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
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
Over an algebraically closed field of characteristic p>2,based on the results on the representation theory of special linear Lie algebra sl(2),restricted simple modules L(λ) of the Schrodinger algebra S(1)are dete...Over an algebraically closed field of characteristic p>2,based on the results on the representation theory of special linear Lie algebra sl(2),restricted simple modules L(λ) of the Schrodinger algebra S(1)are determined,and all derivations of S(1)on L(λ)are also obtained.As an application,the first cohomology of S(1)with the coefficient in L(λ)is determined.展开更多
The Schrodinger equation with a Yukawa type of potential is solved analytically.When different boundary conditions are taken into account,a series of solutions are indicated as a Bessel function,the first kind of Hank...The Schrodinger equation with a Yukawa type of potential is solved analytically.When different boundary conditions are taken into account,a series of solutions are indicated as a Bessel function,the first kind of Hankel function and the second kind of Hankel function,respectively.Subsequently,the scattering processes of K^(*)and D^(*)are investigated.In the K^(*)sector,the f_(1)(1285)particle is treated as a K^(*)bound state,therefore,the coupling constant in the K^(*)Yukawa potential can be fixed according to the binding energy of the f_(1)(1285)particle.Consequently,a K^(*)resonance state is generated by solving the Schrodinger equation with the outgoing wave condition,which lies at 1417-i18 MeV on the complex energy plane.It is reasonable to assume that the K^(*)resonance state at 1417-i18 MeV might correspond to the f_(1)(1420)particle in the review of the Particle Data Group.In the D^(*)sector,since the X(3872)particle is almost located at the D^(*)threshold,its binding energy is approximately equal to zero.Therefore,the coupling constant in the D^(*)Yukawa potential is determined,which is related to the first zero point of the zero-order Bessel function.Similarly to the K^(*)case,four resonance states are produced as solutions of the Schrodinger equation with the outgoing wave condition.It is assumed that the resonance states at 3885~i1 MeV,4029-i108 MeV,4328-i191 MeV and 4772-i267 MeV might be associated with the Zc(3900),the X(3940),theχ_(c1)(4274)andχ_(c1)(4685)particles,respectively.It is noted that all solutions are isospin degenerate.展开更多
The nonlinear Schrodinger equation equation is one of the most important physical models used in optical fiber theory to explain the transmission of an optical soliton.The field of chiral soliton propagation in nuclea...The nonlinear Schrodinger equation equation is one of the most important physical models used in optical fiber theory to explain the transmission of an optical soliton.The field of chiral soliton propagation in nuclear physics is very interesting because of its numerous applications in communications and ultra-fast signal routing systems.The(1+1)-dimensional chiral dynamical structure that describes the soliton behaviour in data transmission is dealt with in this work using a variety of in-depth analytical techniques.This work has applications in particle physics,ionised science,nuclear physics,optics,and other applied mathematical sciences.We are able to develop a variety of solutions to demonstrate the behaviour of solitary wave structures,periodic soliton solutions,chiral soliton solutions,and bell-shaped soliton solutions with the use of applied techniques.Moreover,in order to verify the scientific calculations,the stability analysis for the observed solutions of the governing model is taken into consideration.In addition,the3-dimensional,contour,and 2-dimensional visuals are supplied for a better understanding of the behaviour of the solutions.The employed strategies are dependable,uncomplicated,and effective;yet have not been utilised with the governing model in the literature that is now accessible.The resulting outcomes have impressive applications across a large number of study areas and computational physics phenomena representing real-world scenarios.The methods applied in this model are not utilized on the given models in previous literature so we can say that these describe the novelty of the work.展开更多
We study the dynamics of fundamental and double-pole breathers and solitons for the focusing and defocusing nonlinear Schrodinger equation with the sextic operator under non-zero boundary conditions. Our analysis main...We study the dynamics of fundamental and double-pole breathers and solitons for the focusing and defocusing nonlinear Schrodinger equation with the sextic operator under non-zero boundary conditions. Our analysis mainly focuses onthe dynamical properties of simple- and double-pole solutions. Firstly, through verification, we find that solutions undernon-zero boundary conditions can be transformed into solutions under zero boundary conditions, whether in simple-pole ordouble-pole cases. For the focusing case, in the investigation of simple-pole solutions, temporal periodic breather and thespatial-temporal periodic breather are obtained by modulating parameters. Additionally, in the case of multi-pole solitons,we analyze parallel-state solitons, bound-state solitons, and intersecting solitons, providing a brief analysis of their interactions.In the double-pole case, we observe that the two solitons undergo two interactions, resulting in a distinctive “triangle”crest. Furthermore, for the defocusing case, we briefly consider two situations of simple-pole solutions, obtaining one andtwo dark solitons.展开更多
In this paper,we prove the existence of the scattering operator for the fractional magnetic Schrodinger operators.In order to do this,we construct the fractional distorted Fourier transforms with magnetic potentials.A...In this paper,we prove the existence of the scattering operator for the fractional magnetic Schrodinger operators.In order to do this,we construct the fractional distorted Fourier transforms with magnetic potentials.Applying the properties of the distorted Fourier transforms,the existence and the asymptotic completeness of the wave operators are obtained.Furthermore,we prove the absence of positive eigenvalues for fractional magnetic Schrodinger operators.展开更多
Estimates of the type L1-L∞ for the Schrödinger Equation on the Line and on Half-Line with a regular potential V(x), express the dispersive nature of the Schrödinger Equation and are the essential e...Estimates of the type L1-L∞ for the Schrödinger Equation on the Line and on Half-Line with a regular potential V(x), express the dispersive nature of the Schrödinger Equation and are the essential elements in the study of the problems of initial values, the asymptotic times for large solutions and Scattering Theory for the Schrödinger equation and non-linear in general;for other equations of Non-linear Evolution. In general, the estimates Lp-Lp' express the dispersive nature of this equation. And its study plays an important role in problems of non-linear initial values;likewise, in the study of problems nonlinear initial values;see [1] [2] [3]. On the other hand, following a series of problems proposed by V. Marchenko [4], that we will name Marchenko’s formulation, and relate it to a generalized version of Theorem 1 given in [1], the main theorem (Theorem 1) of this article provides a transformation operator W?that transforms the Reduced Radial Schrödinger Equation (RRSE) (whose main characteristic is the addition a singular term of quadratic order to a regular potential V(x)) in the Schrödinger Equation on Half-Line (RSEHL) under W. That is to say;W?eliminates the singular term of quadratic order of potential V(x) in the asymptotic development towards zero and adds to the potential V(x) a bounded term and a term exponentially decrease fast enough in the asymptotic development towards infinity, which continues guaranteeing the uniqueness of the potential V(x) in the condition of the infinity boundary. Then the L1-L∞ estimates for the (RRSE) are preserved under the transformation operator , as in the case of (RSEHL) where they were established in [3]. Finally, as an open question, the possibility of extending the L1-L∞ estimates for the case (RSEHL), where added to the potential V(x) an analytical perturbation is mentioned.展开更多
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.
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Over an algebraically closed field of characteristic p>2,based on the results on the representation theory of special linear Lie algebra sl(2),restricted simple modules L(λ) of the Schrodinger algebra S(1)are determined,and all derivations of S(1)on L(λ)are also obtained.As an application,the first cohomology of S(1)with the coefficient in L(λ)is determined.
文摘The Schrodinger equation with a Yukawa type of potential is solved analytically.When different boundary conditions are taken into account,a series of solutions are indicated as a Bessel function,the first kind of Hankel function and the second kind of Hankel function,respectively.Subsequently,the scattering processes of K^(*)and D^(*)are investigated.In the K^(*)sector,the f_(1)(1285)particle is treated as a K^(*)bound state,therefore,the coupling constant in the K^(*)Yukawa potential can be fixed according to the binding energy of the f_(1)(1285)particle.Consequently,a K^(*)resonance state is generated by solving the Schrodinger equation with the outgoing wave condition,which lies at 1417-i18 MeV on the complex energy plane.It is reasonable to assume that the K^(*)resonance state at 1417-i18 MeV might correspond to the f_(1)(1420)particle in the review of the Particle Data Group.In the D^(*)sector,since the X(3872)particle is almost located at the D^(*)threshold,its binding energy is approximately equal to zero.Therefore,the coupling constant in the D^(*)Yukawa potential is determined,which is related to the first zero point of the zero-order Bessel function.Similarly to the K^(*)case,four resonance states are produced as solutions of the Schrodinger equation with the outgoing wave condition.It is assumed that the resonance states at 3885~i1 MeV,4029-i108 MeV,4328-i191 MeV and 4772-i267 MeV might be associated with the Zc(3900),the X(3940),theχ_(c1)(4274)andχ_(c1)(4685)particles,respectively.It is noted that all solutions are isospin degenerate.
基金financial support provided by the Hubei University of Automotive Technology,China in the form of a start-up research grant(BK202212)。
文摘The nonlinear Schrodinger equation equation is one of the most important physical models used in optical fiber theory to explain the transmission of an optical soliton.The field of chiral soliton propagation in nuclear physics is very interesting because of its numerous applications in communications and ultra-fast signal routing systems.The(1+1)-dimensional chiral dynamical structure that describes the soliton behaviour in data transmission is dealt with in this work using a variety of in-depth analytical techniques.This work has applications in particle physics,ionised science,nuclear physics,optics,and other applied mathematical sciences.We are able to develop a variety of solutions to demonstrate the behaviour of solitary wave structures,periodic soliton solutions,chiral soliton solutions,and bell-shaped soliton solutions with the use of applied techniques.Moreover,in order to verify the scientific calculations,the stability analysis for the observed solutions of the governing model is taken into consideration.In addition,the3-dimensional,contour,and 2-dimensional visuals are supplied for a better understanding of the behaviour of the solutions.The employed strategies are dependable,uncomplicated,and effective;yet have not been utilised with the governing model in the literature that is now accessible.The resulting outcomes have impressive applications across a large number of study areas and computational physics phenomena representing real-world scenarios.The methods applied in this model are not utilized on the given models in previous literature so we can say that these describe the novelty of the work.
基金the Fundamental Research Funds for the Central Universities(Grant No.2024MS126).
文摘We study the dynamics of fundamental and double-pole breathers and solitons for the focusing and defocusing nonlinear Schrodinger equation with the sextic operator under non-zero boundary conditions. Our analysis mainly focuses onthe dynamical properties of simple- and double-pole solutions. Firstly, through verification, we find that solutions undernon-zero boundary conditions can be transformed into solutions under zero boundary conditions, whether in simple-pole ordouble-pole cases. For the focusing case, in the investigation of simple-pole solutions, temporal periodic breather and thespatial-temporal periodic breather are obtained by modulating parameters. Additionally, in the case of multi-pole solitons,we analyze parallel-state solitons, bound-state solitons, and intersecting solitons, providing a brief analysis of their interactions.In the double-pole case, we observe that the two solitons undergo two interactions, resulting in a distinctive “triangle”crest. Furthermore, for the defocusing case, we briefly consider two situations of simple-pole solutions, obtaining one andtwo dark solitons.
文摘In this paper,we prove the existence of the scattering operator for the fractional magnetic Schrodinger operators.In order to do this,we construct the fractional distorted Fourier transforms with magnetic potentials.Applying the properties of the distorted Fourier transforms,the existence and the asymptotic completeness of the wave operators are obtained.Furthermore,we prove the absence of positive eigenvalues for fractional magnetic Schrodinger operators.
文摘Estimates of the type L1-L∞ for the Schrödinger Equation on the Line and on Half-Line with a regular potential V(x), express the dispersive nature of the Schrödinger Equation and are the essential elements in the study of the problems of initial values, the asymptotic times for large solutions and Scattering Theory for the Schrödinger equation and non-linear in general;for other equations of Non-linear Evolution. In general, the estimates Lp-Lp' express the dispersive nature of this equation. And its study plays an important role in problems of non-linear initial values;likewise, in the study of problems nonlinear initial values;see [1] [2] [3]. On the other hand, following a series of problems proposed by V. Marchenko [4], that we will name Marchenko’s formulation, and relate it to a generalized version of Theorem 1 given in [1], the main theorem (Theorem 1) of this article provides a transformation operator W?that transforms the Reduced Radial Schrödinger Equation (RRSE) (whose main characteristic is the addition a singular term of quadratic order to a regular potential V(x)) in the Schrödinger Equation on Half-Line (RSEHL) under W. That is to say;W?eliminates the singular term of quadratic order of potential V(x) in the asymptotic development towards zero and adds to the potential V(x) a bounded term and a term exponentially decrease fast enough in the asymptotic development towards infinity, which continues guaranteeing the uniqueness of the potential V(x) in the condition of the infinity boundary. Then the L1-L∞ estimates for the (RRSE) are preserved under the transformation operator , as in the case of (RSEHL) where they were established in [3]. Finally, as an open question, the possibility of extending the L1-L∞ estimates for the case (RSEHL), where added to the potential V(x) an analytical perturbation is mentioned.
