images.展开更多
Aim The general arbitrary cracked problem in an elastic plane was discussed. Methods For the purpose of acquiring the solution of the problem, a new formulation on the problem was proposed. Compared with the classic...Aim The general arbitrary cracked problem in an elastic plane was discussed. Methods For the purpose of acquiring the solution of the problem, a new formulation on the problem was proposed. Compared with the classical plane elastic crack model, only the known conditions were revised in the new formulation, which are greatly convenient to solve the problem, and no other new condition was given. Results and Conclusion The general exact analytic solution is given here based on the formulation though the problem is very complicated. Furthermore, the stress intensity factors K Ⅰ, K Ⅱ of the problem are also given.展开更多
In this paper, we obtained a kind of lump solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation with the assistance of Mathematica. Some contour plots with different determinant values are seq...In this paper, we obtained a kind of lump solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation with the assistance of Mathematica. Some contour plots with different determinant values are sequentially made to show that the corresponding lump solutions tend to zero when x2+y2→∞. Particularly, lump solutions with specific values of the include parameters are plotted, as illustrative examples. Finally, a combination of stripe soliton and lump soliton is discussed to the KP-BBM equation, in which such a solution presents two different interesting phenomena: lump-kink and lump-soliton. Simultaneously, breather rational soliton solutions are displayed.展开更多
The(3+1)-dimensional Burgers equation, which describes nonlinear waves in turbulence and the interface dynamics,is considered. Two types of semi-rational solutions, namely, the lump–kink solution and the lump–two ki...The(3+1)-dimensional Burgers equation, which describes nonlinear waves in turbulence and the interface dynamics,is considered. Two types of semi-rational solutions, namely, the lump–kink solution and the lump–two kinks solution, are constructed from the quadratic function ansatz. Some interesting features of interactions between lumps and other solitons are revealed analytically and shown graphically, such as fusion and fission processes.展开更多
In this paper, based on Hirota bilinear form, we aim to show the diversity of interaction solutions to the (2 + 1)-dimensional Sawada-Kotera (SK) equation. By introducing an arbitrary differentiable function in assump...In this paper, based on Hirota bilinear form, we aim to show the diversity of interaction solutions to the (2 + 1)-dimensional Sawada-Kotera (SK) equation. By introducing an arbitrary differentiable function in assumption form, we can obtain abundant interaction solutions which can provide the possibility for exploring the interactions between lump waves and other kinds of waves. By choosing some particular functions and values of the involved parameters, we give four illustrative examples of the resulting solutions, and explore some novel interaction behaviors in (2 + 1)-dimensional SK equation.展开更多
With symbolic computation, some lump solutions are presented to a(3+1)-dimensional nonlinear evolution equation by searching the positive quadratic function from the Hirota bilinear form of equation. The quadratic fun...With symbolic computation, some lump solutions are presented to a(3+1)-dimensional nonlinear evolution equation by searching the positive quadratic function from the Hirota bilinear form of equation. The quadratic function contains six free parameters, four of which satisfy two determinant conditions guaranteeing analyticity and rational localization of the solutions, while the others are free. Then, by combining positive quadratic function with exponential function, the interaction solutions between lump solutions and the stripe solitons are presented on the basis of some conditions. Furthermore, we extend this method to obtain more general solutions by combining of positive quadratic function and hyperbolic cosine function. Thus the interaction solutions between lump solutions and a pair of resonance stripe solitons are derived and asymptotic property of the interaction solutions are analyzed under some specific conditions. Finally, the dynamic properties of these solutions are shown in figures by choosing the values of the parameters.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, we deal with the model with a very general growth law and an M- driven diffusion For the general case of time dependent functions M and #, the existence and uniqueness for positive solution is obtained....In this paper, we deal with the model with a very general growth law and an M- driven diffusion For the general case of time dependent functions M and #, the existence and uniqueness for positive solution is obtained. If M and # are T0-periodic functions in t, then there is an attractive positive periodic solution. Furthermore, if M and # are time-independent, then the non-constant stationary solution M（x） is globally stable. Thus, we can easily formulate the conditions deriving the above behaviors for specific population models with the logistic growth law, Gilpin-Ayala growth law and Gompertz growth law, respectively. We answer an open problem proposed by L. Korobenko and E. Braverman in [Can. Appl. Math. Quart. 17（2009） 85-104].展开更多
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 Nos.11505154,11605156,11775146,and 11975204)the Zhejiang Provincial Natural Science Foundation of China(Grant Nos.LQ16A010003 and LY19A050003)+5 种基金the China Scholarship Council(Grant No.201708330479)the Foundation for Doctoral Program of Zhejiang Ocean University(Grant No.Q1511)the Natural Science Foundation(Grant No.DMS-1664561)the Distinguished Professorships by Shanghai University of Electric Power(China)North-West University(South Africa)King Abdulaziz University(Saudi Arabia)
文摘Through the Hirota bilinear formulation and the symbolic computation software Maple, we construct lump-type solutions for a generalized(3+1)-dimensional Kadomtsev-Petviashvili(KP) equation in three cases of the coefficients in the equation. Then the sufficient and necessary conditions to guarantee the analyticity of the resulting lump-type solutions(or the positivity of the corresponding quadratic solutions to the associated bilinear equation) are discussed. To illustrate the generality of the obtained solutions, two concrete lump-type solutions are explicitly presented, and to analyze the dynamic behaviors of the solutions specifically, the three-dimensional plots and contour profiles of these two lump-type solutions with particular choices of the involved free parameters are well displayed.
基金Project supported by the National Natural Science Foundation of China(Grant No.11971475)。
文摘We study a simplified(3+1)-dimensional model equation and construct a lump solution for the special case of z=y using the Hirota bilinear method.Then,a more general form of lump solution is constructed,which contains more arbitrary autocephalous parameters.In addition,a lumpoff solution is also derived based on the general lump solutions and a stripe soliton.Furthermore,we figure out instanton/rogue wave solutions via introducing two stripe solitons.Finally,one can better illustrate these propagation phenomena of these solutions by analyzing images.
文摘Aim The general arbitrary cracked problem in an elastic plane was discussed. Methods For the purpose of acquiring the solution of the problem, a new formulation on the problem was proposed. Compared with the classical plane elastic crack model, only the known conditions were revised in the new formulation, which are greatly convenient to solve the problem, and no other new condition was given. Results and Conclusion The general exact analytic solution is given here based on the formulation though the problem is very complicated. Furthermore, the stress intensity factors K Ⅰ, K Ⅱ of the problem are also given.
文摘In this paper, we obtained a kind of lump solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation with the assistance of Mathematica. Some contour plots with different determinant values are sequentially made to show that the corresponding lump solutions tend to zero when x2+y2→∞. Particularly, lump solutions with specific values of the include parameters are plotted, as illustrative examples. Finally, a combination of stripe soliton and lump soliton is discussed to the KP-BBM equation, in which such a solution presents two different interesting phenomena: lump-kink and lump-soliton. Simultaneously, breather rational soliton solutions are displayed.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11501323,11701323,and 11605102)。
文摘The(3+1)-dimensional Burgers equation, which describes nonlinear waves in turbulence and the interface dynamics,is considered. Two types of semi-rational solutions, namely, the lump–kink solution and the lump–two kinks solution, are constructed from the quadratic function ansatz. Some interesting features of interactions between lumps and other solitons are revealed analytically and shown graphically, such as fusion and fission processes.
文摘In this paper, based on Hirota bilinear form, we aim to show the diversity of interaction solutions to the (2 + 1)-dimensional Sawada-Kotera (SK) equation. By introducing an arbitrary differentiable function in assumption form, we can obtain abundant interaction solutions which can provide the possibility for exploring the interactions between lump waves and other kinds of waves. By choosing some particular functions and values of the involved parameters, we give four illustrative examples of the resulting solutions, and explore some novel interaction behaviors in (2 + 1)-dimensional SK equation.
基金Supported by National Natural Science Foundation of China under Grant Nos.11271211,11275072,and 11435005Ningbo 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.xkzw11502K.C.Wong Magna Fund in Ningbo University
文摘With symbolic computation, some lump solutions are presented to a(3+1)-dimensional nonlinear evolution equation by searching the positive quadratic function from the Hirota bilinear form of equation. The quadratic function contains six free parameters, four of which satisfy two determinant conditions guaranteeing analyticity and rational localization of the solutions, while the others are free. Then, by combining positive quadratic function with exponential function, the interaction solutions between lump solutions and the stripe solitons are presented on the basis of some conditions. Furthermore, we extend this method to obtain more general solutions by combining of positive quadratic function and hyperbolic cosine function. Thus the interaction solutions between lump solutions and a pair of resonance stripe solitons are derived and asymptotic property of the interaction solutions are analyzed under some specific conditions. Finally, the dynamic properties of these solutions are shown in figures by choosing the values of the parameters.
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(11271342)
文摘In this paper, we deal with the model with a very general growth law and an M- driven diffusion For the general case of time dependent functions M and #, the existence and uniqueness for positive solution is obtained. If M and # are T0-periodic functions in t, then there is an attractive positive periodic solution. Furthermore, if M and # are time-independent, then the non-constant stationary solution M（x） is globally stable. Thus, we can easily formulate the conditions deriving the above behaviors for specific population models with the logistic growth law, Gilpin-Ayala growth law and Gompertz growth law, respectively. We answer an open problem proposed by L. Korobenko and E. Braverman in [Can. Appl. Math. Quart. 17（2009） 85-104].
