Exact solutions to three-dimensional generalized Gross-Pitaevskii - - PowerPoint PPT Presentation

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

1

Introduction 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

1

Introduction Bose-Einstein condensates: Gross-Pitaevskii (GP) equation 1D GP equation with varying coefficients 3D generalized GP equation with varying coefficients

2

Similarity reductions and solutions 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

1

Introduction Bose-Einstein condensates: Gross-Pitaevskii (GP) equation 1D GP equation with varying coefficients 3D generalized GP equation with varying coefficients

2

Similarity reductions and solutions Similarity reduction and determining equations

3

Surfaces and stationary solutions 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

1

Introduction Bose-Einstein condensates: Gross-Pitaevskii (GP) equation 1D GP equation with varying coefficients 3D generalized GP equation with varying coefficients

2

Similarity reductions and solutions Similarity reduction and determining equations

3

Surfaces and stationary solutions Amplitude and phase surfaces Solutions: cubic GP equations with varying coefficients Extension of stationary solutions

4

Time-dependent surfaces and solutions 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

1

Introduction Bose-Einstein condensates: Gross-Pitaevskii (GP) equation 1D GP equation with varying coefficients 3D generalized GP equation with varying coefficients

2

Similarity reductions and solutions Similarity reduction and determining equations

3

Surfaces and stationary solutions Amplitude and phase surfaces Solutions: cubic GP equations with varying coefficients Extension of stationary solutions

4

Time-dependent surfaces and solutions Different surfaces depending on time Time-dependent Solutions

5

Conclusions

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Bose-Einstein condensates: Gross-Pitaevskii (GP) equation 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 Time-dependent surfaces and solutions Conclusions Bose-Einstein condensates: Gross-Pitaevskii (GP) equation 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 Time-dependent surfaces and solutions Conclusions Bose-Einstein condensates: Gross-Pitaevskii (GP) equation 1D GP

• 1. Introduction

Quasi-one dimensional (1D) GP equation 1D GP equation with space-modulated coefficients iψt = −ψxx + v(x)ψ + g(x)|ψ|2ψ, (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 Time-dependent surfaces and solutions Conclusions Bose-Einstein condensates: Gross-Pitaevskii (GP) equation 1D GP

• 1. Introduction

Quasi-one dimensional (1D) GP equation 1D GP equation with space-modulated coefficients iψt = −ψxx + v(x)ψ + g(x)|ψ|2ψ, (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 Time-dependent surfaces and solutions Conclusions Bose-Einstein condensates: Gross-Pitaevskii (GP) equation 1D GP

• 1. Introduction

Quasi-one dimensional (1D) GP equation 1D GP equation with space-modulated coefficients iψt = −ψxx + v(x)ψ + g(x)|ψ|2ψ, (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 iψt = −ψxx + v(x, t)ψ + g(x, t)|ψ|2ψ, (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 Time-dependent surfaces and solutions Conclusions Bose-Einstein condensates: Gross-Pitaevskii (GP) equation 1D GP

• 1. Introduction

3D generalized GP equation with varying coefficients i∂ψ ∂t = −1 2∇2ψ + v(r, t)ψ +

• gp(r, t)|ψ|p−1 + gq(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 Time-dependent surfaces and solutions Conclusions Bose-Einstein condensates: Gross-Pitaevskii (GP) equation 1D GP

• 1. Introduction

3D generalized GP equation with varying coefficients i∂ψ ∂t = −1 2∇2ψ + v(r, t)ψ +

• gp(r, t)|ψ|p−1 + gq(r, t)|ψ|q−1

ψ, (1.3) where ψ ≡ ψ(r, t), r ∈ R3, ∇2 ≡ ∂2

x + ∂2 y + ∂2 z, q > p ≥ 3 are

integers, the linear potential v(r, t) and the nonlinear coefficients gp,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 Time-dependent surfaces and solutions Conclusions Similarity reduction and determining equations

2.1 Similarity reductions

Consider the similarity transformation ψ(r, t) = ρ(r, t)eiϕ(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 Time-dependent surfaces and solutions Conclusions Similarity reduction and determining equations

2.1 Similarity reductions

Consider the similarity transformation ψ(r, t) = ρ(r, t)eiϕ(r,t)Φ(η(r, t)), (2.1) Requiring Φ(η) to satisfy the generalized stationary GP equa- tion with constant coefficients µΦ = −Φηη + Gp|Φ|p−1Φ + Gq|Φ|q−1Φ. (2.2)

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Similarity reduction and determining equations

2.1 Similarity reductions

Consider the similarity transformation ψ(r, t) = ρ(r, t)eiϕ(r,t)Φ(η(r, t)), (2.1) Requiring Φ(η) to satisfy the generalized stationary GP equa- tion with constant coefficients µΦ = −Φηη + Gp|Φ|p−1Φ + Gq|Φ|q−1Φ. (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 Gp,q are constants. we, without loss of generality, focus on the cases where Gp = 0, ±1 and Gq = 0, ±1.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Similarity reduction and determining equations

We thus obtain the set of equations ∇ · (ρ2∇η) = 0, (2.3a) (ρ2)t + ∇ · (ρ2∇ϕ) = 0, (2.3b) ηt + ∇ϕ · ∇η = 0, (2.3c) 2gj(r, t)ρj−1 − Gj|∇η|2 = 0 (j = p, q), (2.3d) 2v(r, t) + µ|∇η|2 + |∇ϕ|2 − ρ−1∇2ρ + 2ϕt = 0. (2.3e)

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Similarity reduction and determining equations

We thus obtain the set of equations ∇ · (ρ2∇η) = 0, (2.3a) (ρ2)t + ∇ · (ρ2∇ϕ) = 0, (2.3b) ηt + ∇ϕ · ∇η = 0, (2.3c) 2gj(r, t)ρj−1 − Gj|∇η|2 = 0 (j = p, q), (2.3d) 2v(r, t) + µ|∇η|2 + |∇ϕ|2 − ρ−1∇2ρ + 2ϕt = 0. (2.3e) These equations lead to several immediate conclusions. First, it follows from (2.3d) that gp,q(r, t) are sign definite, and Gj = sign–gj(r, t)˝. Moreover, comparing the equations in (2.3d) for j = p and j = q we find that either |gp| = ρq−p|gq|, or one of the nonlinear coefficients is zero, i.e. either |gp| ≡ 0 or |gq| ≡ 0. Respectively, we define the function g(r, t) ≡ 2gjρj−1/Gj, where j = p, q.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

3.1 Surfaces and stationary solutions

We start with the stationary solutions of Eq. (1.3), ρt = ηt = ϕt = 0. We here take ρ = 1. Moreover, we have v(r, t) ≡ v(r) and g(r, t) ≡ g(r).

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

3.1 Surfaces and stationary solutions

We start with the stationary solutions of Eq. (1.3), ρt = ηt = ϕt = 0. We here take ρ = 1. Moreover, we have v(r, t) ≡ v(r) and g(r, t) ≡ g(r). We rewrite the system (2.3) in the stationary case as

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

3.1 Surfaces and stationary solutions

We start with the stationary solutions of Eq. (1.3), ρt = ηt = ϕt = 0. We here take ρ = 1. Moreover, we have v(r, t) ≡ v(r) and g(r, t) ≡ g(r). We rewrite the system (2.3) in the stationary case as ∇2η = 0, ∇2ϕ = 0, ∇η · ∇ϕ = 0, (3.1a) g(r) = |∇η|2, v(r) = −1 2(|∇ϕ|2 + µ|∇η|2). (3.1b)

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

Now we consider surfaces of the constant amplitude and phase, i.e. η(r) = η0 = const and ϕ(r) = ϕ0 = const. (3.2) In what follows we restrict our consideration to the finite power (N-order) surfaces, i.e. depending on terms like xn1yn2zn3 with nj being finite positive integers This allows us to list in the Table I all admissible coordinate and phase surfaces which appear to be not higher than the third-order, i.e. N ≤ 3.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

3.2 The velocity fields v = ∇ϕ

Figure: (Color online) The velocity fields v = ∇ϕ corresponding to the phases listed in Table I for a = a1,2 = 1. (a) (y, z)-plane for Case II, (b) 3D-space for Case III , (c) 3D-space for Case IV.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

3.3 Solution: cubic GP equation

Following the algorithm described above, in order to construct the exact solutions of Eq. (1.3), as the last step we have to address solutions of Eq. (2.2). They depend on the particular choice of the

• model. In the present work we consider two the most relevant

physical cases. Thus starting with the case p = 3, gq(r) ≡ 0 and hence Gq = 0 we have to deal with the cubic GP equation µΦ = −Φηη + G3Φ3. (3.3) The respective periodic and localized solutions are very well

• known. Below we consider the simplest ones for attractive and

repulsive nonlinearities G3.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

3.3.1 Attractive nonlinearity G3 = −1

The bright solitons ψbs(r) =

• −2µ sech[√−µ η(r)] exp[iϕ],

where µ < 0 and the amplitude η(r) and phase ϕ(r) are defined by Table I.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

3.3.1 Attractive nonlinearity G3 = −1

The bright solitons ψbs(r) =

• −2µ sech[√−µ η(r)] exp[iϕ],

where µ < 0 and the amplitude η(r) and phase ϕ(r) are defined by Table I.

Figure: Cross-sections of the density distribution of the “bright soliton” |ψbs(r)|2 with η given in Table I for µ = −1. (a) c = 1 with η for Case II, (b) c = 0.5 with η for Case III; here the cross-section at z = 0 shows the peak intensity, (c) η for Case IV.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

The periodic cn-wave solution: ψcn(r) =

• 2µk2

1 − 2k2 cn

• 2µ

1 − 2k2 η(r), k

• eiϕ(r),

(3.4) where k ∈ [0, 1] is the modulus of the Jacobi elliptic function and µ satisfies the condition µ(1 − 2k2) > 0.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

The periodic cn-wave solution: ψcn(r) =

• 2µk2

1 − 2k2 cn

• 2µ

1 − 2k2 η(r), k

• eiϕ(r),

(3.4) where k ∈ [0, 1] is the modulus of the Jacobi elliptic function and µ satisfies the condition µ(1 − 2k2) > 0.

Figure: Cross-sections of the density distribution of the “bright soliton” |ψbs(r)|2 with η given in Table I for µ = −1. (a) c = 1 with η for Case II, (b) c = 0.5 with η for Case III; here the cross-section at z = 0 shows the peak intensity, (c) η for Case IV.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

3.3.2 Repulsive nonlinearity G3 = 1

The dark soliton solution: ψds(r) = √µ tanh[

• µ/2 η(r)] exp(iϕ(r))

.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

3.3.2 Repulsive nonlinearity G3 = 1

The dark soliton solution: ψds(r) = √µ tanh[

• µ/2 η(r)] exp(iϕ(r))

.

Figure: Cross-sections of the density distribution of the “dark soliton” |ψds(r)|2 with η listed in Table I for µ = 2. (a) c = 1 with η for Case II, (b) c = 0.5 with η for Case III, the minimum intensity is shown in the cross-section with z = 0, (c) η for Case IV.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

The “periodic” sn-wave solutions ψsn(r) =

• 2µk2

1 + k2 sn

• µ

1 + k2 η(r), k

• eiϕ(r)

(3.5) with a positive chemical potential µ.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

The “periodic” sn-wave solutions ψsn(r) =

• 2µk2

1 + k2 sn

• µ

1 + k2 η(r), k

• eiϕ(r)

(3.5) with a positive chemical potential µ.

Figure: Cross-sections of the density distribution of the sn-wave solution |ψsn(r)|2 with η given in Table I for µ = 2. (a) c = 1 with η for Case II, (b) c = 0.5 with η for Case III, the minimum intensity is shown in the cross-section at z = 0, (c) η for Case IV. In all the panels k = 0.8.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

3.4 Solutions: cubic-quintic GP equation

The cubic-quintic GP equation, i.e., p = 3, q = 5 for which Eq. (2.2) becomes µΦ = −Φηη + G3Φ3 + G5Φ5. (3.6) Its solutions for the condition G3G5 < 0 are also known (some nontrivial examples are listed in Table III. While the amplitude and the phase surfaces are now the same as in the case of the cubic GP equation, the density distribution is described by different periodic and localized functions. We in particular emphasize possibility of the algebraic solutions, like the ones given by the cases 4 and 7 in Table III.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

3.5 Extension of stationary solutions

In fact, we can also consider the general case ρ(r) depending

• n the spatial variable r, for which system (2.3) reduces to the

system

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

3.5 Extension of stationary solutions

In fact, we can also consider the general case ρ(r) depending

• n the spatial variable r, for which system (2.3) reduces to the

system ∇ · (ρ∇η) = 0, ∇ · (ρ∇ϕ) = 0, ∇η · ∇ϕ = 0, (3.7a) gj(r) = 1 2ρ1−jGj|∇η|2 (j = p, q), (3.7b) v(r) = 1 2

• ρ−1∇2ρ − µ|∇η|2 − |∇ϕ|2

. (3.7c)

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

Table: IV. Admissible r-modulated amplitude surface and function ρ(r) (f(ξ) being an arbitrary function, f ′(ξ) > 0)

Case Amplitude surface Function ρ(r) I η(r) = f(ξ), ξ = c · r

• 1/f′(ξ)

II η(r) = f(ξ), ξ = x + c(y2 − z2)

• 1/f′(ξ)

III η(r) = f(ξ), ξ = cx2 + (1 − c)y2 − z2

• 1/f′(ξ)

IV η(r) = f(ξ), ξ = xyz

• 1/f′(ξ)

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Amplitude and phase surfaces Solutions: cubic GP equations with

Table: V. Admissible r-modulated phase surfaces and function ρ(r) (f(ς) being an arbitrary function with f ′(ς) > 0)

Case Phase surface Function ρ(r) I ϕ(r) = f(ς), ς = a · r, (c · a = 0)

• 1/f′(ς)

II ϕ(r) = f(ς), ς = yz

• 1/f′(ς)

III ϕ(r) = f(ς), ς = xyz

• 1/f′(ς)

IV ϕ(r) = f(ς), ς = ax2 + (1 − a)y2 − z2

• 1/f′(ς)

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Different surfaces depending on time Time-dependent Solutions

4 Time-dependent surfaces and solutions

Now we allow ρ(t), η(r, t) and ϕ(r, t) to depend on spatial and temporal variables. Then the system (2.3) is simplified:

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Different surfaces depending on time Time-dependent Solutions

4 Time-dependent surfaces and solutions

Now we allow ρ(t), η(r, t) and ϕ(r, t) to depend on spatial and temporal variables. Then the system (2.3) is simplified: ∇2η = 0, 2ρt + ρ∇2ϕ = 0, ηt + ∇ϕ · ∇η = 0, (4.1a) g(r, t) − |∇η|2 = 0, (4.1b) 2v(r, t) + µ|∇η|2 + |∇ϕ|2 + 2ϕt = 0. (4.1c)

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Different surfaces depending on time Time-dependent Solutions

4 Time-dependent surfaces and solutions

Now we allow ρ(t), η(r, t) and ϕ(r, t) to depend on spatial and temporal variables. Then the system (2.3) is simplified: ∇2η = 0, 2ρt + ρ∇2ϕ = 0, ηt + ∇ϕ · ∇η = 0, (4.1a) g(r, t) − |∇η|2 = 0, (4.1b) 2v(r, t) + µ|∇η|2 + |∇ϕ|2 + 2ϕt = 0. (4.1c) As before we focus on the finite power surfaces, i.e. depending

• n terms like fn1n2n3(t)xn1yn2zn3 with nj being finite positive

integers and fn1n2n3(t) being functions on time t. In what follows we consider all admissible coordinate and phase surfaces which appear to be not higher than the third-order, i.e. N ≤ 3.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Different surfaces depending on time Time-dependent Solutions

4.1 Plane surface depending on time

Based on the Case I of the Table I and Eq. (4.1a) we consider η parameterizing moving plains η(r, t) = c(t) · r (4.2) where c(t) = (cx(t), cy(t), cz(t)) subject to the constrain cx(t)cy(t)cz(t) > 0 for time.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Different surfaces depending on time Time-dependent Solutions

4.1 Plane surface depending on time

Based on the Case I of the Table I and Eq. (4.1a) we consider η parameterizing moving plains η(r, t) = c(t) · r (4.2) where c(t) = (cx(t), cy(t), cz(t)) subject to the constrain cx(t)cy(t)cz(t) > 0 for time. The nontrivial phase and function ρ(t) now read ϕ(r, t) = rˆ Ω(t)r + a(t) · r, ρ(t) =

• cx(t)cy(t)cz(t), (4.3)

where ˆ Ω denotes the diagonal matrix, ˆ Ω =diag(Ωx, Ωy, Ωz) with Ωσ = −˙ cσ(t)/(2cσ(t)) (hereafter σ = x, y, z) and a(t) = (ax(t), ay(t), az(t)) is a time dependent vector-function, such that the condition c(t) · a(t) = 0 is satisfied.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Different surfaces depending on time Time-dependent Solutions

Now, from Eqs. (4.1b) and (4.1c) we obtain v(r, t) = r ˆ A(t)r + b(t) · r − 1 2(µ|c(t)|2 + |a(t)|2), and g(r, t) = |c(t)|2. Here ˆ A =diag(Ax, Ay, Az) with the entries Aσ = ¨ cσ(t) 2cσ(t) − ˙ c2

σ(t)

c2

σ(t),

(4.4) and the vector function b(t) = (bx, by, bz) with bσ = ˙ cσ(t)aσ(t) cσ(t) − ˙ aσ(t). (4.5)

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Different surfaces depending on time Time-dependent Solutions

4.2 Paraboloid depending on time

Consider the generalization of the parabolic Case II from the Table I for which the amplitude η(r, t) and the phase ϕ(r, t) are as follows η(r, t) = cx(t)x + cy(t)(y2 − z2), ϕ(r, t) = r˜ Ω(t)r + a(t)yz,(4.6) where cx(t)cy(t) > 0, and ˜ Ω =diag(Ωx, Ωy/2, Ωz/2).

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Different surfaces depending on time Time-dependent Solutions

4.2 Paraboloid depending on time

Consider the generalization of the parabolic Case II from the Table I for which the amplitude η(r, t) and the phase ϕ(r, t) are as follows η(r, t) = cx(t)x + cy(t)(y2 − z2), ϕ(r, t) = r˜ Ω(t)r + a(t)yz,(4.6) where cx(t)cy(t) > 0, and ˜ Ω =diag(Ωx, Ωy/2, Ωz/2). We have ρ(t) =

• cx(t)cy(t), and linear and nonlinear potentials given by

v(r, t) = r ˜ A(t)r + by(t)yz − µ 2 c2

x(t),

g(r, t) = c2

x(t) + 4c2 y(t)(y2 + z2),

where by(t) is given by Eq. (4.5), and ˜ A =diag(Ax, Cy−c2

x/2, Cy−

c2

x/2) with Ax and Cσ being defined by Eq. (4.4) and Cσ = 2cσ(t)¨ cσ(t)−3˙ c2

σ(t)

8c2

σ(t)

− 2µc2

σ(t), respectively.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Different surfaces depending on time Time-dependent Solutions

4.3 Hyperboloid depending on time

Consider the generalization of the hyperbolic Case III from Table I for which the amplitude η(r, t) and phase ϕ(r, t) are of the forms η(r, t) = rˆ c(t)r, ϕ(r, t) = 1 2rˆ Ω(t)r + a(t)xyz, (4.7) where cx(t)cy(t)cz(t) > 0, and the condition Tr ˆ c(t) = 0 is re- quired, ˆ c =diag(cx, cy, cz).

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions Different surfaces depending on time Time-dependent Solutions

4.3 Hyperboloid depending on time

Consider the generalization of the hyperbolic Case III from Table I for which the amplitude η(r, t) and phase ϕ(r, t) are of the forms η(r, t) = rˆ c(t)r, ϕ(r, t) = 1 2rˆ Ω(t)r + a(t)xyz, (4.7) where cx(t)cy(t)cz(t) > 0, and the condition Tr ˆ c(t) = 0 is re- quired, ˆ c =diag(cx, cy, cz). Now we have ρ(t) = [cx(t)cy(t)cz(t)]1/4, and the nonlinearity g(r, t) and potential v(r, t) are given by g(r, t) = 4rˆ c(t)r, (4.8) v(r, t) = r ˆ C(t)r −

• a(t)Tr ˆ

Ω(t) + ˙ a(t)

• xyz

+1 2a2(t)(y2z2 + x2z2 + x2y2). (4.9)

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4.4 Third order surface depending on time

Consider the time-dependent generalization of the third order surface in Case IV from the Table I for which the amplitude η(r, t) and phase ϕ(r, t) are as follows η(r, t) = c(t)xyz, ϕ(r, t) = rˆ a(t)r, (4.10) where cx(t) > 0 and the condition ˙ c(t) + 2c(t)Tr ˆ a(t) = 0 is required, ˆ a =diag(ax, ay, az).

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4.4 Third order surface depending on time

Consider the time-dependent generalization of the third order surface in Case IV from the Table I for which the amplitude η(r, t) and phase ϕ(r, t) are as follows η(r, t) = c(t)xyz, ϕ(r, t) = rˆ a(t)r, (4.10) where cx(t) > 0 and the condition ˙ c(t) + 2c(t)Tr ˆ a(t) = 0 is required, ˆ a =diag(ax, ay, az). Now we have ρ(t) =

• c(t), the nonlinearity g(r, t) and potential

v(r, t) are as follows g(r, t) = c2(t)(y2z2 + x2z2 + x2y2), (4.11) v(r, t) = r ˆ D(t)r − µ 2 c2(t)(y2z2 + x2z2 + x2y2), (4.12) where ˆ D =diag(Dx, Dy, Dz)) with Dσ = −˙ aσ(t) − 2a2

σ(t).

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4.5 Time-dependent Solutions

In the following we display the profiles of some solutions of Eq. (1.3) with η(r, t) given in subsection 4.2 in which we choose the function cx(t) and cy(t) as the positive periodic functions cx(t) = dn(t, k1), cy(t) = sn(t, k2) + 2 (4.13) where k1, k2 ∈ [0, 1] are the moduli of the Jacobi elliptic functions.

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Cubic GP equation with attractive nonlinearity G3 = −1

The bright soliton of Eq. (1.3) can be written as ψbst(r, t) =

• −2µ ρ(t) sech[√−µ η(r, t)] eiϕ(r,t)

(4.14) where µ < 0. Here we only consider the case that the amplitude η(r, t) is given by Eq. (4.6).

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Cubic GP equation with attractive nonlinearity G3 = −1

The bright soliton of Eq. (1.3) can be written as ψbst(r, t) =

• −2µ ρ(t) sech[√−µ η(r, t)] eiϕ(r,t)

(4.14) where µ < 0. Here we only consider the case that the amplitude η(r, t) is given by Eq. (4.6).

Figure: The density distribution of the bright soliton solution |ψbst(r, t)|2 given by Eq. (4.14) with η given by Eq. (4.6) with µ = −1, k1 = 0.6, k2 = 0.8. (a) (x, t)-space with y = 1, z = 0, (b) (y, t)-space with x = 1, z = 0, (c) (z, t)-space with x = y = 1.

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The periodic cn-wave solution of Eq. (1.3) ψcnt(r, t) = ρ(t)

• 2µk2

1 − 2k2 cn

• 2µ

1 − 2k2 η(r, t), k

• eiϕ(r,t),

(4.15) where k ∈ (0, 1] and µ(1 − 2k2) > 0.

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The periodic cn-wave solution of Eq. (1.3) ψcnt(r, t) = ρ(t)

• 2µk2

1 − 2k2 cn

• 2µ

1 − 2k2 η(r, t), k

• eiϕ(r,t),

(4.15) where k ∈ (0, 1] and µ(1 − 2k2) > 0.

Figure: The density distribution of the periodic cn-soliton solution |ψcnt(r, t)|2 given by Eq. (4.15) with η given given by Eq. (4.6) with µ = −1, k = 0.9, cx(t) = dn(t, 0.6) and cy(t) = 2 + sn(t, 0.8). (a) (x, t)-space with y = 1, z = 0, (b) (y, t)-space with x = 1, z = 0, (c) (z, t)-space with x = y = 1.

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Cubic GP equation with repulsive nonlinearity G3 = 1

The dark soliton solution of Eq. (1.3) can be written as ψdst(r, t) = √µ ρ(t) tanh[

• µ/2 η(r, t)] eiϕ(r,t),

µ > 0, (4.16)

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Cubic GP equation with repulsive nonlinearity G3 = 1

The dark soliton solution of Eq. (1.3) can be written as ψdst(r, t) = √µ ρ(t) tanh[

• µ/2 η(r, t)] eiϕ(r,t),

µ > 0, (4.16)

Figure: The density distribution of the dark soliton solution |ψdst(r, t)|2 given by Eq. (4.16) with η given by Eq. (4.6) with µ = 2, cx(t) = dn(t, 0.6) and cy(t) = 2 + sn(t, 0.8). (a) (x, t)-space with y = 1, z = 0, (b) (y, t)-space with x = 1, z = 0, (c) (z, t)-space with x = y = 1.

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One can also construct the periodic sn-wave solutions of Eq. (1.3) ψsnt(r, t) = ρ(t)

• 2µk2

1 + k2 sn

• µ

1 + k2 η(r, t), k

• eiϕ(r,t), (4.17)

where µ > 0.

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One can also construct the periodic sn-wave solutions of Eq. (1.3) ψsnt(r, t) = ρ(t)

• 2µk2

1 + k2 sn

• µ

1 + k2 η(r, t), k

• eiϕ(r,t), (4.17)

where µ > 0.

Figure: The density distribution of the periodic sn-soliton solution |ψsnt(r, t)|2 given by Eq. (4.17) with η given by Eq. (4.6) with µ = 1, k = 0.9, cx(t) = dn(t, 0.6) and cy(t) = 2 + sn(t, 0.8). (a) (x, t)-space with y = 1, z = 0, (b) (y, t)-space with x = 1, z = 0, (c) (z, t)-space with x = y = 1.

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4.6 ρ(r, t) depending on both time t and space r

We can, in fact, consider the general case ρ(r, t) in Eq. (2.1) depending on both time t and space r, then based on Eq. (2.3), i.e., ∇ · (ρ2∇η) = 0, (ρ2)t + ∇ · (ρ2∇ϕ) = 0, ηt + ∇ϕ · ∇η = 0, 2gj(r, t)ρj−1 − Gj|∇η|2 = 0 (j = p, q), 2v(r, t) + µ|∇η|2 + |∇ϕ|2 − ρ−1∇2ρ + 2ϕt = 0. we can obtain the general amplitude η(r, t) and ρ(r, t) including an arbitrary differentiable function Γ(ζ) are listed in Table VI, and the corresponding phase ϕ(r, t) are same as ones given Sec. II, for which the corresponding general linear and nonlinear potentials, i.e. gj(r, t) and v(r, t), can be obtained from Eqs. (2.3d) and (2.3e).

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Table: VI. Admissible (r, t)-modulated amplitude and phase surfaces and ρ(r, t) (Γ(ζ), arbitrary differentiable function)

Case Amplitude surface Function ρ(r, t) I η = Γ(ζ), ζ = c(t) · r

• cx(t)cy(t)cz(t)

Γ′(ζ)

II η = Γ(ζ), ζ = cx(t)x + cy(t)(y2 − z2)

• cx(t)cy(t)/Γ′(ζ)

III η = Γ(ζ), ζ = cx(t)x2 +cy(t)y2 +cz(t)z2

4

• cx(t)cy(t)cz(t)

Γ′2(ζ)

IV η = Γ(ζ), ζ = cx(t)xyz

• cx(t)/Γ′(ζ)

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Table: VII. Admissible (r, t)-modulated amplitude and phase surfaces and ρ(r, t) (Γ(ζ), arbitrary differentiable function)

Case Amplitude surface Function ρ(r, t) I ϕ(r, t) = Γ(ζ), ζ = rˆ Ω(t)r+a(t)·r, (c(t) · a(t) = 0)

• cx(t)cy(t)cz(t)/Γ′(ζ)

II ϕ(r, t) = Γ(ζ), ζ = r˜ Ω(t)r + a(t)yz

• cx(t)cy(t)/Γ′(ζ)

III ϕ(r, t) = Γ(ζ), ζ =

1 2rˆ

Ω(t)r + a(t)xyz, (cz(t)=− cx(t)−cy(t))

4

• cx(t)cy(t)cz(t)/Γ′2(ζ)

IV ϕ(r, t) = Γ(ζ), ζ = rˆ a(t)r, (˙ c(t)= −2c(t)Tr ˆ a(t))

• cx(t)/Γ′(ζ)

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Conclusions

3D generalized GP equations with (time, space)-modulated coefficients mapped by the proper similarity transformation into 1D models allowing for exact solutions.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions

Conclusions

3D generalized GP equations with (time, space)-modulated coefficients mapped by the proper similarity transformation into 1D models allowing for exact solutions. We considered power surfaces, which give origin to parabolic and quartic linear and nonlinear potentials. Such potentials are typical for the physical applications in the nonlinear optics and in the mean-field theory of Bose-Einstein condensates, what determines the large range of the possible applications of the found solutions, as well as of the method itself.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions

Conclusions

3D generalized GP equations with (time, space)-modulated coefficients mapped by the proper similarity transformation into 1D models allowing for exact solutions. We considered power surfaces, which give origin to parabolic and quartic linear and nonlinear potentials. Such potentials are typical for the physical applications in the nonlinear optics and in the mean-field theory of Bose-Einstein condensates, what determines the large range of the possible applications of the found solutions, as well as of the method itself. The reported method can be also extended to the 3D (or N-dimensional) generalized GP equation (or coupled GP equations) with varying potentials, nonlinearities, dispersions and gain/loss terms.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions

Ref.: Z. Y. Yan and V. V. Konotop, Phys. Rev. E 80 (2009) 036607.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions

Ref.: Z. Y. Yan and V. V. Konotop, Phys. Rev. E 80 (2009) 036607.

Zhenya Yan 3D GP equation with varying coefficients

Introduction Similarity reductions and solutions Surfaces and stationary solutions Time-dependent surfaces and solutions Conclusions

Thanks for your attention !

Zhenya Yan 3D GP equation with varying coefficients