We investigate normalized solutions to a class of Chern-Simons-Schrödinger systems with combined nonlinearities f(u)=|u|p−2u+µ|u|q−2u in R2,whereµ∈{±1}and 2<p,q<∞.The solutions correspond t...We investigate normalized solutions to a class of Chern-Simons-Schrödinger systems with combined nonlinearities f(u)=|u|p−2u+µ|u|q−2u in R2,whereµ∈{±1}and 2<p,q<∞.The solutions correspond to critical points of the underlying energy functional subject to the L2-norm constraint,namely,∫R2|u|2dx=c for c>0 given.Of particular interest is the competing and double L2-supercritical case,i.e.,µ=−1 and min{p,q}>4.We prove several existence and multiplicity results depending on the size of the exponents p and q.It is worth emphasizing that some of them are also new even in the study of the Schrödinger equations.In addition,the asymptotic behavior of the solutions and the associated Lagrange multipliersλas c→0 is described.展开更多
In this paper,we are concerned with the existence of multiple solutions to the critical magnetic Schr?dinger equation(-i▽-a(x))^(2)u+⒂λV(x)u=p|u|^(p-2)u+(∫R(n)|u(y)|^(2)_(a)^(*)/|x-y|^(a)dy)|u|2_(a)^(*)-2_(u)in R^...In this paper,we are concerned with the existence of multiple solutions to the critical magnetic Schr?dinger equation(-i▽-a(x))^(2)u+⒂λV(x)u=p|u|^(p-2)u+(∫R(n)|u(y)|^(2)_(a)^(*)/|x-y|^(a)dy)|u|2_(a)^(*)-2_(u)in R^(N),(0.1)where N≥4,2≤p<2^(*),2_α^(*)=(2N-α)/(N-2)with 0<α<4,λ>0,μ∈R,A(x)=(A_(1)(x),A_(2)(x),…,A_(N)(x))is a real local Hölder continuous vector function,i is the imaginary unit,and V(x)is a real valued potential function on R^(N).Supposing thatΩ=int V^(-1)(0)■R^(N)is bounded,we show that problem(0.1)possesses at least cat_(Ω)(Ω)nontrivial solutions ifλis large.展开更多
In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al...In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al.in J Sci Comput 66:321–345,2016;Dong and Wang in J Comput Appl Math 380:1–11,2020)for a one-dimensional stationary Schrödinger equation.Previous work showed that penalty parameters were required to be positive in error analysis,but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes.In this work,by performing extensive numerical experiments,we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods,and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.展开更多
In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercriti...In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercritical case, we obtain the existence and stability of standing waves. Our results are complements to the results of Carles and Il’yasov’s artical, where orbital stability of standing waves have been studied for the 2D Schrödinger equation with combined nonlinearities and harmonic potential.展开更多
This paper is devoted to studying the existence of solutions for the following logarithmic Schrödinger problem: −div(a(x)∇u)+V(x)u=ulogu2+k(x)| u |q1−2u+h(x)| u |q2−2u, x∈ℝN.(1)We first prove that the correspon...This paper is devoted to studying the existence of solutions for the following logarithmic Schrödinger problem: −div(a(x)∇u)+V(x)u=ulogu2+k(x)| u |q1−2u+h(x)| u |q2−2u, x∈ℝN.(1)We first prove that the corresponding functional I belongs to C1(HV1(ℝN),ℝ). Furthermore, by using the variational method, we prove the existence of a sigh-changing solution to problem (1).展开更多
In this paper,we investigate a class of nonlinear Chern-Simons-Schr?dinger systems with a steep well potential.By using variational methods,the mountain pass theorem and Nehari manifold methods,we prove the existence ...In this paper,we investigate a class of nonlinear Chern-Simons-Schr?dinger systems with a steep well potential.By using variational methods,the mountain pass theorem and Nehari manifold methods,we prove the existence of a ground state solution forλ>0 large enough.Furthermore,we verify the asymptotic behavior of ground state solutions asλ→+∞.展开更多
In this paper, we study the Schrodinger equations (-△)^(s)u + V(x)u = a(x)|u|^(p-2)u + b(x)|u|^(q-2)u, x∈R^(N),where 0 < s < 1, 2 < q < p < 2_(s)^(*), 2_(s)^(*) is the fractional Sobolev critical expo...In this paper, we study the Schrodinger equations (-△)^(s)u + V(x)u = a(x)|u|^(p-2)u + b(x)|u|^(q-2)u, x∈R^(N),where 0 < s < 1, 2 < q < p < 2_(s)^(*), 2_(s)^(*) is the fractional Sobolev critical exponent. Under suitable assumptions on V, a and b for which there may be no ground state solution, the existence of positive solutions are obtained via variational methods.展开更多
We make a quantitative study on the soliton interactions in the nonlinear Schro¨dinger equation(NLSE) and its variable–coefficient(vc) counterpart. For the regular two-soliton and double-pole solutions of the NL...We make a quantitative study on the soliton interactions in the nonlinear Schro¨dinger equation(NLSE) and its variable–coefficient(vc) counterpart. For the regular two-soliton and double-pole solutions of the NLSE, we employ the asymptotic analysis method to obtain the expressions of asymptotic solitons, and analyze the interaction properties based on the soliton physical quantities(especially the soliton accelerations and interaction forces);whereas for the bounded two-soliton solution, we numerically calculate the soliton center positions and accelerations, and discuss the soliton interaction scenarios in three typical bounded cases. Via some variable transformations, we also obtain the inhomogeneous regular two-soliton and double-pole solutions for the vcNLSE with an integrable condition. Based on the expressions of asymptotic solitons, we quantitatively study the two-soliton interactions with some inhomogeneous dispersion profiles,particularly discuss the influence of the variable dispersion function f(t) on the soliton interaction dynamics.展开更多
We study a coupled Schrödinger equation which is started from the Boussinesq equation of atmospheric gravity waves by using multiscale analysis and reduced perturbation method.For the coupled Schrödinger equ...We study a coupled Schrödinger equation which is started from the Boussinesq equation of atmospheric gravity waves by using multiscale analysis and reduced perturbation method.For the coupled Schrödinger equation,we obtain the Manakov model of all-focusing,all-defocusing and mixed types by setting parameters value and apply the Hirota bilinear approach to provide the two-soliton and three-soliton solutions.Especially,we find that the all-defocusing type Manakov model admits bright-bright soliton solutions.Furthermore,we find that the all-defocusing type Manakov model admits bright-bright-bright soliton solutions.Therefrom,we go over how the free parameters affect the Manakov model’s allfocusing type’s two-soliton and three-soliton solutions’collision locations,propagation directions,and wave amplitudes.These findings are useful for setting a simulation scene in Rossby waves research.The answers we have found are helpful for studying physical properties of the equation in Rossby waves.展开更多
基金supported by National Natural Science Foundation of China (Grant No.12071486)supported by National Natural Science Foundation of China (Grant No.11671236)Shandong Provincial Natural Science Foundation (Grant No. ZR2020JQ01)。
文摘We investigate normalized solutions to a class of Chern-Simons-Schrödinger systems with combined nonlinearities f(u)=|u|p−2u+µ|u|q−2u in R2,whereµ∈{±1}and 2<p,q<∞.The solutions correspond to critical points of the underlying energy functional subject to the L2-norm constraint,namely,∫R2|u|2dx=c for c>0 given.Of particular interest is the competing and double L2-supercritical case,i.e.,µ=−1 and min{p,q}>4.We prove several existence and multiplicity results depending on the size of the exponents p and q.It is worth emphasizing that some of them are also new even in the study of the Schrödinger equations.In addition,the asymptotic behavior of the solutions and the associated Lagrange multipliersλas c→0 is described.
基金supported by the National Natural Science Foundation of China(12171212)。
文摘In this paper,we are concerned with the existence of multiple solutions to the critical magnetic Schr?dinger equation(-i▽-a(x))^(2)u+⒂λV(x)u=p|u|^(p-2)u+(∫R(n)|u(y)|^(2)_(a)^(*)/|x-y|^(a)dy)|u|2_(a)^(*)-2_(u)in R^(N),(0.1)where N≥4,2≤p<2^(*),2_α^(*)=(2N-α)/(N-2)with 0<α<4,λ>0,μ∈R,A(x)=(A_(1)(x),A_(2)(x),…,A_(N)(x))is a real local Hölder continuous vector function,i is the imaginary unit,and V(x)is a real valued potential function on R^(N).Supposing thatΩ=int V^(-1)(0)■R^(N)is bounded,we show that problem(0.1)possesses at least cat_(Ω)(Ω)nontrivial solutions ifλis large.
基金supported by the National Science Foundation grant DMS-1818998.
文摘In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al.in J Sci Comput 66:321–345,2016;Dong and Wang in J Comput Appl Math 380:1–11,2020)for a one-dimensional stationary Schrödinger equation.Previous work showed that penalty parameters were required to be positive in error analysis,but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes.In this work,by performing extensive numerical experiments,we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods,and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.
文摘In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercritical case, we obtain the existence and stability of standing waves. Our results are complements to the results of Carles and Il’yasov’s artical, where orbital stability of standing waves have been studied for the 2D Schrödinger equation with combined nonlinearities and harmonic potential.
文摘This paper is devoted to studying the existence of solutions for the following logarithmic Schrödinger problem: −div(a(x)∇u)+V(x)u=ulogu2+k(x)| u |q1−2u+h(x)| u |q2−2u, x∈ℝN.(1)We first prove that the corresponding functional I belongs to C1(HV1(ℝN),ℝ). Furthermore, by using the variational method, we prove the existence of a sigh-changing solution to problem (1).
基金supported by National Natural Science Foundation of China(11971393)。
文摘In this paper,we investigate a class of nonlinear Chern-Simons-Schr?dinger systems with a steep well potential.By using variational methods,the mountain pass theorem and Nehari manifold methods,we prove the existence of a ground state solution forλ>0 large enough.Furthermore,we verify the asymptotic behavior of ground state solutions asλ→+∞.
基金supported by the NNSF of China(12171014, 12271539, 12171326)the Beijing Municipal Commission of Education (KZ202010028048)the Research Foundation for Advanced Talents of Beijing Technology and Business University (19008022326)。
文摘In this paper, we study the Schrodinger equations (-△)^(s)u + V(x)u = a(x)|u|^(p-2)u + b(x)|u|^(q-2)u, x∈R^(N),where 0 < s < 1, 2 < q < p < 2_(s)^(*), 2_(s)^(*) is the fractional Sobolev critical exponent. Under suitable assumptions on V, a and b for which there may be no ground state solution, the existence of positive solutions are obtained via variational methods.
基金Project supported by the Natural Science Foundation of Beijing Municipality (Grant No.1212007)the National Natural Science Foundation of China (Grant No.11705284)the Open Project Program of State Key Laboratory of Petroleum Resources and Prospecting,China University of Petroleum (Grant No.PRP/DX-2211)。
文摘We make a quantitative study on the soliton interactions in the nonlinear Schro¨dinger equation(NLSE) and its variable–coefficient(vc) counterpart. For the regular two-soliton and double-pole solutions of the NLSE, we employ the asymptotic analysis method to obtain the expressions of asymptotic solitons, and analyze the interaction properties based on the soliton physical quantities(especially the soliton accelerations and interaction forces);whereas for the bounded two-soliton solution, we numerically calculate the soliton center positions and accelerations, and discuss the soliton interaction scenarios in three typical bounded cases. Via some variable transformations, we also obtain the inhomogeneous regular two-soliton and double-pole solutions for the vcNLSE with an integrable condition. Based on the expressions of asymptotic solitons, we quantitatively study the two-soliton interactions with some inhomogeneous dispersion profiles,particularly discuss the influence of the variable dispersion function f(t) on the soliton interaction dynamics.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.12102205 and 12161065)the Scientific Research Ability of Youth Teachers of Inner Mongolia Agricultural University(Grant Nos.JC2021001 and BR220126)+1 种基金the Natural Science Foundation of Inner Mongolia Autonomous Region of China(Grant No.2022QN01003)the Research Program of Inner Mongolia Autonomous Region Education Department(Grant Nos.NJYT23099 and NMGIRT2208).
文摘We study a coupled Schrödinger equation which is started from the Boussinesq equation of atmospheric gravity waves by using multiscale analysis and reduced perturbation method.For the coupled Schrödinger equation,we obtain the Manakov model of all-focusing,all-defocusing and mixed types by setting parameters value and apply the Hirota bilinear approach to provide the two-soliton and three-soliton solutions.Especially,we find that the all-defocusing type Manakov model admits bright-bright soliton solutions.Furthermore,we find that the all-defocusing type Manakov model admits bright-bright-bright soliton solutions.Therefrom,we go over how the free parameters affect the Manakov model’s allfocusing type’s two-soliton and three-soliton solutions’collision locations,propagation directions,and wave amplitudes.These findings are useful for setting a simulation scene in Rossby waves research.The answers we have found are helpful for studying physical properties of the equation in Rossby waves.
