We gave the localized solutions,the interaction solutions and the mixed solutions to a reduced(3+1)-dimensional nonlinear evolution equation.These solutions were characterized by superposition formulas of positive qua...We gave the localized solutions,the interaction solutions and the mixed solutions to a reduced(3+1)-dimensional nonlinear evolution equation.These solutions were characterized by superposition formulas of positive quadratic functions,the exponential and hyperbolic functions.According to the known lump solution in the outset,we obtained the superposition formulas of positive quadratic functions by plausible reasoning.Next,we constructed the interaction solutions between the localized solutions and the exponential function solutions with the similar theory.These two kinds of solutions contained superposition formulas of positive quadratic functions,which were turned into general ternary quadratic functions,the coefficients of which were all rational operation of vector inner product.Then we obtained linear superposition formulas of exponential and hyperbolic function solutions.Finally,for aforementioned various solutions,their dynamic properties were showed by choosing specific values for parameters.From concrete plots,we observed wave characteristics of three kinds of solutions.Especially,we could observe distinct generation and separation situations when the localized wave and the stripe wave interacted at different time points.展开更多
Starting from the step-by-step iterative method, the analytical formulas of solutions of the geometrically nonlinear equations of the axisymmetric plates and shallow shells, have been obtained. The uniform convergence...Starting from the step-by-step iterative method, the analytical formulas of solutions of the geometrically nonlinear equations of the axisymmetric plates and shallow shells, have been obtained. The uniform convergence of the iterative method, is used to prove the convergence of the analytical formulas of the exact solutions of the equa- tions.展开更多
The method combining the function transformation with the auxiliary equation is presented and the new infinite sequence complexion solutions of a class of nonlinear evolutionary equations are constructed. Step one, ac...The method combining the function transformation with the auxiliary equation is presented and the new infinite sequence complexion solutions of a class of nonlinear evolutionary equations are constructed. Step one, according to two function transformations, a class of nonlinear evolutionary equations is changed into two kinds of ordinary differential equations. Step two, using the first integral of the ordinary differential equations, two first order nonlinear ordinary differential equations are obtained. Step three, using function transformation, two first order nonlinear ordinary differential equations are changed to the ordinary differential equation that could be integrated. Step four, the new solutions, B?cklund transformation and the nonlinear superposition formula of solutions of the ordinary differential equation that could be integrated are applied to construct the new infinite sequence complexion solutions of a class of nonlinear evolutionary equations. These solutions are consisting of two-soliton solutions, two-period solutions and solutions composed of soliton solutions and period solutions.展开更多
The solutions to a linear wave equation can satisfy the principle of superposition, i.e., the linear superposition of two or more known solutions is still a solution of the linear wave equation. We show in this articl...The solutions to a linear wave equation can satisfy the principle of superposition, i.e., the linear superposition of two or more known solutions is still a solution of the linear wave equation. We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic, triangle, and exponential functions, and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics. The linear superposition solutions to the generalized KdV equation K（2,2,1）, the Oliver water wave equation, and the k（n, n） equation are given. The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed, and the reason why the solutions with the forms of hyperbolic, triangle, and exponential functions can form the linear superposition solutions is also discussed.展开更多
New forms of different-periodic travelling wave solutions for the （2＋1）-dimensional Zakharov-Kuznetsov （ZK） equation and the Davey-Stewartson （DS） equation are obtained by the linear superposition approach of J...New forms of different-periodic travelling wave solutions for the （2＋1）-dimensional Zakharov-Kuznetsov （ZK） equation and the Davey-Stewartson （DS） equation are obtained by the linear superposition approach of Jacobi elliptic function. A sequence of cyclic identities plays an important role in these procedures.展开更多
Using Jacobi elliptic function linear superposition approach for the(1+1)-dimensional Caudrey-Dodd-Gibbon-Sawada-Kotera(CDGSK)equation and the(2+ 1)-dimensional Nizhnik-Novikov-Veselov(NNV)equation,many new periodic t...Using Jacobi elliptic function linear superposition approach for the(1+1)-dimensional Caudrey-Dodd-Gibbon-Sawada-Kotera(CDGSK)equation and the(2+ 1)-dimensional Nizhnik-Novikov-Veselov(NNV)equation,many new periodic travelling wave solutions with different periods and velocities are obtained based on the known periodic solutions.This procedure is crucially dependent on a sequence of cyclic identities involving Jacobi elliptic functions sn(ζ,m),cn(ζ,m),and dn(ζ,m).展开更多
In this paper,a Boussinesq hierarchy in the bilinear form is proposed. A Backlund transformation for this hierarchy is presented and the nonlinear superposition formula is proved rigorously.
A monotonicity formula for stable solutions to a class of weighted semilinear elliptic equations with "negative exponent" is established.It is well known that such a monotonicity formula plays an essential r...A monotonicity formula for stable solutions to a class of weighted semilinear elliptic equations with "negative exponent" is established.It is well known that such a monotonicity formula plays an essential role in the study of finite Morse index solutions of equations with "positive exponent".Unlike the positive exponent case,we will see that both the monotonicity formula and the sub-harmonicity play crucial roles in classifying positive finite Morse index solutions to the equations with negative exponent and obtaining sharp results for their asymptotic behaviors.展开更多
In this paper,we investigate a class of nonlinear backward stochastic differential equations(BSDEs)arising from financial economics,and give the sign of corresponding solution.Furthermore,we are able to obtain explici...In this paper,we investigate a class of nonlinear backward stochastic differential equations(BSDEs)arising from financial economics,and give the sign of corresponding solution.Furthermore,we are able to obtain explicit solutions to an interesting class of nonlinear BSDEs,including the k-ignorance BSDE arising from the modeling of ambiguity of asset pricing.Moreover,we show its applications in PDEs and contingent pricing in an incomplete market.展开更多
基金the National Natural Science Foundation of China(Grant No.12061054)Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region of China(Grant No.NJYT-20A06)。
文摘We gave the localized solutions,the interaction solutions and the mixed solutions to a reduced(3+1)-dimensional nonlinear evolution equation.These solutions were characterized by superposition formulas of positive quadratic functions,the exponential and hyperbolic functions.According to the known lump solution in the outset,we obtained the superposition formulas of positive quadratic functions by plausible reasoning.Next,we constructed the interaction solutions between the localized solutions and the exponential function solutions with the similar theory.These two kinds of solutions contained superposition formulas of positive quadratic functions,which were turned into general ternary quadratic functions,the coefficients of which were all rational operation of vector inner product.Then we obtained linear superposition formulas of exponential and hyperbolic function solutions.Finally,for aforementioned various solutions,their dynamic properties were showed by choosing specific values for parameters.From concrete plots,we observed wave characteristics of three kinds of solutions.Especially,we could observe distinct generation and separation situations when the localized wave and the stripe wave interacted at different time points.
文摘Starting from the step-by-step iterative method, the analytical formulas of solutions of the geometrically nonlinear equations of the axisymmetric plates and shallow shells, have been obtained. The uniform convergence of the iterative method, is used to prove the convergence of the analytical formulas of the exact solutions of the equa- tions.
文摘The method combining the function transformation with the auxiliary equation is presented and the new infinite sequence complexion solutions of a class of nonlinear evolutionary equations are constructed. Step one, according to two function transformations, a class of nonlinear evolutionary equations is changed into two kinds of ordinary differential equations. Step two, using the first integral of the ordinary differential equations, two first order nonlinear ordinary differential equations are obtained. Step three, using function transformation, two first order nonlinear ordinary differential equations are changed to the ordinary differential equation that could be integrated. Step four, the new solutions, B?cklund transformation and the nonlinear superposition formula of solutions of the ordinary differential equation that could be integrated are applied to construct the new infinite sequence complexion solutions of a class of nonlinear evolutionary equations. These solutions are consisting of two-soliton solutions, two-period solutions and solutions composed of soliton solutions and period solutions.
文摘The solutions to a linear wave equation can satisfy the principle of superposition, i.e., the linear superposition of two or more known solutions is still a solution of the linear wave equation. We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic, triangle, and exponential functions, and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics. The linear superposition solutions to the generalized KdV equation K（2,2,1）, the Oliver water wave equation, and the k（n, n） equation are given. The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed, and the reason why the solutions with the forms of hyperbolic, triangle, and exponential functions can form the linear superposition solutions is also discussed.
基金Project supported by the National Natural Science Foundation of China （Grant No 10575087） and the Natural Foundation of Zhejiang Province （Grant No 102053）.
文摘New forms of different-periodic travelling wave solutions for the （2＋1）-dimensional Zakharov-Kuznetsov （ZK） equation and the Davey-Stewartson （DS） equation are obtained by the linear superposition approach of Jacobi elliptic function. A sequence of cyclic identities plays an important role in these procedures.
文摘Using Jacobi elliptic function linear superposition approach for the(1+1)-dimensional Caudrey-Dodd-Gibbon-Sawada-Kotera(CDGSK)equation and the(2+ 1)-dimensional Nizhnik-Novikov-Veselov(NNV)equation,many new periodic travelling wave solutions with different periods and velocities are obtained based on the known periodic solutions.This procedure is crucially dependent on a sequence of cyclic identities involving Jacobi elliptic functions sn(ζ,m),cn(ζ,m),and dn(ζ,m).
文摘In this paper,a Boussinesq hierarchy in the bilinear form is proposed. A Backlund transformation for this hierarchy is presented and the nonlinear superposition formula is proved rigorously.
基金supported by National Natural Science Foundation of China(Grant Nos.11171092 and 11271133)Innovation Scientists and Technicians Troop Construction Projects of Henan Province(Grant No.114200510011)
文摘A monotonicity formula for stable solutions to a class of weighted semilinear elliptic equations with "negative exponent" is established.It is well known that such a monotonicity formula plays an essential role in the study of finite Morse index solutions of equations with "positive exponent".Unlike the positive exponent case,we will see that both the monotonicity formula and the sub-harmonicity play crucial roles in classifying positive finite Morse index solutions to the equations with negative exponent and obtaining sharp results for their asymptotic behaviors.
基金This paper was originally exhibited in 2020(arXiv:2006.00222)。
文摘In this paper,we investigate a class of nonlinear backward stochastic differential equations(BSDEs)arising from financial economics,and give the sign of corresponding solution.Furthermore,we are able to obtain explicit solutions to an interesting class of nonlinear BSDEs,including the k-ignorance BSDE arising from the modeling of ambiguity of asset pricing.Moreover,we show its applications in PDEs and contingent pricing in an incomplete market.
