Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Exact solutions to three-dimensional generalized Gross-Pitaevskii equations with varying potential and nonlinearities Zhenya Yan KLMM, Chinese Academy of Sciences, Beijing, China Email: zyyan@mmrc.iss.ac.cn Joint work with V. V. Konotop CFTC, Universidade de Lisboa, Lisboa 1649-003, Portugal Oct. 30, 2010 The 4th International Workshop on Differential Algebra and Related Topics, Beijing Zhenya Yan 3D GP equation with varying coefficients
Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Outline Introduction 1 Bose-Einstein condensates: Gross-Pitaevskii (GP) equation 1D GP equation with varying coefficients 3D generalized GP equation with varying coefficients Zhenya Yan 3D GP equation with varying coefficients
Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Outline Introduction 1 Bose-Einstein condensates: Gross-Pitaevskii (GP) equation 1D GP equation with varying coefficients 3D generalized GP equation with varying coefficients Similarity reductions and solutions 2 Similarity reduction and determining equations Zhenya Yan 3D GP equation with varying coefficients
Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Outline Introduction 1 Bose-Einstein condensates: Gross-Pitaevskii (GP) equation 1D GP equation with varying coefficients 3D generalized GP equation with varying coefficients Similarity reductions and solutions 2 Similarity reduction and determining equations Surfaces and stationary solutions 3 Amplitude and phase surfaces Solutions: cubic GP equations with varying coefficients Extension of stationary solutions Zhenya Yan 3D GP equation with varying coefficients
Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Outline Introduction 1 Bose-Einstein condensates: Gross-Pitaevskii (GP) equation 1D GP equation with varying coefficients 3D generalized GP equation with varying coefficients Similarity reductions and solutions 2 Similarity reduction and determining equations Surfaces and stationary solutions 3 Amplitude and phase surfaces Solutions: cubic GP equations with varying coefficients Extension of stationary solutions Time-dependent surfaces and solutions 4 Different surfaces depending on time Time-dependent Solutions Zhenya Yan 3D GP equation with varying coefficients
Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Outline Introduction 1 Bose-Einstein condensates: Gross-Pitaevskii (GP) equation 1D GP equation with varying coefficients 3D generalized GP equation with varying coefficients Similarity reductions and solutions 2 Similarity reduction and determining equations Surfaces and stationary solutions 3 Amplitude and phase surfaces Solutions: cubic GP equations with varying coefficients Extension of stationary solutions Time-dependent surfaces and solutions 4 Different surfaces depending on time Time-dependent Solutions Conclusions 5 Zhenya Yan 3D GP equation with varying coefficients
Introduction Similarity reductions and solutions Surfaces and stationary solutions Bose-Einstein condensates: Gross-Pitaevskii (GP) equation Time-dependent surfaces and solutions Conclusions 1D GP 1. Introduction Bose-Einstein condensates: Gross-Pitaevskii (GP) equation Zhenya Yan 3D GP equation with varying coefficients
Introduction Similarity reductions and solutions Surfaces and stationary solutions Bose-Einstein condensates: Gross-Pitaevskii (GP) equation Time-dependent surfaces and solutions Conclusions 1D GP 1. Introduction Quasi-one dimensional (1D) GP equation Zhenya Yan 3D GP equation with varying coefficients
Introduction Similarity reductions and solutions Surfaces and stationary solutions Bose-Einstein condensates: Gross-Pitaevskii (GP) equation Time-dependent surfaces and solutions Conclusions 1D GP 1. Introduction Quasi-one dimensional (1D) GP equation 1D GP equation with space-modulated coefficients iψ t = − ψ xx + v ( x ) ψ + g ( x ) | ψ | 2 ψ, (1.1) where v ( x ) denotes the external potential and g ( x ) stands for the nonlinearity. Zhenya Yan 3D GP equation with varying coefficients
Introduction Similarity reductions and solutions Surfaces and stationary solutions Bose-Einstein condensates: Gross-Pitaevskii (GP) equation Time-dependent surfaces and solutions Conclusions 1D GP 1. Introduction Quasi-one dimensional (1D) GP equation 1D GP equation with space-modulated coefficients iψ t = − ψ xx + v ( x ) ψ + g ( x ) | ψ | 2 ψ, (1.1) where v ( x ) denotes the external potential and g ( x ) stands for the nonlinearity. [J. Belmonte-Beitia, et al. , Phys. Rev. Lett. 98 (2007) 064102.] Zhenya Yan 3D GP equation with varying coefficients
Introduction Similarity reductions and solutions Surfaces and stationary solutions Bose-Einstein condensates: Gross-Pitaevskii (GP) equation Time-dependent surfaces and solutions Conclusions 1D GP 1. Introduction Quasi-one dimensional (1D) GP equation 1D GP equation with space-modulated coefficients iψ t = − ψ xx + v ( x ) ψ + g ( x ) | ψ | 2 ψ, (1.1) where v ( x ) denotes the external potential and g ( x ) stands for the nonlinearity. [J. Belmonte-Beitia, et al. , Phys. Rev. Lett. 98 (2007) 064102.] 1D GP equation with (time, space)-modulated coefficients iψ t = − ψ xx + v ( x, t ) ψ + g ( x, t ) | ψ | 2 ψ, (1.2) [ J. Belmonte-Beitia, et al. , Phys. Rev. Lett. 100 (2008) 164102.] Zhenya Yan 3D GP equation with varying coefficients
Introduction Similarity reductions and solutions Surfaces and stationary solutions Bose-Einstein condensates: Gross-Pitaevskii (GP) equation Time-dependent surfaces and solutions Conclusions 1D GP 1. Introduction 3D generalized GP equation with varying coefficients i∂ψ ∂t = − 1 2 ∇ 2 ψ + v ( r , t ) ψ g p ( r , t ) | ψ | p − 1 + g q ( r , t ) | ψ | q − 1 � � + ψ, (1.3) Zhenya Yan 3D GP equation with varying coefficients
Introduction Similarity reductions and solutions Surfaces and stationary solutions Bose-Einstein condensates: Gross-Pitaevskii (GP) equation Time-dependent surfaces and solutions Conclusions 1D GP 1. Introduction 3D generalized GP equation with varying coefficients i∂ψ ∂t = − 1 2 ∇ 2 ψ + v ( r , t ) ψ g p ( r , t ) | ψ | p − 1 + g q ( r , t ) | ψ | q − 1 � � + ψ, (1.3) where ψ ≡ ψ ( r , t ), r ∈ R 3 , ∇ 2 ≡ ∂ 2 x + ∂ 2 y + ∂ 2 z , q > p ≥ 3 are integers, the linear potential v ( r , t ) and the nonlinear coefficients g p,q ( r , t ) are all real-valued functions of time and space. Zhenya Yan 3D GP equation with varying coefficients
Introduction Similarity reductions and solutions Surfaces and stationary solutions Similarity reduction and determining equations Time-dependent surfaces and solutions Conclusions 2.1 Similarity reductions Consider the similarity transformation ψ ( r , t ) = ρ ( r , t ) e iϕ ( r ,t ) Φ( η ( r , t )) , (2.1) Zhenya Yan 3D GP equation with varying coefficients
Introduction Similarity reductions and solutions Surfaces and stationary solutions Similarity reduction and determining equations Time-dependent surfaces and solutions Conclusions 2.1 Similarity reductions Consider the similarity transformation ψ ( r , t ) = ρ ( r , t ) e iϕ ( r ,t ) Φ( η ( r , t )) , (2.1) Requiring Φ( η ) to satisfy the generalized stationary GP equa- tion with constant coefficients µ Φ = − Φ ηη + G p | Φ | p − 1 Φ + G q | Φ | q − 1 Φ . (2.2) Zhenya Yan 3D GP equation with varying coefficients
Introduction Similarity reductions and solutions Surfaces and stationary solutions Similarity reduction and determining equations Time-dependent surfaces and solutions Conclusions 2.1 Similarity reductions Consider the similarity transformation ψ ( r , t ) = ρ ( r , t ) e iϕ ( r ,t ) Φ( η ( r , t )) , (2.1) Requiring Φ( η ) to satisfy the generalized stationary GP equa- tion with constant coefficients µ Φ = − Φ ηη + G p | Φ | p − 1 Φ + G q | Φ | q − 1 Φ . (2.2) Here Φ ≡ Φ( η ) is a function of the variable η ≡ η ( r , t ) whose relation to the original variables ( r , t ) is to be determined, µ is the eigenvalue of the nonlinear equation, and G p,q are constants. we, without loss of generality, focus on the cases where G p = 0 , ± 1 and G q = 0 , ± 1. Zhenya Yan 3D GP equation with varying coefficients
Introduction Similarity reductions and solutions Surfaces and stationary solutions Similarity reduction and determining equations Time-dependent surfaces and solutions Conclusions We thus obtain the set of equations ∇ · ( ρ 2 ∇ η ) = 0 , (2.3a) ( ρ 2 ) t + ∇ · ( ρ 2 ∇ ϕ ) = 0 , (2.3b) η t + ∇ ϕ · ∇ η = 0 , (2.3c) 2 g j ( r , t ) ρ j − 1 − G j |∇ η | 2 = 0 ( j = p, q ) , (2.3d) 2 v ( r , t ) + µ |∇ η | 2 + |∇ ϕ | 2 − ρ − 1 ∇ 2 ρ + 2 ϕ t = 0 . (2.3e) Zhenya Yan 3D GP equation with varying coefficients
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