Analytical solutions in terms of rational-like functions are presented for a （3＋1）-dimensional nonlinear Schrodinger equation with time-varying coefficients and a harmonica potential using the similarity transforma...Analytical solutions in terms of rational-like functions are presented for a （3＋1）-dimensional nonlinear Schrodinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz. Several free functions of time t are involved to generate abundant wave structures. Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations.展开更多
Let L = -△Hn ＋ V be a SchrSdinger operator on Heisenberg group Hn, where AHn is the sublaplacian and the nonnegative potential V belongs to the reverse HSlder class BQ/2 where Q is the homogeneous dimension of Hn. L...Let L = -△Hn ＋ V be a SchrSdinger operator on Heisenberg group Hn, where AHn is the sublaplacian and the nonnegative potential V belongs to the reverse HSlder class BQ/2 where Q is the homogeneous dimension of Hn. Let T1 = （--△Hn ＋V）-1V, T2 = （-△Hn ＋V）-1/2V1/2, and T3 = （--AHn ＋V）-I/2△Hn, then we verify that [b, Ti], i = 1, 2, 3 are bounded on some LP（Hn）, where b ∈ BMO（Hn）. Note that the kernel of Ti, i = 1, 2, 3 has no smoothness.展开更多
In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of SchrSdinger-Kirchhoff type -εpM（εp-N∫RN｜△u｜p）△pu＋v（x｜u｜p-2u=f（u）in RN, where ...In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of SchrSdinger-Kirchhoff type -εpM（εp-N∫RN｜△u｜p）△pu＋v（x｜u｜p-2u=f（u）in RN, where △p is the p-Laplacian operator, 1 〈 p 〈 N, M ： R＋ → R＋ and V ： RN →R＋ are continuous functions, ε is a positive parameter, and f is a continuous function with subcritical growth. We assume that V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and Lyusternik- Schnirelmann theory, we prove the existence, multiplicity, and concentration of solutions for the above equation.展开更多
The authors prove the existence of nontrivial solutions for the SchrSdinger equation -△u ＋ V（x）u =λf（x, u） in R^N, where f is superlinear, subcritical and critical at infinity, respectively, V is periodic.
The coupled nonlinear Schodinger equations （CNLSEs） of two symmetrical optical fibres are nonintegrable, however the transformed CNLSEs have integrability. Integrability of the transformed CNLSEs is proved by the Ha...The coupled nonlinear Schodinger equations （CNLSEs） of two symmetrical optical fibres are nonintegrable, however the transformed CNLSEs have integrability. Integrability of the transformed CNLSEs is proved by the Hamilton dynamics theory and Galilei transform. Making use of a transform for CNLSEs and using the ansatz with Jacobi elliptic function form, this paper obtains the exact optical pulse solutions.展开更多
In this article, a nonconforming quadrilateral element （named modified quasi- Wilson element） is applied to solve the nonlinear schrSdinger equation （NLSE）. On the basis of a special character of this element, tha...In this article, a nonconforming quadrilateral element （named modified quasi- Wilson element） is applied to solve the nonlinear schrSdinger equation （NLSE）. On the basis of a special character of this element, that is, its consistency error is of order O（ha） for broken Ha-norm on arbitrary quadrilateral meshes, which is two order higher than its interpolation error, the optimal order error estimate and superclose property are obtained. Moreover, the global superconvergence result is deduced with the help of interpolation postprocessing technique. Finally, some numerical results are provided to verify the theoretical analysis.展开更多
In this article, we study constrained minimizers of the following variational problem ε（p）：={u∈H1 inf（R3）,｜｜u｜｜22=p} E（u）,ρ〉0,where E（u） is the SchrSdinger-Poisson-Slater （SPS） energy functional E（...In this article, we study constrained minimizers of the following variational problem ε（p）：={u∈H1 inf（R3）,｜｜u｜｜22=p} E（u）,ρ〉0,where E（u） is the SchrSdinger-Poisson-Slater （SPS） energy functional E（u）：1/2∫R3｜△u（x）｜2dx-1/4∫R3∫R3u2（y）u2（x）/｜x-y｜dydx-1/p∫R3｜u（x）∫pdx in R3,and p ∈ （2,6）. We prove the existence of minimizers for the cases 2 〈 p 〈10/3, p 〉 0, and P =10/3, 0 〈 p 〈 p＊, and show that e（ρ） = -∞ for the other cases, where p＊ = ｜｜φ｜｜22 and φ（x） is the unique （up to translations） positive radially symmetric solution of -△u ＋ u = u7/3 in R3. Moreover, when e（ρ＊） = -∞, the blow-up behavior of minimizers as p/p＊ is also analyzed rigorously.展开更多
基金Project supported by the National Natural Science Foundation of China （Grant No. 10772110） and the Natural Science Foundation of Zhejiang Province, China （Grant Nos. Y606049, Y6090681, and Y6100257）.
文摘Analytical solutions in terms of rational-like functions are presented for a （3＋1）-dimensional nonlinear Schrodinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz. Several free functions of time t are involved to generate abundant wave structures. Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations.
基金Supported by the National Natural Science Foundation under Grant(11147180)Science and Technology Agency Foundation of Hubei Province under Grant(2011CDC005,D20122804)
基金supported by NSFC 11171203, S2011040004131STU Scientific Research Foundation for Talents TNF 10026+1 种基金supported by NSFC No.10990012,10926179RFDP of China No.200800010009
文摘Let L = -△Hn ＋ V be a SchrSdinger operator on Heisenberg group Hn, where AHn is the sublaplacian and the nonnegative potential V belongs to the reverse HSlder class BQ/2 where Q is the homogeneous dimension of Hn. Let T1 = （--△Hn ＋V）-1V, T2 = （-△Hn ＋V）-1/2V1/2, and T3 = （--AHn ＋V）-I/2△Hn, then we verify that [b, Ti], i = 1, 2, 3 are bounded on some LP（Hn）, where b ∈ BMO（Hn）. Note that the kernel of Ti, i = 1, 2, 3 has no smoothness.
基金supported by Natural Science Foundation of China(11371159 and 11771166)Hubei Key Laboratory of Mathematical Sciences and Program for Changjiang Scholars and Innovative Research Team in University#IRT_17R46
文摘In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of SchrSdinger-Kirchhoff type -εpM（εp-N∫RN｜△u｜p）△pu＋v（x｜u｜p-2u=f（u）in RN, where △p is the p-Laplacian operator, 1 〈 p 〈 N, M ： R＋ → R＋ and V ： RN →R＋ are continuous functions, ε is a positive parameter, and f is a continuous function with subcritical growth. We assume that V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and Lyusternik- Schnirelmann theory, we prove the existence, multiplicity, and concentration of solutions for the above equation.
文摘The authors prove the existence of nontrivial solutions for the SchrSdinger equation -△u ＋ V（x）u =λf（x, u） in R^N, where f is superlinear, subcritical and critical at infinity, respectively, V is periodic.
文摘The coupled nonlinear Schodinger equations （CNLSEs） of two symmetrical optical fibres are nonintegrable, however the transformed CNLSEs have integrability. Integrability of the transformed CNLSEs is proved by the Hamilton dynamics theory and Galilei transform. Making use of a transform for CNLSEs and using the ansatz with Jacobi elliptic function form, this paper obtains the exact optical pulse solutions.
基金supported by the National Natural Science Foundation of China(11271340,11101381)
文摘In this article, a nonconforming quadrilateral element （named modified quasi- Wilson element） is applied to solve the nonlinear schrSdinger equation （NLSE）. On the basis of a special character of this element, that is, its consistency error is of order O（ha） for broken Ha-norm on arbitrary quadrilateral meshes, which is two order higher than its interpolation error, the optimal order error estimate and superclose property are obtained. Moreover, the global superconvergence result is deduced with the help of interpolation postprocessing technique. Finally, some numerical results are provided to verify the theoretical analysis.
基金partially supported by National Natural Science Foundation of China(11671394)
文摘In this article, we study constrained minimizers of the following variational problem ε（p）：={u∈H1 inf（R3）,｜｜u｜｜22=p} E（u）,ρ〉0,where E（u） is the SchrSdinger-Poisson-Slater （SPS） energy functional E（u）：1/2∫R3｜△u（x）｜2dx-1/4∫R3∫R3u2（y）u2（x）/｜x-y｜dydx-1/p∫R3｜u（x）∫pdx in R3,and p ∈ （2,6）. We prove the existence of minimizers for the cases 2 〈 p 〈10/3, p 〉 0, and P =10/3, 0 〈 p 〈 p＊, and show that e（ρ） = -∞ for the other cases, where p＊ = ｜｜φ｜｜22 and φ（x） is the unique （up to translations） positive radially symmetric solution of -△u ＋ u = u7/3 in R3. Moreover, when e（ρ＊） = -∞, the blow-up behavior of minimizers as p/p＊ is also analyzed rigorously.
