Numerical approximation to solutions of linear and

Numericalapproximationto solutions of

linear andnonlinear

Schrodingerequations

EnriquePereiraBatista

Erwin SuazoMartınez

Introduction

Soliton-likesolutions

NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Numerical approximation to solutions of linearand nonlinear Schrodinger equations

Prairie Analysis Seminar 2015Kansas State University

Enrique Pereira BatistaErwin Suazo Martınez

University of Puerto Rico at Mayaguez

September 2015


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Contents

1 Introduction

2 Soliton-like solutions

3 Numerical MethodsImplementations

4 ResultsLinear caseNonlinear case

5 Future Work

6 References


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

The LSE

The linear Schrodinger equation (LSE) plays an important rolein quantum physics. It describes the evolution of the wave func-tion of a physical system with respect to the time (Fevens andJiang, 1999) (Griffiths, 2004). The LSE is utilized in many ap-plications, such as light propagation in lens-like media, trappingof small particles, molecular spectroscopy and quantum electro-dynamics (Gutierrez-Vega, 2007) (Guo, 2009) (Suslov, 2013a)(Suslov, 2015). Its standard form is given by

i∂ψ

∂t+ a

∂2ψ

∂x2+ Vψ = 0, (1)

where a is a parameter depending on the modeled system andV is the coefficient related to the potential energy.


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

The NLSE

The nonlinear Schrodinger equation (NLSE) has the classicalform:

i∂ψ

∂t+ a

∂2ψ

∂x2+ Vψ + s|ψ|2ψ = 0 (2)

and it describes modulated wave propagation. The NLSE arisesin the studying of nonlinear optical fibers, Bose-Einstein con-densation, superfluids, propagation of soliton waves and plasmaphysics (Bradley, 1995) (Bao, 2003) (Torres, 2004) (Agrawal,2007) (Suslov, 2012).When V 6= 0 the equation is also known as the Gross-Pitaevskiiequation (Zoller, 1997) (Carretero-Gonzalez, 2013).


linear andnonlinear




Introduction


NumericalMethods

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Results

Linear case

Nonlinear case

Future Work

References

References

Quadratic Hamiltonian Operator

Of great interest in quantum physics (Suslov, 2011a) (Suslov,2013a) (Suslov, 2013b) (Suslov, 2015) are the nonautonomousversions of the LSE and NLSE involving the quadratic Hamilto-nian operator H given, in terms of x and p = −i ∂∂x , as

H(x , p; t) = −a(t)p2+b(t)x2−ic(t)xp−id(t)−f (t)x−ig(t)p, (3)

with a, b, c , d , f and g are suitable real-valued functions.


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Nonautonomous LSE

The nonautonomous linear Schrodinger equation has the form

i∂ψ

∂t= H(x , p)ψ. (4)

Substituting (3) in (4) yields

iψt = −a(t)ψxx + b(t)x2ψ − ic(t)xψx (5)

−id(t)ψ − f (t)xψ + ig(t)ψx .


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Nonautonomous NLSE

The nonautonomous nonlinear Schrodinger equation has the form

i∂ψ

∂t= H(x , p)ψ + h(t)|ψ|2ψ. (6)

Substituting (3) in (6) yields


−id(t)ψ − f (t)xψ + ig(t)ψx + h(t)|ψ|2ψ.


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Riccati/Ermakov-type system

The following system of equations and its solution was providedin (Suslov, 2008) (Suslov, 2011a) (Suslov, 2011b) with c0 = 0, 1

dα

dt+ b + 2cα + 4aα2 = c0aβ

4 (8)

dβ

dt+ (c + 4aα)β = 0 (9)

dγ

dt+ aβ2 = 0 (10)


linear andnonlinear




Introduction


NumericalMethods

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Results

Linear case

Nonlinear case

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References

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Riccati/Ermakov-typesystem(cont.)

dδ

dt+ (c + 4aα)δ = f + 2gα + 2c0aβ

3ε (11)

dε

dt= (g − 2aδ)β (12)

dκ

dt= gδ − aδ2 + c0aβ

2ε2. (13)

If c0 = 0, the system (8)-(13) is called Riccati-type system.If c0 = 1, the system (8)-(13) is known as Ermakov-type system.


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Ansatz for the nonautonomousLSE

Lemma (Suslov (2011a))The substitution

ψ(x , t) =1√µ(t)

e i(α(t)x2+δ(t)x+κ(t))u(ξ, τ), (14)

where ξ = β(t)x + ε(t) and τ = γ(t), transforms thenonautonomous equation


−id(t)ψ − f (t)xψ + ig(t)ψx .

into an autonomous form

− iuτ = −uξξ + c0ξ2u (c0 = 0, 1) (16)

provided that the system (8)-(13) holds with α = 14aµ′

µ −d2a .


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Ansatz for the nonautonomousNLSE

Lemma (Suslov (2015))The nonlinear parabolic equation


−id(t)ψ − f (t)xψ + ig(t)ψx + h(t)|ψ|2ψ.

can be transformed to

− iuτ = −uξξ + c0ξ2u + h0|ψ|sψ (c0 = 0, 1) (18)

by the ansatz

ψ(x , t) =1√µ(t)

e i(α(t)x2+δ(t)x+κ(t))u(ξ, τ), (19)

where ξ = β(t)x + ε(t) and τ = γ(t), h = h0aβ2µs (h0

constant) provided that the system (8)-(13) holds with

α = 14aµ′

µ −d2a .


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Mixed norms

According to (Tao, 2006), let us define the mixed Lebesguenorms

Lqt Lrx (Rn × I , C)

for any interval I as the space of all functions u : Rn × I → Cwith norm

‖u‖Lqt Lrx (Rn×I ,C) :=

(∫I‖u(t)‖qLrx (Rn)dt

)1/q

=

(∫I

(∫Rn

|u(x , t)|rdx)q/r

dt

)1/q

.


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Uniqueness of solutions

For the case c0 = 0, the following result guarantees the unique-ness of the Cauchy problem involving the autonomous form ofthe LSE and NLSE.

Proposition (Tao (2006))

Let I be a time interval containing t0, and letu, u′ ∈ C 2

x ,t (Rn × I , C) be two classical solutions to−iuτ = −uξξ + c0ξ

2u + h0|ψ|sψ (c0 = 0, 1) with the same initialdatum u0 for some h0 and s. Assume that we have the milddecay hypothesis u, u′ ∈ L∞t Lqx (R× I ) for q = 2,∞. Thenu = u′.


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Description of finite differencemethods

In order to avoid conditionality in the stability for the linear case,an implicit scheme (Crank-Nicolson) in time is used in order toapproximate solutions for the autonomous form of the LSE.In the case of the autonomous form of the NLSE, an explicitscheme (Fourth order Runge-Kutta) is used for evolution in timein order to avoid the treatment of the nonlinear term with animplicit scheme. This makes the implementation conditionallystable and so, the number and size of time steps depend on thespace discretization. A detailed analysis of the stability of thismethod for NLSE is carried on (Carretero-Gonzalez, 2013).


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Split process

Following (Agrawal, 2007),for the Split-Step Fourier method theautonomous NLSE, with ξ = x and τ = t, is viewed as

∂ψ

∂t= (D + N)ψ, (20)

where the dispersive operator D and the nonlinear operator Nare given by:

D = −i ∂2

∂x2(21)

N = i(c0ξ

2 + h0|ψ|2). (22)


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Crank-Nicolson

After discretizing in time and space, the following matrix equa-tion is obtained(

I +ik

2h2∆− ik

2[ξ]n)

Ψn+1 =

(I− ik

2h2∆ +

ik

2[ξ]n)

Ψn,

where h and k are the space step-size and time step-size, respec-tively; Ψn is the discrete approximation of ψ at the n-th timestep, I is the identity matrix, ∆ is the discrete representation ofthe second derivative in space, and [ξ]n is the diagonal matrixthat represents the operator of external potential, with entries[ξ]nii = ξ(xi , tn).


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Fourth-order Runge-Kutta

According to (Carretero-Gonzalez, 2013), the Runge-Kutta schemefor NLSE is given by:

Ψn+1 = Ψn +1

6k(m1 + 2m2 + 2m3 + m4)

m1 = F (Ψn)

m2 = F (Ψn +1

2km1)

m3 = F (Ψn +1

2km2)

m4 = F (Ψn + km3)

where F (Ψn) =(−ih2 ∆ + i

h2 [ξ]n + ih0h2 [|Ψn|2]

)Ψn.

I is the identity matrix and [|Ψn|2] is a diagonal matrix where[|Ψn|2]i ,i = |Ψn

xi|2.


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Split-step Fourier method

The method assumes that, over a small time-step, the operatorsD and N act independently (Agrawal, 2007),

Ψn+1 = exp

(k

2D

)exp(τ N)exp

(k

2D

)Ψn. (23)

The linear step has a solution in the Fourier domain. So, Fouriertransform is used:

exp(kD)Ψ = F−1T exp[ikD(ω2)]FTΨ,

where FT is the Fourier transform. The term i D(ω2) is obtained

from ∂2

∂x2 by replacing ∂∂x by −iω. ω is the frequency in the

Fourier domain (wave number).Discrete Fast Fourier Transform is used to compute the Fouriertransform and inverse Fourier transform.


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

(Suslov, 2013b)

Some exact solutions to the following paraxial wave equation are givenin (Suslov, 2013b)

2iψt = ψxx + x2ψ. (24)

Exact solutions are expressed by

ψn(x , t) = e i(αx2+δx+κ)+i(2n+1)γ

√β

2nn!√πe−(βx+ε)2/2Hn(βx + ε),

whereα(t) = α(0)cos 2t+sin 2t(β4(0)+4α2(0)−1)/4

β4(0)sin t+(2α(0)sin t+cos t)2 ,

β(t) = α(0)cos 2t+sin 2t(β4(0)+4α2(0)−1)/4β4(0)sin t+(2α(0)sin t+cos t)2 ,

γ(t) = − 12arctan

β2(0)tan t1+2α(0)tan t ,

δ(t) = δ(0)(2α(0)sin t+cos t)+ε(0)β3(0)sin tβ4(0)sin2 t+(2α(0)sin t+cos t)2 ,

ε(t) = ε(0)(2α(0)sin t+cos t)−β(0)δ(0)sin t√β4(0)sin2t+(2α(0)sin t+cos t)2

,

κ(t) = sin2t ε(0)β2(0)(α(0)ε(0)−β(0)δ(0))−α(0)δ2(0)β4(0)sint2t+(2α(0)sin t+cos t)2 ,

with α(0) = γ(0) = ε(0) = κ(0) = 0, n = 0, δ(0) = 0, 1, β(0) =4/9.


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Numerical solution. Case δ(0) = 0

Numerical solution to the equation 2iψt = ψxx + x2ψ with the parameterδ(0) = 0.


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Numerical solution. Case δ(0) = 1

Numerical solution to the equation 2iψt = ψxx + x2ψ with the parameterδ(0) = 1.


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

(Rajendran, 2010)

The following equation was considered for the NLSE

iψt = −aψxx − |ψ|2ψ. (25)

The exact solution is given in (Rajendran, 2010) with

ψ(x , t) = A sech

[A√

2(x − Ωt)

]exp

[i

(Ω

2x +

2A2 − Ω2

4t

)],

corresponding to a bright soliton solution.If A =

√2 and Ω = 0, then it yields

ψ(x , t) =√

2 sech [x ] e it .


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Numerical solution. Case A =√

2,Ω = 0

Numerical solution to the equation iψt = −aψxx − |ψ|2ψ with the parametersA =

√2 and Ω = 0.


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Numerical solution. CaseA = −1.8, Ω = 4

Numerical solution to the equation iψt = −aψxx − |ψ|2ψ with the parametersA = −1.8 and Ω = 4.


linear andnonlinear




Introduction


NumericalMethods

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Results

Linear case

Nonlinear case

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(Escorcia, 2014)

For the equation

iψt = −aψxx + |ψ|2ψ, (26)

with solution given in (Escorcia, 2014) by

ψ(x , t) =1√2

[v − 2iA tanh A(x − Ω t)] exp

[− i

2

(v2 + 4A2

)t

],

corresponding to a dark soliton-type solution.If A = 1/2 and Ω = 0, this yields

− 1√2tanh

(1

2x

)exp

(− i

2t

).


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Numerical solution. Case A = 1/2,Ω = 0

Numerical solution to the equation iψt = −aψxx + |ψ|2ψ with the parametersA = 1/2 and Ω = 0.


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Ongoing work

• To explore more methods to get an optimal approximationto the nonautonomous NLSE.

• To validate the results obtained when considering moregeneral forms of the coefficients in the nonautonomousform of the LSE and NLSE (nonexplicit solutions andgeneralized functions).

• To extend the previous results for the LSE and the NLSEon two-dimensional domains, and compare them withexact solutions like...


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

T. Fevens and H. Jiang. Absorbing boundary conditions for theSchrodinger equation. SIAM Journal on Scientific Computing,21(1):255–282, July 1999.

D. J. Griffiths. Introduction to Quantum Mechanics. PrenticeHall, 2nd edition, 2004.

M.A. Bandres, J.C. Gutierrez-Vega. Airy-Gauss beams and theirtransformation by paraxial optical systems. Physiscal ReviewLetters, 15(25):16719–16728, December 2007.

D. Deng, Q. Guo. Airy complex variable function Gaussianbeams. New Journal of Physics, 11(10):103029+07, Octo-ber 2009.

C. Koutschan, E. Suazo, S. Suslov. Multi-parameter laser modesin paraxial optics. ACM Commun. Comput. Algebra, 49(1):2763–2766, March 2015.


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

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References

References

C.A. Sackett, J.J. Tollett, R.G. Hulet, C.C. Bradley. Evidenceof Bose-Einstein condensation in an atomic gas with attrac-tive interactions. Physiscal Review Letters, 75(9):1687–1690,August 1995.

D. Jaksch,P. A. Markowich, W. Bao. Numerical solution ofthe Gross-Pitaevskii equation for Bose-Einstein condensation.Journal of Computational Physics, 187(1):318–342, February2003.

G.D. Montesinos, V.M. Perez-Garcia, P.J. Torres. Stabiliza-tion of solitons of the multidimensional nonlinear Schrodingerequation: matter-wave breathers. Phisyca D: Nonlinear Phe-nomena, 191(3-4):193–210, May 2004.

Govind P. Agrawal. Nonlinear Fiber Optics. Academic Press,4th edition, 2007.

V.M. Perez-Garcia, H. Michinel, J.I. Cirac, M. Lewenstein, P.Zoller. Dynamics of Bose-Einstein condensates: Variationalsolutions of the Gross-Pitaevskii equaitions. Physiscal ReviewA, 56(2):1424–1432, August 1997.


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Introduction


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Results

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Nonlinear case

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References

References

R.M. Caplan, R. Carretero-Gonzalez. Numerical stability of ex-plicit Runge-Kutta finite-difference schemes for the nonlinearSchrodinger equation. Applied Numerical Mathematics, 13(71):20–40, April 2013.

N. Lanfear, R.M. Lopez, S.K. Suslov. Exact wave functionsfor generalized harmonic oscillators. Journal of Russian LaserResearch, 32(4):352–361, July 2011a.

R. M. Lopez, J.M. Vega-Guzman, S.K. Suslov. Reconstructingthe Schrodinger groups. Physica Scripta, 87(3):038112+06,February 2013a.

A. Mahalov, E. Suazo, S. Suslov. Spiral laser beams in inho-mogeneous media. Optics Letters, 38(15):2763–2766, August2013b.

E.Suazo, S.K. Suslov. Soliton-like solutions for the nonlin-ear Schrodinger equation with variable quadratic Hamiltoni-ans. Journal of Russian Laser Research, 33(1):63–83, January2012.


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

R. Cordero-Soto, R.M. Lopez, E. Suazo, S.K. Suslov. Propagatorof a charged particle with a spin in uniform magnetic andperpendicular electric fields. Letters in Mathematical Physics,84(2-3):159–178, June 2008.

S.K. Suslov. On integrability of nonautonomous nonlinearSchrodinger equations. Proc. Amer. Math Soc., 140:3067–3082, December 2011b.

T. Tao. Nonlinear Dispersive Equations: local and Global Anal-ysis. American Mathematical Soc., 2006.

P. Muruganandam, M. Lakshmanan, S. Rajendran. Bright anddark solitons in a quasi 1D Bose-Einstein condensates mod-elled by 1D Gross-Pitaevskii equation with time-dependent pa-rameters. Phisyca D: Nonlinear Phenomena, 239(7):366–386,April 2010.

J.M. Escorcia. Soluciones para la ecuacion de Schrodinger nolineal con coeficientes variables: Existencia de solitones ysu dinamica. Master’s thesis, University of Puerto Rico atMayaguez, 2014.


linear andnonlinear




Introduction


NumericalMethods

Implementations

Results

Linear case

Nonlinear case

Future Work

References

References

Thank you!

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Numerical approximation to solutions of linear and