The coupled nonlocal nonlinear Schrödinger equations with variable coefficients are researched using the nonstandard Hirota bilinear method.The two-soliton and double-hump one-soliton solutions for the equations ...The coupled nonlocal nonlinear Schrödinger equations with variable coefficients are researched using the nonstandard Hirota bilinear method.The two-soliton and double-hump one-soliton solutions for the equations are first obtained.By assigning different functions to the variable coefficients,we obtain V-shaped,Y-shaped,wave-type,exponential solitons,and so on.Next,we reveal the influence of the real and imaginary parts of the wave numbers on the double-hump structure based on the soliton solutions.Finally,by setting different wave numbers,we can change the distance and transmission direction of the solitons to analyze their dynamic behavior during collisions.This study establishes a theoretical framework for controlling the dynamics of optical fiber in nonlocal nonlinear systems.展开更多
本文研究了半耗散格点非线性Schrödinger方程组解的拉回渐近行为及其概率分布。该方程组描述带有杂质的Bose-Einstein浓缩模型,模型中的Bose波函数具有耗散性,杂质波函数的能量守恒。作者首先证明该问题的整体适定性,然后研究Bose...本文研究了半耗散格点非线性Schrödinger方程组解的拉回渐近行为及其概率分布。该方程组描述带有杂质的Bose-Einstein浓缩模型,模型中的Bose波函数具有耗散性,杂质波函数的能量守恒。作者首先证明该问题的整体适定性,然后研究Bose波函数在适当意义下拉回吸引子的存在性,接着应用该拉回吸引子和广义Banach极限构造统计解,并证明统计解满足Liouville型定理。In this paper, the pullback asymptotic behavior of solutions to the nonlinear Schrödinger system of equations with semi-dissipative lattices and their probability distributions are studied. The equations describe the Bose-Einstein condensation model with impurities, in which the Bose wave function is dissipative, and the energy of the impurity wave function is conserved. The authors first prove the global well-posed of the problem and then investigate the existence of a pullback attractor for the Bose wave function in a suitable sense. The authors then apply the pullback attractor and the generalized Banach limit to construct a statistical solution and show that the statistical solution satisfies the Liouville-type theorem.展开更多
The heavy quarks present in the quark-gluon plasma(QGP)can act as a probe of relativistic heavy ion collisions as they retain the memory of their interaction history.In a previous study,a stochastic Schrödinger e...The heavy quarks present in the quark-gluon plasma(QGP)can act as a probe of relativistic heavy ion collisions as they retain the memory of their interaction history.In a previous study,a stochastic Schrödinger equation(SSE)has been applied to describe the evolution of heavy quarks,where an external field with random phases is used to simulate the thermal medium.In this work,we study the connection between the SSE and the Boltzmann transport equation(BE)approach in the Keldysh Green’s function formalism.By comparing the Green’s function of the heavy quark from the SSE and the Keldysh Green’s functions leading to the Boltzmann equation,we demonstrate that the SSE is consistent with the Boltzmann equation in the weak coupling limit.We subsequently confirm their consistency through numerical calculations.展开更多
In this paper,we study the following Schrödinger-Poisson system{-ε^(p)Δ_(p)u+V(x)|u|^(p-2)u+ϕ|u|^(p-2)u=f(u)+|u|^(p*-2)u in R^(3),-ε^(2)Δϕ=|u|^(p)in R^(3),whereε>0 is a parameter,3/2<p<3,Δ_(p)u=div...In this paper,we study the following Schrödinger-Poisson system{-ε^(p)Δ_(p)u+V(x)|u|^(p-2)u+ϕ|u|^(p-2)u=f(u)+|u|^(p*-2)u in R^(3),-ε^(2)Δϕ=|u|^(p)in R^(3),whereε>0 is a parameter,3/2<p<3,Δ_(p)u=div(|∇u|^(p-2)∇u),p^(*)=3p/3-p,V:R^(3)→R is a potential function with a local minimum and f is subcritical growth.Based on the penalization method,Nehari manifold techniques and Ljusternik-Schnirelmann category theory,we obtain the multiplicity and concentration of positive solutions to the above system.展开更多
In this paper,we consider the following logarithmic Schrödinger equation:−Δu+ωu=u log|u|^(2),u∈H^(1)(R^(N)),where N≥3,and ω>0 is a constant.With an auxiliary equation,we obtain the existence of normalized...In this paper,we consider the following logarithmic Schrödinger equation:−Δu+ωu=u log|u|^(2),u∈H^(1)(R^(N)),where N≥3,and ω>0 is a constant.With an auxiliary equation,we obtain the existence of normalized solutions by using the constrained variational method.展开更多
In this paper,we investigate the following fractional Schrödinger-Poisson system with concave-convex nonlinearities and a steep potential well{(-Δ)^(s)u+V_(λ)(x)u+ϕu=f(x)|u|^(q-2)u+|u|^(p-2)u,in R^(3),(-Δ)^(t)...In this paper,we investigate the following fractional Schrödinger-Poisson system with concave-convex nonlinearities and a steep potential well{(-Δ)^(s)u+V_(λ)(x)u+ϕu=f(x)|u|^(q-2)u+|u|^(p-2)u,in R^(3),(-Δ)^(t)ϕ=u^(2),in R^(3),where s∈(3/4,1),t∈(0,1),q∈(1,2),p∈(4,2_(s)^(*)),2_(s)^(*):=6/3-2s is the fractional critical exponent in dimension 3,V_(λ)(x)=λV(x)+1 withλ>0.Under the case of steep potential well,we obtain the existence of the sign-changing solutions for the above system by using the constraint variational method and the quantitative deformation lemma.Furthermore,we prove that the energy of ground state sign-changing solution is strictly more than twice of the energy of the ground state solution.Our results improve the recent results in the literature.展开更多
本文借助变分法和临界点理论研究一类次线性Schrödinger-Maxwell方程无穷多非平凡解的存在性问题{ −Δu+V(x)u+αϕf(u)=g(x,u),x∈R3,−Δϕ=2αF(u),x∈R3.其中α>0,V(x)∈C1(R3,R),V(x)>0。在f,g符合相关条件下,p∈(1,2)。In...本文借助变分法和临界点理论研究一类次线性Schrödinger-Maxwell方程无穷多非平凡解的存在性问题{ −Δu+V(x)u+αϕf(u)=g(x,u),x∈R3,−Δϕ=2αF(u),x∈R3.其中α>0,V(x)∈C1(R3,R),V(x)>0。在f,g符合相关条件下,p∈(1,2)。In this paper, we discuss the existence of infinitely many nontrivial solutions for the following kind of sublinear Schrödinger-Maxwell equation by using the variational method and critical point theory. { −Δu+V(x)u+αϕf(u)=g(x,u),x∈R3,−Δϕ=2αF(u),x∈R3.where α>0, V(x)∈C1(R3,R), V(x)>0. Under certain assumptions on f,gand p∈(1,2).展开更多
In this paper,we consider the p-Laplacian Schrödinger-Poisson equation with L^(2)-norm constraint-Δ_(p)u+|u|^(p-2)u+λu+(1/4π|x|*|u|^(2))u=|u|^(q-2)u,x∈R^(3),where 2≤p<3,5p/3<q<p*=3p/3-p,λ>0 is a...In this paper,we consider the p-Laplacian Schrödinger-Poisson equation with L^(2)-norm constraint-Δ_(p)u+|u|^(p-2)u+λu+(1/4π|x|*|u|^(2))u=|u|^(q-2)u,x∈R^(3),where 2≤p<3,5p/3<q<p*=3p/3-p,λ>0 is a Lagrange multiplier.We obtain the critical point of the corresponding functional of the problem on mass constraint by the variational method and the Mountain pass lemma,and then find a normalized solution to this equation.展开更多
The(2+1)-dimensional generalized coupled nonlinear Schrödinger equations with a four-wave mixing term are studied in this paper,which describe optical solitons in birefringent fibers.Utilizing the Hirota bilinear...The(2+1)-dimensional generalized coupled nonlinear Schrödinger equations with a four-wave mixing term are studied in this paper,which describe optical solitons in birefringent fibers.Utilizing the Hirota bilinear method,we systematically construct single-and double-periodic lump solutions.To provide a detailed insight into the dynamic behavior of the nonlinear waves,we explore diverse mixed solutions,including bright-dark,W-shaped,multi-peak,and bright soliton solutions.Building upon single-periodic lump solutions,we analyze the dynamics of lump waves on both plane-wave and periodic backgrounds using the long-wave limit method.Moreover,we obtain the interaction solutions involving lumps,periodic lumps,and solitons.The interactions among two solitons,multiple lumps,and mixed waves are illustrated and analyzed.Comparative analysis reveals that these multi-lump solutions exhibit richer dynamical properties than conventional single-lump ones.These results contribute to a deeper understanding of nonlinear systems and may facilitate solving nonlinear problems in nature.展开更多
In this paper,we present some properties of scattering data for the derivative nonlinear Schrödinger equation in H^(S)(R)(s≥1/2)starting from the Lax pair.We show that the reciprocal of the transmission coeffici...In this paper,we present some properties of scattering data for the derivative nonlinear Schrödinger equation in H^(S)(R)(s≥1/2)starting from the Lax pair.We show that the reciprocal of the transmission coefficient can be expressed as the sum of some iterative integrals,and its logarithm can be written as the sum of some connected iterative integrals.We provide the asymptotic properties of the first few iterative integrals of the reciprocal of the transmission coefficient.Moreover,we provide some regularity properties of the reciprocal of the transmission coefficient related to scattering data in H^(S)(R).展开更多
基金supported by the National Key R&D Program of China(Grant No.2022YFA1604200)the National Natural Science Foundation of China(Grant No.12261131495)Institute of Systems Science,Beijing Wuzi University(Grant No.BWUISS21).
文摘The coupled nonlocal nonlinear Schrödinger equations with variable coefficients are researched using the nonstandard Hirota bilinear method.The two-soliton and double-hump one-soliton solutions for the equations are first obtained.By assigning different functions to the variable coefficients,we obtain V-shaped,Y-shaped,wave-type,exponential solitons,and so on.Next,we reveal the influence of the real and imaginary parts of the wave numbers on the double-hump structure based on the soliton solutions.Finally,by setting different wave numbers,we can change the distance and transmission direction of the solitons to analyze their dynamic behavior during collisions.This study establishes a theoretical framework for controlling the dynamics of optical fiber in nonlocal nonlinear systems.
文摘本文研究了半耗散格点非线性Schrödinger方程组解的拉回渐近行为及其概率分布。该方程组描述带有杂质的Bose-Einstein浓缩模型,模型中的Bose波函数具有耗散性,杂质波函数的能量守恒。作者首先证明该问题的整体适定性,然后研究Bose波函数在适当意义下拉回吸引子的存在性,接着应用该拉回吸引子和广义Banach极限构造统计解,并证明统计解满足Liouville型定理。In this paper, the pullback asymptotic behavior of solutions to the nonlinear Schrödinger system of equations with semi-dissipative lattices and their probability distributions are studied. The equations describe the Bose-Einstein condensation model with impurities, in which the Bose wave function is dissipative, and the energy of the impurity wave function is conserved. The authors first prove the global well-posed of the problem and then investigate the existence of a pullback attractor for the Bose wave function in a suitable sense. The authors then apply the pullback attractor and the generalized Banach limit to construct a statistical solution and show that the statistical solution satisfies the Liouville-type theorem.
文摘The heavy quarks present in the quark-gluon plasma(QGP)can act as a probe of relativistic heavy ion collisions as they retain the memory of their interaction history.In a previous study,a stochastic Schrödinger equation(SSE)has been applied to describe the evolution of heavy quarks,where an external field with random phases is used to simulate the thermal medium.In this work,we study the connection between the SSE and the Boltzmann transport equation(BE)approach in the Keldysh Green’s function formalism.By comparing the Green’s function of the heavy quark from the SSE and the Keldysh Green’s functions leading to the Boltzmann equation,we demonstrate that the SSE is consistent with the Boltzmann equation in the weak coupling limit.We subsequently confirm their consistency through numerical calculations.
文摘In this paper,by using the method of Lyapunov-Schmidt reduction,we obtain the existence of multi-bump solutions for planar Schrödinger-Poisson system.
基金supported by the Natural Science Foundation of Gansu Province(No.24JRRP001)。
文摘In this paper,we study the following Schrödinger-Poisson system{-ε^(p)Δ_(p)u+V(x)|u|^(p-2)u+ϕ|u|^(p-2)u=f(u)+|u|^(p*-2)u in R^(3),-ε^(2)Δϕ=|u|^(p)in R^(3),whereε>0 is a parameter,3/2<p<3,Δ_(p)u=div(|∇u|^(p-2)∇u),p^(*)=3p/3-p,V:R^(3)→R is a potential function with a local minimum and f is subcritical growth.Based on the penalization method,Nehari manifold techniques and Ljusternik-Schnirelmann category theory,we obtain the multiplicity and concentration of positive solutions to the above system.
基金supported by the Natural Science Research Project of Department of Education of Guizhou Province(No.QJJ2023062)the National Natural Science Foundation of China(No.52174184)。
文摘In this paper,we consider the following logarithmic Schrödinger equation:−Δu+ωu=u log|u|^(2),u∈H^(1)(R^(N)),where N≥3,and ω>0 is a constant.With an auxiliary equation,we obtain the existence of normalized solutions by using the constrained variational method.
基金supported by the Natural Science Foundation of Sichuan(No.2023NSFSC0073)。
文摘In this paper,we investigate the following fractional Schrödinger-Poisson system with concave-convex nonlinearities and a steep potential well{(-Δ)^(s)u+V_(λ)(x)u+ϕu=f(x)|u|^(q-2)u+|u|^(p-2)u,in R^(3),(-Δ)^(t)ϕ=u^(2),in R^(3),where s∈(3/4,1),t∈(0,1),q∈(1,2),p∈(4,2_(s)^(*)),2_(s)^(*):=6/3-2s is the fractional critical exponent in dimension 3,V_(λ)(x)=λV(x)+1 withλ>0.Under the case of steep potential well,we obtain the existence of the sign-changing solutions for the above system by using the constraint variational method and the quantitative deformation lemma.Furthermore,we prove that the energy of ground state sign-changing solution is strictly more than twice of the energy of the ground state solution.Our results improve the recent results in the literature.
文摘本文借助变分法和临界点理论研究一类次线性Schrödinger-Maxwell方程无穷多非平凡解的存在性问题{ −Δu+V(x)u+αϕf(u)=g(x,u),x∈R3,−Δϕ=2αF(u),x∈R3.其中α>0,V(x)∈C1(R3,R),V(x)>0。在f,g符合相关条件下,p∈(1,2)。In this paper, we discuss the existence of infinitely many nontrivial solutions for the following kind of sublinear Schrödinger-Maxwell equation by using the variational method and critical point theory. { −Δu+V(x)u+αϕf(u)=g(x,u),x∈R3,−Δϕ=2αF(u),x∈R3.where α>0, V(x)∈C1(R3,R), V(x)>0. Under certain assumptions on f,gand p∈(1,2).
基金supported by the National Natural Science Foundation of China(No.12461024)the Natural Science Research Project of Department of Education of Guizhou Province(Nos.QJJ2023012,QJJ2023061,QJJ2023062)the Natural Science Research Project of Guizhou Minzu University(No.GZMUZK[2022]YB06)。
文摘In this paper,we consider the p-Laplacian Schrödinger-Poisson equation with L^(2)-norm constraint-Δ_(p)u+|u|^(p-2)u+λu+(1/4π|x|*|u|^(2))u=|u|^(q-2)u,x∈R^(3),where 2≤p<3,5p/3<q<p*=3p/3-p,λ>0 is a Lagrange multiplier.We obtain the critical point of the corresponding functional of the problem on mass constraint by the variational method and the Mountain pass lemma,and then find a normalized solution to this equation.
基金supported by the Applied Basic Research Program of Shanxi Province,China(Grant Nos.202403021212253 and 202203021221217).
文摘The(2+1)-dimensional generalized coupled nonlinear Schrödinger equations with a four-wave mixing term are studied in this paper,which describe optical solitons in birefringent fibers.Utilizing the Hirota bilinear method,we systematically construct single-and double-periodic lump solutions.To provide a detailed insight into the dynamic behavior of the nonlinear waves,we explore diverse mixed solutions,including bright-dark,W-shaped,multi-peak,and bright soliton solutions.Building upon single-periodic lump solutions,we analyze the dynamics of lump waves on both plane-wave and periodic backgrounds using the long-wave limit method.Moreover,we obtain the interaction solutions involving lumps,periodic lumps,and solitons.The interactions among two solitons,multiple lumps,and mixed waves are illustrated and analyzed.Comparative analysis reveals that these multi-lump solutions exhibit richer dynamical properties than conventional single-lump ones.These results contribute to a deeper understanding of nonlinear systems and may facilitate solving nonlinear problems in nature.
基金W.W.was supported by the China Postdoctoral Science Foundation(Grant No.2023M741992)Z.Y.was supported by the National Natural Science Foundation of China(Grant No.11925108).
文摘In this paper,we present some properties of scattering data for the derivative nonlinear Schrödinger equation in H^(S)(R)(s≥1/2)starting from the Lax pair.We show that the reciprocal of the transmission coefficient can be expressed as the sum of some iterative integrals,and its logarithm can be written as the sum of some connected iterative integrals.We provide the asymptotic properties of the first few iterative integrals of the reciprocal of the transmission coefficient.Moreover,we provide some regularity properties of the reciprocal of the transmission coefficient related to scattering data in H^(S)(R).
