带混合色散项的四阶Hartree方程驻波解的存在性与稳定性
Existence and Stability of Standing Wavesfor the Fourth-Order Hartree Equation withMixed Dispersion Terms
DOI: 10.12677/PM.2024.148307, PDF,    科研立项经费支持
作者: 颜春阳：西北师范大学数学与统计学院,甘肃 兰州
关键词: 波形分解；驻波解；轨道稳定性；Profile Decomposition； Standing Waves； Orbital Stability
摘要: 本文主要研究了如下带有混合色散项的四阶 Hartree 方程驻波解的存在性与轨道稳定性iψt-Δ2ψ+uΔψ+(|x|-γ*|ψ|2ψ=0,其中 0<γ< 4, μ€ℝ, ψ =ψ(t,x) : ℝ × ℝd →ℂ 是复值函数。在L2-次临界情况下,基于波形分解和广义 Gagliardo-Nirenberg 不等式,证明了该方程驻波解的存在性与轨道稳定性。
Abstract: In this paper, we study the existence and orbital stability of standing waves for the fourth-order Hartree equation with mixed dispersion terms iψt-Δ2ψ+uΔψ+(|x|-γ*|ψ|2ψ=0, where 0<γ< 4, μ€ℝ, ψ =ψ(t,x) : ℝ × ℝd →ℂ is the complex-valued wave function. In the L2-subcritical case, based on the generalized Gagliardo-Nirenberg inequality and the profile decomposition, we prove existence and orbital stability of standing waves for this equation.
文章引用：颜春阳. 带混合色散项的四阶Hartree方程驻波解的存在性与稳定性[J]. 理论数学, 2024, 14(8): 80-92. https://doi.org/10.12677/PM.2024.148307

参考文献

[1] Jeanjean, L., Jendrej, J., Le, T.T. and Visciglia, N. (2022) Orbital Stability of Ground States for a Sobolev Critical Schrodinger Equation. Journal de Mathematiques Pures et Appliquees, 164, 158-179.
https://doi.org/10.1016/j.matpur.2022.06.005
[2] Lieb, E.H. and Yau, H. (1987) The Chandrasekhar Theory of Stellar Collapse as the Limit of Quantum Mechanics. Communications in Mathematical Physics, 112, 147-174.
https://doi.org/10.1007/bf01217684
[3] Banquet, C. and J. Villamizar-Roa, E. (2020) On the Management Fourth-Order Schrodinger- Hartree Equation. Evolution Equations & Control Theory, 9, 865-889.
https://doi.org/10.3934/eect.2020037
[4] Shi, C. (2022) Existence of Stable Standing Waves for the Nonlinear Schrodinger Equation with Mixed Power-Type and Choquard-Type Nonlinearities. AIMS Mathematics, 7, 3802-3825.
https://doi.org/10.3934/math.2022211
[5] Tarek, S. (2020) Non-Linear Bi-Harmonic Choquard Equations. Communications on Pure and Applied Analysis, 11, 5033-5057.
[6] Karpman, V.I. (1996) Stabilization of Soliton Instabilities by Higher-Order Dispersion: Fourth- Order Nonlinear Schrodinger-Type Equations. Physical Review E, 53, R1336-R1339.
https://doi.org/10.1103/physreve.53.r1336
[7] Karpman, V.I. and Shagalov, A.G. (2000) Stability of Solitons Described by Nonlinear Schrodinger-Type Equations with Higher-Order Dispersion. Physica D: Nonlinear Phenom- ena, 144, 194-210.
https://doi.org/10.1016/s0167-2789(00)00078-6
[8] Fernandez, A.J., Jeanjean, L., Mandel, R. and Maris, M. (2022) Non-Homogeneous Gagliardo- Nirenberg Inequalities in RN and Application to a Biharmonic Non-Linear Schrodinger Equation. Journal of Differential Equations, 330, 1-65.
https://doi.org/10.1016/j.jde.2022.04.037
[9] Luo, T., Zheng, S. and Zhu, S. (2022) The Existence and Stability of Normalized Solutions for a Bi-Harmonic Nonlinear Schrodinger Equation with Mixed Dispersion. Acta Mathematica Scientia, 43, 539-563.
https://doi.org/10.1007/s10473-023-0205-5
[10] Cho, Y., Hajaiej, H., Hwang, G. and Ozawa, T. (2013) On the Cauchy Problem of Fractional Schrodinger Equation with Hartree Type Nonlinearity. Funkcialaj Ekvacioj, 56, 193-224.
https://doi.org/10.1619/fesi.56.193
[11] Feng, B. and Zhang, H. (2018) Stability of Standing Waves for the Fractional Schrodinger- Hartree Equation. Journal of Mathematical Analysis and Applications, 460, 352-364.
https://doi.org/10.1016/j.jmaa.2017.11.060
[12] Cazenave, T. and Lions, P.L. (1982) Orbital Stability of Standing Waves for Some Nonlinear Schroodinger Equations. Communicati
[13] Bonheure, D., Casteras, J., dos Santos, E.M. and Nascimento, R. (2018) Orbitally Stable Standing Waves of a Mixed Dispersion Nonlinear Schrodinger Equation. SIAM Journal on Mathematical Analysis, 50, 5027-5071.
https://doi.org/10.1137/17m1154138
[14] Feng, W., Stanislavova, M. and Stefanov, A. (2018) On the Spectral Stability of Ground States of Semi-Linear Schrodinger and Klein-Gordon Equations with Fractional Dispersion. Communications on Pure & Applied Analysis, 17, 1371-1385.
https://doi.org/10.3934/cpaa.2018067
[15] Posukhovskyi, I. and G. Stefanov, A. (2020) On the Normalized Ground States for the Kawahara Equation and a Fourth Order NLS. Discrete & Continuous Dynamical Systems|A, 40, 4131-4162. https://doi.org/10.3934/dcds.2020175
[16] Gerard, P. (1998) Description of the Lack of Compactness for the Sobolev Imbedding. ESAIM: Control, Optimisation and Calculus of Variations, 3, 213-233.
https://doi.org/10.1051/cocv:1998107
[17] Hmidi, T. and Keraani, S. (2005) Blowup Theory for the Critical Nonlinear Schrodinger Equations Revisited. International Mathematics Research Notices, 2005, 2815-2828.
https://doi.org/10.1155/imrn.2005.2815
[18] Yang, H., Zhang, J. and Zhu, S. (2010) Limiting Profile of the Blow-Up Solutions for the Fourth-Order Nonlinear Schrodinger Equation. Dynamics of Partial Differential Equations, 7, 187-205.
https://doi.org/10.4310/dpde.2010.v7.n2.a4
[19] Carles, R., Markowich, P.A. and Sparber, C. (2008) On the Gross-Pitaevskii Equation for Trapped Dipolar Quantum Gases. Nonlinearity, 21, 2569-2590.
https://doi.org/10.1088/0951-7715/21/11/006
[20] Lieb, E.H. (1983) Sharp Constants in the Hardy-Littlewood-Sobolev and Related Inequalities. The Annals of Mathematics, 118, 349-374.
https://doi.org/10.2307/2007032

为你推荐