We mainly investigate the variable-coefficient 3-coupled nonlinear Schrodinger(NLS)system,which describes soli- ton dynamics in the three-spineα-helical protein with inhomogeneous effect.The variable-coefficient NLS ...We mainly investigate the variable-coefficient 3-coupled nonlinear Schrodinger(NLS)system,which describes soli- ton dynamics in the three-spineα-helical protein with inhomogeneous effect.The variable-coefficient NLS equation is transformed into the constant coefficient NLS equation by similarity transformation firstly.The Hirota method is used to solve the constant coefficient NLS equation,and then we get the one-and two-breather solutions of the variable-coefficient NLS equation.The results show that,in the background of plane waves and periodic waves,the breather can be transformed into some forms of combined soliton solutions.The influence of different parameters on the soliton solution and the collision between two solitons are discussed by some graphs in detail.Our results are helpful to study the soliton dynamics inα-helical protein.展开更多
Optical fibers are seen in the optical sensing and optical fiber communication. Simultaneous propagation of optical pulses in an inhomogeneous optical fiber is described by a coupled time-dependent coefficient fourth-...Optical fibers are seen in the optical sensing and optical fiber communication. Simultaneous propagation of optical pulses in an inhomogeneous optical fiber is described by a coupled time-dependent coefficient fourth-order nonlinear Schr?dinger system, which is discussed in this paper. For such a system, we work out the Lax pair, Darboux transformation, and corresponding vector semi-rational nonautonomous rogue wave solutions. When the group velocity dispersion(GVD) and fourth-order dispersion(FOD) coefficients are the constants, we exhibit the first-and second-order vector semirational rogue waves which are composed of the four-petalled rogue waves and eye-shaped breathers. Both the width of the rogue wave along the time axis and temporal separation between the adjacent peaks of the breather decrease with the GVD coefficient or FOD coefficient. With the GVD and FOD coefficients as the linear, cosine, and exponential functions, we respectively present the first-and second-order periodic vector semi-rational rogue waves, first-and second-order asymmetry vector semi-rational rogue waves, and interactions between the eye-shaped breathers and the composite rogue waves.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
We give sufficient and necessary conditions for the existence and nonexistence of positive ground state solutions of a class of coupled nonlinear Schroedinger equations.
In this article,we prove that there exists a unique local smooth solution for the Cauchy problem of the Navier–Stokes–Schrodinger system.Our methods rely upon approximating the system with a sequence of perturbed sy...In this article,we prove that there exists a unique local smooth solution for the Cauchy problem of the Navier–Stokes–Schrodinger system.Our methods rely upon approximating the system with a sequence of perturbed system and parallel transport and are closer to the one in Ding and Wang(Sci China 44(11):1446–1464,2001)and McGahagan(Commun Partial Differ Equ 32(1–3):375–400,2007).展开更多
For the Schrodinger system{-△uj+λjuj+k∑i=1βijui^2uj in R^N,uj(x)→0 as|x|→∞,j=1,…,k where k≥2 and N=2,3,we prove that for anyλj>0 andβjj>0 and any positive integers pj,j=1,2,…,k,there exists b>0 su...For the Schrodinger system{-△uj+λjuj+k∑i=1βijui^2uj in R^N,uj(x)→0 as|x|→∞,j=1,…,k where k≥2 and N=2,3,we prove that for anyλj>0 andβjj>0 and any positive integers pj,j=1,2,…,k,there exists b>0 such that ifβij=βji≤b for all i≠j then there exists a radial solution(u1,u2,…uk)with uj having exactly Pj-1 zeroes.Moreover,there exists a positive constant Co such that ifβij=βji≤b(i≠j)then any solution obtained satisfies k∑i,j=1|βij|∫R^Nui^2uj^2≤C0.Therefore,the solutions exhibit a trend of phase separations asβij→-∞for i≠j.展开更多
For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, classical symmetry algebra is found and 1-dimensional optimal system, up to conjugacy, is constructed. Its symmetry reductions are performed for each...For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, classical symmetry algebra is found and 1-dimensional optimal system, up to conjugacy, is constructed. Its symmetry reductions are performed for each class, and someexamples of exact invainvariant solutions are given.展开更多
In this paper, we consider cooperative hyperbolic systems involving Schr?dinger operator defined on ?Rn. First we prove the existence and uniqueness of the state for these systems. Then we find the necessary and suffi...In this paper, we consider cooperative hyperbolic systems involving Schr?dinger operator defined on ?Rn. First we prove the existence and uniqueness of the state for these systems. Then we find the necessary and sufficient conditions of optimal control for such systems of the boundary type. We also find the necessary and sufficient conditions of optimal control for same systems when the observation is on the boundary.展开更多
We obtain the global smooth solution of a nonlinear SchrSdinger equations in atomic Bose-Einstein condensates with two-dimensional spaces. By using the Galerkin method and a priori estimates, we establish the global e...We obtain the global smooth solution of a nonlinear SchrSdinger equations in atomic Bose-Einstein condensates with two-dimensional spaces. By using the Galerkin method and a priori estimates, we establish the global existence and uniqueness of the smooth solution.展开更多
When the microscopic particles was depicted by linear Schrodinger equation, we find that the particles have only a wave feature, thus, a series of difficulties and intense disputations occur in quantum mechanics. Thes...When the microscopic particles was depicted by linear Schrodinger equation, we find that the particles have only a wave feature, thus, a series of difficulties and intense disputations occur in quantum mechanics. These problems excite us to consider the nonlinear interactions among the particles or between the particle and background field, which is completely ignored in quantum mechanics. Thus we use the nonlinear Schrodinger equation to describe the natures of microscopic particles. In this case the natures and features of microscopic particles are considerably different from those in quantum mechanics, where the microscopic particles are localized and have truly a wave-particle duality. Meanwhile, they satisfy both the classical dynamics equation and Lagrangian and Hamilton equations and obey the conservation laws of mass, energy and momentum. These natures and features are due to the nonlinear interactions, which are generated in virtue of the interaction between the moved particles and background field through the mechanisms of self-trapping, self-focus and self-condensation. Finally, we verified experimentally the localization and wave-corpuscle features of microscopic particles described by the nonlinear Schrodinger equation using the properties of water soliton and optical-soliton depicted also by the nonlinear Schrodinger equation in water and optical fiber, respectively. Therefore, the new nonlinear quantum theory established on the basis of nonlinear Schrodinger equation is correct and credible. From this investigation we can not only solve difficulties and problems disputed for about a century by plenty of scientists in quantum mechanics but also promote the development of physics and enhance the knowledge and recognition levels to the essences of microscopic matter.展开更多
For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, the infinite dimensional Lie algebra of the classical symmetry group is found and the one-dimensional optimal system of an 8-dimensional subalgebra o...For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, the infinite dimensional Lie algebra of the classical symmetry group is found and the one-dimensional optimal system of an 8-dimensional subalgebra of the infinite Lie algebra is constructed. The reduced equations of the equations with respect to the optimal system are derived. Furthermore, the one-dimensional optimal systems of the Lie algebra admitted by the reduced equations are also constructed. Consequently, the classification of the twice optimal symmetry reductions of the equations with respect to the optimal systems is presented. The reductions show that the (1 + 2)-dimensional nonlinear Schrodinger equations can be reduced to a group of ordinary differential equations which is useful for solving the related problems of the equations.展开更多
We point out that a suitable scale of time for the Schrödinger perturbation process is a closed line having rather a circular and not a conventional straight-linear character. A circular nature of the scale c...We point out that a suitable scale of time for the Schrödinger perturbation process is a closed line having rather a circular and not a conventional straight-linear character. A circular nature of the scale concerns especially the time associated with a particular order N of the perturbation energy which provides us with a full number of the perturbation terms predicted by Huby and Tong. On the other hand, a change of the order N—connected with an increased number of the special time points considered on the scale—requires a progressive character of time. A classification of the perturbation terms is done with the aid of the time-point contractions present on a scale characteristic for each N. This selection of terms can be simplified by a partition procedure of the integer numbers representing N-1. The detailed calculations are performed for the perturbation energy of orders N=7 and N=8 .展开更多
The primary objective of this paper is to investigate the well-posedness theories associated with the discrete nonlinear Schrodinger and Klein-Gordon equations.These theories encompass both local and global well-posed...The primary objective of this paper is to investigate the well-posedness theories associated with the discrete nonlinear Schrodinger and Klein-Gordon equations.These theories encompass both local and global well-posedness,as well as the existence of blowing-up solutions for large and irregular initial data.The main results presented in this paper can be summarized as follows:(1)Discrete Nonlinear Schrodinger Equation:Global well-posedness in l^(p) spaces for all1≤p≤∞,regardless of whether it is in the defocusing or focusing cases.(2)Discrete Klein-Gordon Equation:Local well-posedness in l^(p) spaces for all 1≤p≤∞.Furthermore,in the defocusing case,we establish global well-posedness in l^(p) spaces for any2≤p≤2σ+2(σ>0).In contrast,in the focusing case,we show that solutions with negative energy blow up within a finite time.These conclusions reveal the distinct dynamic behaviors exhibited by the solutions of the equations in discrete settings compared to their continuous setting.Additionally,they illuminate the significant role that discretization plays in preventing ill-posedness,and collapse for the nonlinear Schrodinger equation.展开更多
In this paper,we study high energy normalized solutions for the following Schr?dinger equation{-Δu+V(x)u+λu=f(u),in R^(2),∫_(R^(2))|u|^(2)dx=c,where c>0,λ∈R will appear as a Lagrange multiplier,V(x)=ω|x|2 rep...In this paper,we study high energy normalized solutions for the following Schr?dinger equation{-Δu+V(x)u+λu=f(u),in R^(2),∫_(R^(2))|u|^(2)dx=c,where c>0,λ∈R will appear as a Lagrange multiplier,V(x)=ω|x|2 represents a trapping potential,and f has an exponential critical growth.Under the appropriate assumptions of f,we have obtained the existence of normalized solutions to the above Schr?dinger equation by introducing a variational method.And these solutions are also high energy solutions with positive energy.展开更多
This paper is concerned with the positive ground state solutions for a quasilinear Schrodinger equation with a Hardy-type term.We obtain positive ground state solutions for the given quasilinear Schrodinger equation b...This paper is concerned with the positive ground state solutions for a quasilinear Schrodinger equation with a Hardy-type term.We obtain positive ground state solutions for the given quasilinear Schrodinger equation by using a change of variables and variational method.展开更多
基金supported by the Scientific and Technological Innovation Programs of Higher Education Institution in Shanxi,China(Grant Nos.2020L0525 and 2019L0782)the National Natural Science Foundation of China(Grant Nos.11805141 and 12075210)+2 种基金Applied Basic Research Program of Shanxi Province,China(Grant No.201901D211424)“1331 Project”Key Innovative Research Team of Taiyuan Normal University(Grant No.I0190364)Key Research and Development program of Shanxi Province,China(Grant No.201903D421042).
文摘We mainly investigate the variable-coefficient 3-coupled nonlinear Schrodinger(NLS)system,which describes soli- ton dynamics in the three-spineα-helical protein with inhomogeneous effect.The variable-coefficient NLS equation is transformed into the constant coefficient NLS equation by similarity transformation firstly.The Hirota method is used to solve the constant coefficient NLS equation,and then we get the one-and two-breather solutions of the variable-coefficient NLS equation.The results show that,in the background of plane waves and periodic waves,the breather can be transformed into some forms of combined soliton solutions.The influence of different parameters on the soliton solution and the collision between two solitons are discussed by some graphs in detail.Our results are helpful to study the soliton dynamics inα-helical protein.
基金Project supported by the BUPT Excellent Ph.D.Students Foundation(Grant No.CX2019201)the National Natural Science Foundation of China(Grant Nos.11772017 and 11805020)+1 种基金the Fund of State Key Laboratory of Information Photonics and Optical Communications(Beijing University of Posts and Telecommunications),China(Grant No.IPOC:2017ZZ05)the Fundamental Research Funds for the Central Universities of China(Grant No.2011BUPTYB02)。
文摘Optical fibers are seen in the optical sensing and optical fiber communication. Simultaneous propagation of optical pulses in an inhomogeneous optical fiber is described by a coupled time-dependent coefficient fourth-order nonlinear Schr?dinger system, which is discussed in this paper. For such a system, we work out the Lax pair, Darboux transformation, and corresponding vector semi-rational nonautonomous rogue wave solutions. When the group velocity dispersion(GVD) and fourth-order dispersion(FOD) coefficients are the constants, we exhibit the first-and second-order vector semirational rogue waves which are composed of the four-petalled rogue waves and eye-shaped breathers. Both the width of the rogue wave along the time axis and temporal separation between the adjacent peaks of the breather decrease with the GVD coefficient or FOD coefficient. With the GVD and FOD coefficients as the linear, cosine, and exponential functions, we respectively present the first-and second-order periodic vector semi-rational rogue waves, first-and second-order asymmetry vector semi-rational rogue waves, and interactions between the eye-shaped breathers and the composite rogue waves.
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.
文摘We give sufficient and necessary conditions for the existence and nonexistence of positive ground state solutions of a class of coupled nonlinear Schroedinger equations.
基金This work was supported by the National Natural Science Foundation of China(Grant No.11771415).
文摘In this article,we prove that there exists a unique local smooth solution for the Cauchy problem of the Navier–Stokes–Schrodinger system.Our methods rely upon approximating the system with a sequence of perturbed system and parallel transport and are closer to the one in Ding and Wang(Sci China 44(11):1446–1464,2001)and McGahagan(Commun Partial Differ Equ 32(1–3):375–400,2007).
基金supported by the National Natural Science Foundation of China with grand numbers Nos.11671272,11331010,11771324 and 11831009
文摘For the Schrodinger system{-△uj+λjuj+k∑i=1βijui^2uj in R^N,uj(x)→0 as|x|→∞,j=1,…,k where k≥2 and N=2,3,we prove that for anyλj>0 andβjj>0 and any positive integers pj,j=1,2,…,k,there exists b>0 such that ifβij=βji≤b for all i≠j then there exists a radial solution(u1,u2,…uk)with uj having exactly Pj-1 zeroes.Moreover,there exists a positive constant Co such that ifβij=βji≤b(i≠j)then any solution obtained satisfies k∑i,j=1|βij|∫R^Nui^2uj^2≤C0.Therefore,the solutions exhibit a trend of phase separations asβij→-∞for i≠j.
基金supported by the Natural science foundation of China(NSF),under grand number 11071159.
文摘For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, classical symmetry algebra is found and 1-dimensional optimal system, up to conjugacy, is constructed. Its symmetry reductions are performed for each class, and someexamples of exact invainvariant solutions are given.
文摘In this paper, we consider cooperative hyperbolic systems involving Schr?dinger operator defined on ?Rn. First we prove the existence and uniqueness of the state for these systems. Then we find the necessary and sufficient conditions of optimal control for such systems of the boundary type. We also find the necessary and sufficient conditions of optimal control for same systems when the observation is on the boundary.
基金Acknowledgements This work was supported in part by the National Natural Science Foundation of China （Grant No. 11571254）.
文摘We obtain the global smooth solution of a nonlinear SchrSdinger equations in atomic Bose-Einstein condensates with two-dimensional spaces. By using the Galerkin method and a priori estimates, we establish the global existence and uniqueness of the smooth solution.
文摘When the microscopic particles was depicted by linear Schrodinger equation, we find that the particles have only a wave feature, thus, a series of difficulties and intense disputations occur in quantum mechanics. These problems excite us to consider the nonlinear interactions among the particles or between the particle and background field, which is completely ignored in quantum mechanics. Thus we use the nonlinear Schrodinger equation to describe the natures of microscopic particles. In this case the natures and features of microscopic particles are considerably different from those in quantum mechanics, where the microscopic particles are localized and have truly a wave-particle duality. Meanwhile, they satisfy both the classical dynamics equation and Lagrangian and Hamilton equations and obey the conservation laws of mass, energy and momentum. These natures and features are due to the nonlinear interactions, which are generated in virtue of the interaction between the moved particles and background field through the mechanisms of self-trapping, self-focus and self-condensation. Finally, we verified experimentally the localization and wave-corpuscle features of microscopic particles described by the nonlinear Schrodinger equation using the properties of water soliton and optical-soliton depicted also by the nonlinear Schrodinger equation in water and optical fiber, respectively. Therefore, the new nonlinear quantum theory established on the basis of nonlinear Schrodinger equation is correct and credible. From this investigation we can not only solve difficulties and problems disputed for about a century by plenty of scientists in quantum mechanics but also promote the development of physics and enhance the knowledge and recognition levels to the essences of microscopic matter.
基金supported by the Natural science foundation of China(NSF),under grand number 11071159.
文摘For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, the infinite dimensional Lie algebra of the classical symmetry group is found and the one-dimensional optimal system of an 8-dimensional subalgebra of the infinite Lie algebra is constructed. The reduced equations of the equations with respect to the optimal system are derived. Furthermore, the one-dimensional optimal systems of the Lie algebra admitted by the reduced equations are also constructed. Consequently, the classification of the twice optimal symmetry reductions of the equations with respect to the optimal systems is presented. The reductions show that the (1 + 2)-dimensional nonlinear Schrodinger equations can be reduced to a group of ordinary differential equations which is useful for solving the related problems of the equations.
文摘We point out that a suitable scale of time for the Schrödinger perturbation process is a closed line having rather a circular and not a conventional straight-linear character. A circular nature of the scale concerns especially the time associated with a particular order N of the perturbation energy which provides us with a full number of the perturbation terms predicted by Huby and Tong. On the other hand, a change of the order N—connected with an increased number of the special time points considered on the scale—requires a progressive character of time. A classification of the perturbation terms is done with the aid of the time-point contractions present on a scale characteristic for each N. This selection of terms can be simplified by a partition procedure of the integer numbers representing N-1. The detailed calculations are performed for the perturbation energy of orders N=7 and N=8 .
基金in part supported by the NSFC(12171356,12494544)supported by the National Key R&D Program of China(2020 YFA0713300)+1 种基金the NSFC(12531006)the Nankai Zhide Foundation。
文摘The primary objective of this paper is to investigate the well-posedness theories associated with the discrete nonlinear Schrodinger and Klein-Gordon equations.These theories encompass both local and global well-posedness,as well as the existence of blowing-up solutions for large and irregular initial data.The main results presented in this paper can be summarized as follows:(1)Discrete Nonlinear Schrodinger Equation:Global well-posedness in l^(p) spaces for all1≤p≤∞,regardless of whether it is in the defocusing or focusing cases.(2)Discrete Klein-Gordon Equation:Local well-posedness in l^(p) spaces for all 1≤p≤∞.Furthermore,in the defocusing case,we establish global well-posedness in l^(p) spaces for any2≤p≤2σ+2(σ>0).In contrast,in the focusing case,we show that solutions with negative energy blow up within a finite time.These conclusions reveal the distinct dynamic behaviors exhibited by the solutions of the equations in discrete settings compared to their continuous setting.Additionally,they illuminate the significant role that discretization plays in preventing ill-posedness,and collapse for the nonlinear Schrodinger equation.
基金Supported by National Natural Science Foundation of China(Grant Nos.11671403 and 11671236)Henan Provincial General Natural Science Foundation Project(Grant No.232300420113)。
文摘In this paper,we study high energy normalized solutions for the following Schr?dinger equation{-Δu+V(x)u+λu=f(u),in R^(2),∫_(R^(2))|u|^(2)dx=c,where c>0,λ∈R will appear as a Lagrange multiplier,V(x)=ω|x|2 represents a trapping potential,and f has an exponential critical growth.Under the appropriate assumptions of f,we have obtained the existence of normalized solutions to the above Schr?dinger equation by introducing a variational method.And these solutions are also high energy solutions with positive energy.
基金Supported by Research Start-up Fund of Jianghan University(06050001).
文摘This paper is concerned with the positive ground state solutions for a quasilinear Schrodinger equation with a Hardy-type term.We obtain positive ground state solutions for the given quasilinear Schrodinger equation by using a change of variables and variational method.
