Open access peer-reviewed chapter

Three Solutions to the Nonlinear Schrödinger Equation for a Constant Potential

By Gabino Torres Vega

Submitted: April 25th 2018Reviewed: August 16th 2018Published: November 5th 2018

DOI: 10.5772/intechopen.80938

Downloaded: 268

Abstract

We introduce three sets of solutions to the nonlinear Schrödinger equation for the free particle case. A well-known solution is written in terms of Jacobi elliptic functions, which are the nonlinear versions of the trigonometric functions sin, cos, tan, cot, sec, and csc. The nonlinear versions of the other related functions like the real and complex exponential functions and the linear combinations of them is the subject of this chapter. We also illustrate the use of these functions in Quantum Mechanics as well as in nonlinear optics.

Keywords

  • new nonlinear exponential-like functions
  • superpositions of nonlinear functions
  • nonlinear optics
  • nonlinear quantum mechanics

1. Introduction

Since the nonlinear Schrödinger equation appears in many fields of physics, including nonlinear optics, thus, there is interest in finding its solutions, in particular, its eigenfunctions. A set of eigenfunctions, for the free particle, is given in terms of Jacobi’s elliptic functions [1, 2, 3, 4], which are real periodic functions, and they have been used in order to find the eigenstates of the particle in a box [5, 6] and in a double square well [7].

Jacobi’s elliptic functions are needed in subjects like the description of pulse narrowing nonlinear transmission lines [8].

Interestingly, there is a way to linearly superpose Jacobi’s elliptic functions by means of adding constant terms to their arguments [3]. So, we ask ourselves if there are other ways to achieve nonlinear superposition of nonlinear functions.

Besides, the linear equation has complex solutions with a current density flux different from zero, and we expect that the nonlinear equation should also have this type of solutions at least for small nonlinear interaction.

In this chapter, we introduce three other sets of functions which are also solutions to the Gross-Pitaevskii equation; they all are nonlinear superpositions of functions. The modification of the elliptic functions allows us to consider the nonlinear equivalent of the linear superposition of exponential, real and complex, and trigonometric functions found in nonrelativistic linear quantum mechanics.

The functions we are about to introduce can be used, for instance, in the case of a free Bose-Einstein condensate reflected by a potential barrier. One might be able to further analyze nonlinear tunneling [7] and nonlinear optics phenomena with the help of these functions.

2. Nonlinear complex exponential functions

The definitions of the functions and their properties are similar to those used in Jacobi’s elliptic functions [1, 2, 4]. Let us start with the definition of our complex exponential nonlinear functions:

cncuα=aeix+beix,sncuα=aeixbeix,E1
dncuα=1αcncu2,nccuα=1cncuα,E2
nscuα=1sncuα,ndcuα=1dncuα,E3
tacuα=sncuαcncuα,cocuα=cncuαsncuα.E4

where α,a,bR, and they are such that α<1/maxa±b2. With these choices, the function dnc is always positive, and we do not have to worry about branch points in the relation between the variables xand u. The variables uand xare related as

u=0xdt1αcnctα2.E5

A plot of these functions is found in Figure 1 for a particular set of values of the parameters. These functions behave like the usual superposition of complex exponential functions (α=0), changing behavior as the value of αincreases until it reaches the soliton value, α=1/maxa±b2. The functions become concentrated around the origin for the soliton value of α.

Figure 1.

Nonlinear complex exponential functions with a = 0.1 , b = 0.9 , and α = 0.9 . The curves correspond to 1, c nc u α 2 ; 2, s nc u α 2 ; and 3, d nc u α .

The quarter period of these functions is defined as

Kc=0π/2dt1αcnctα2.E6

If we call

n0=a2+b2,n1=12αa2+b2,E7
n2=132αa2+b2,n3=1αa2+b2,E8
n4=1α2a2+b2,n5=1+α2a2+b2,E9
n6=1+αa2+b2,n7=1+32αa2+b2,E10

the squares of the nonlinear functions are written as

cnc2uαsnc2uα=4ab,E11
cncuα2+snc(uα)2=2n0,E12
dnc2uα=1αcnc(uα)2E13
=n1+αsncuα2,E14
tac2uα=14abncc2uα,E15
coc2uα=1+4abnsc2uα.E16

Some derivatives of these functions are

cnc'uα=isncuαdncuα,E17
snc'uα=icncuαdncuα,E18
dnc'uα=αcncuαsncuα,E19
ncc'uα=itacuαnccuαdncuα,E20
nsc'uα=icocuαnscuαdncuα,E21
ndc'uα=αndc2uαcncuαsncuα,E22
tac'uα=i1+tac2uαdncuαE23
coc'uα=i4abnsc2uαdncuα,E24

where indicates to take the imaginary part of the quantity.

We also have that the derivative of the inverse functions is given by

ddycnc1y=±iy24ab1αy2,E25
ddysnc1y=±i4ab+y2n1+αy2,E26
ddyncc1y=±iy14aby21α/y2,E27
ddynsc1y=±iy1+4aby2n1+α/y2.E28

Now, the second derivatives are as follows

cnc''uα=[2αcnc(uα)2n6cncuα2αabcncuα,E29
snc''uα=3αa2+b212αsnc(uα)2sncuα2αabsncuα,E30
dnc''uα=2dnc2uααn0dncuα,E31
ncc''uα=n6nccuα2αcncuα+2αabncc2uαcncuα,E32
nsc''uα=n3+8abn1nsc2uαnscuαα1+10abnsc2uαsncuα,E33
ndc''uα=2α2ndc3uα{cncuαsnc(uα)}2+2αa2+b2ndcuα,E34
tac''uα=1+tac2uαα2tacuα+i{tac(uα)}cncuα22tacuα,E35
coc''uα=21coc2uαcocuαdnc2uα2αabcncuαcncuαsncuαsncuαcocuα.E36

The first three of the above equations can be thought of as modifications of the Gross-Pitaevskii equation, which allows for solutions of the form cnc uα, snc uα, and dnc uα. However, when aor bvanishes, we get the Gross-Pitaevskii form.

With these results at hand, we can see that the probability current densities associated with cnc uαand snc uαare given by

jcu=Recncuαidducncuα=a2b2dncuα,E37
jsu=Resncuαiddusncuα=a2b2dncuα,E38

respectively. The nonlinear term causes that the quantum flux be no longer constant (as is the case for linear interaction) but modulated by dnc uαinstead.

The differential equations for cnc uαand snc uαwould have the Gross-Pitaevskii equation form if any of α, a, or bbecomes zero or when a=b(which is the case of real functions, i.e., Jacobi’s functions). The case of α, a, or bzero corresponds to the cases when there is no nonlinear interaction or when there is total reflection or only transmission in a quantum system.

2.1 The potential step

A straight forward application of the functions introduced in this section is the finding of the eigenfunctions of the Gross-Pitaevskii equation for a step potential:

Vu=0,whenu<0,V0,whenu0,E39

and a chemical potential μlarger than the potential height V0. The Gross-Pitaevskii equation is written as

d2ψudu2+2ML2ћ2μV0ψu2ML2ћ2A2NU0ψu2ψu=0,E40

where ψuis the unnormalized eigenfunction for the Bose-Einstein condensate (BEC), Mis the mass of a single atom, Nis the number of atoms in the condensate, U0=4πћ2a/Mcharacterizes the atom-atom interaction, ais the scattering length, Lis a scaling length, Ais the integral of the magnitude squared of the wave function, uis a dimensionless length, μis the chemical potential, and V0is an external constant potential.

For u<0(we call it the region I, V0=0), we use the cnc function with a=1, i.e.,

ψIuα=cnckIuαI,E41

with parameters

kI2=2ML2μћ21+αIa2+b2,E42
αI=ML2NU0ћ2A2kI2E43

From these equations, we obtain

αI=NU02μA2NU0a2+b2,E44

and

μ=ћ2kI22ML2+NU02A2a2+b2.E45

This last result for μis in agreement with the conjecture formulated by D’Agosta et al. in Ref. [9], with the last term being the self-energy of the condensate, which is independent of kI.

For u>0, we use the nonlinear plane wave (a=T, b=0)

ψIIu=cnckIIuαII,E46

with

1+αIIT2=2ML2ћ2kII2μV0,2ML2NU0ћ2A2kII2=2αII,E47

i.e.,

μ=V0+ћ2kII22ML2+NU02A2T2,kII2=2ML2μV0ћ21+αIIT2.E48

By combining the expressions for the αs in both regions, we find that

αIkI2=αIIkII2,E49

and since the chemical potential should be the same on both regions, we also get

V0=ћ2kI2kII22ML2+NU02A2a2+b2T2.E50

The equal flux condition results in

kIa2b2=kIIT2.E51

Now, equating the functions and their derivatives at u=0, we find two relations for the parameters:

a+b=T,E52
abkI1αIa+b2=TkII1αIIT2,E53

i.e.,

kIIkI=aba+b1αIa+b21αIa+b2kI2/kII2.E54

We show these values in Figure 2 . We observe a behavior similar to the linear system; when μV0(kIIkI), which means very high energies, the step is just a small perturbation on the evolution of the wave.

Figure 2.

A three-dimensional plot of the values of k II / k I for the potential step. Dimensionless units.

3. Nonlinear superposition of trigonometric functions

A second set of nonlinear functions is the nonlinear version of the superposition of trigonometric functions, which is the subject of this section. We only mention some results; more details are found in Ref. [10].

Let us consider the change of variable from θto udefined by the Jacobian

dnau=du=1+α2a2b2cos2θ+αabsin2θ,E55

where α.a,bR, and α<4ab/a2+b22, a plot of 4ab/a2+b22, is shown in Figure 3 . Thus, the relationship between θand uis

u=0θ1+α2a2b2cos2θ+αabsin2θ.E56

Figure 3.

Three-dimensional plot of 4 ∣ ab ∣ / a 2 + b 2 2 .

We also define the nonlinear functions

snauasinθbcosθ,E57
cnauacosθ+bsinθ,E58
osau1snau,ocau1cnau,odau1dnau,E59
csaucnausnau,scausnaucnau,dsaudnausnau,E60
dcaudnaucnau,sdausnaudnau,cdaucnaudnau.E61

A plot of these functions can be found in Figure 4 , for a set of values of α,a,b.

Figure 4.

Plots of the nonlinear functions for a = 0.1 , b = 0.9 , and α = 1.2 . Note that the functions cna and sna have different shapes, and, thus, they are not just the other function shifted by some amount.

The algebraic relationships between the above functions are

a2+b2=sna2u+cna2u,E62
dna2u=1α2sna2ucna2uE63
=n4+αcna2uE64
=n5αsna2u,E65
sda2u=n0oda2ucda2u,E66
1oda2u=α2cda2usda2u,E67
1+αsda2u=n5oda2u,E68
1αcda2u=n4oda2u,E69
sca2u=n0oca2u1,E70
dca2u=n4oca2u+α,E71
csa2u=n0osa2u1.E72

The derivatives of these functions are

sna'u=cnaudnau,E73
cna'u=snaudnau,E74
dna'u=αsnaucnau,E75
osa'u=cnaudnauosa2u,E76
oca'u=snaudnauoca2u,E77
oda'u=αcnausnauoda2u.E78

Another property is the eliminant equation, also known as energy or Liapunov function,

sna'u2+n7sna2uαsna4u=n0n5,E79
[cna'u]2+n2cna2u+αcna4u=n0n4,E80
[dna'u]22dna2u+dna4u=n4n5,E81
[osa'u]2+n7osa2un0n5osa4u=α,E82
[oca'u]2+n2oca2un0n4oca4u=α,E83
[oda'u]22oda2u+n4n5oda4u=1.E84

Second derivatives of the functions lead to the differential equations similar to the Gross-Pitaevskii nonlinear differential equation. For sna, cna, and dna, we have that

sna''u+n7snau2αsna3u=0,E85
cna''u+n2cnau+2αcna3u=0,E86
dna''u+2dnaudna2u1=0,E87
osa''u+n7osau2n0n5osa3u=0,E88
oca''u+n2ocau2n0n4oca3u=0,E89
oda''u2odau+2n4n5oda3u=0.E90

Quarter period of these functions is defined as

Kaαab=0π/2dt1+αa2b2cos2t/2+αabsin2t.E91

A plot of Ka αabcan be found in Figure 5 for α=1.2.

Figure 5.

Some of the values of nonlinear quarter period Ka α a b , for α = 1.2 .

The derivatives of the inverse functions are

ddysna1y=±1n0y2n5αy2,E92
ddycna1y=±1n0y2n4+αy2,E93
ddydna1y=±1n5y2y2n4,E94
ddyosa1y=±1n0y21n5y2α,E95
ddyoca1y=±1n0y21n4y2+α,E96
ddyoda1y=±1n5y211n4y2.E97

Then, as expected, we can see that these functions also invert the same integrals that Jacobi’s functions invert.

We also introduce the integral

Eau=0udvdna2vE98
=n5uα0udvsna2vE99
=n4u+α0udvcna2v,E100

which resembles Jacobi’s elliptic integral of the second kind. This function is shown in Figure 6 , for a set of values of the parameters.

Figure 6.

Plot of Ea u for A = 0.1 , B = 0.9 , and α = 1.2 .

This is the minimum set of properties for these functions. Fortunately, we can still introduce another set of nonlinear functions.

4. Nonlinear exponential-like functions

It is possible to define still another set of nonlinear functions inspired on Jacobi’s elliptic functions [11]. Let us consider the following set of nonlinear functions of exponential type:

pnu=ex,mnu=ex,fnu=aex+bex,E101
gnu=aexbex,rnu=1+maexbex2,E102
nfu=1fnu,ngu=1gnu,nru=1rnu,E103

with uand xrelated as

u=0xdt1+maetbet2,E104

where a,bRand m>0. The required values of a,b,mcauses that the radical is positive and then there is no need to consider branching points.

Note that rn u, rn u, and, then, mnuis not the mirror image of pnu, i.e., mnupnuunless a=b. A plot of these functions is found in Figure 7 for a set of values of the parameters a, b, and m. The values of aand bare related to the mirror symmetry between the functions pn uand mn u, being b=athe more symmetric case (which would be the case of Jacobi’s elliptic functions with complex arguments). The value of mcauses that these functions decay or increase more rapidly with respect to the regular exponential functions. The domain of these functions is finite unless m=0; in fact, increasing the magnitude of xbeyond, for instance, ln104/2am, does not increase the magnitude of usignificantly. One can extend the domain of these functions by setting the value of the function to zero or infinity for larger u, making them nonperiodic functions on the real axes. We also note that some of these functions are actually bounded.

Figure 7.

Nonlinear exponential-like functions for m = 1 , a = 0.1 , and b = 0.9 .

We can verify easily the following properties which are similar to those for the elliptic functions. The square of these functions are related as

4ab=fn2ugn2u,E105
rn2u1=mgn2u=mfn2u4abE106
fnugnu=a2pn2ub2mn2u,E107
fn2u+gn2u=2b2mn2u+a2pn2u,E108

whereas the derivatives of them are

pn'u=pnurnu,mn'u=mnurnu,E109
fn'u=gnurnu,gn'u=fnurnu,E110
rn'u=mfnugnu,nf'u=gnunf2urnu,E111
ng'u=fnung2urnu,nr'u=mfnugnunr2u.E112

As we can see from these derivatives, the rate of increase or decrease of the functions is modulated by the rn function; it would be the same as that for the usual exponential functions for the case m=0.

We also have that

dpn1ydy=1y2+may2b2,E113
dmn1ydy=1y2+maby22,E114
dfn1ydy=±1y24abc2+my2,E115
dgn1ydy=1y2+4ab1+my2,E116
drn1ydy=±11y2c2y2,E117
dnf1ydy=114aby2c2y2+m,E118
dng1ydy=11+4aby2y2+m,E119
dnr1ydy=11y21c2y2.E120

As expected, from these derivatives, we can see that these functions also invert the same integral functions that Jacobi was interested on [1, 4].

The second derivatives are

pn''upnuc3+2ma2pn2u=0,E121
mn''umnuc3+2mb2mn2u=0,E122
fn''ufnuc1+2mfn2u=0,E123
gn''ugnuc4+2mgn2u=0,E124
rn''u+2rnuc3rn2u=0,E125
nf''unfuc18abc2nf2u=0,E126
ng''unguc4+8abng2u=0,E127
nr''u2nruc2nr2uc3=0.E128

where

c1=18mab,c2=14mab,c3=12mab,E129
c4=1+4mab.E130

Then, the functions that we have just introduced are solutions of nonlinear second-order differential equations with the one-dimensional Gross-Pitaevskii equation form, for a constant potential and real functions.

Additionally, the energy or Liapunov functions are given by

pn'u2pnu2c3+ma2pn2u=mb2,E131
mn'u2mnu2c3+mb2mn2u=ma2,E132
fn'u2fnu2c1+mfn2u=4abc2,E133
gn'u2gnu2c4+mgn2u=4ab,E134
rn'u2+2rnu2c3rn2u=c2,E135
nf'u2nfu2c14abc2nf2u=m,E136
ng'u2ngu2c4+4abng2u=m,E137
nr'u2nru22c3+c2nr2u=1,E138

where we have made use of the relationships between the squares of the functions. Note that, the functions nf and ng have the same energy, whereas that the functions pn uand mn uwould have the same energy if b=a.

Some particular cases are the following. When 4mab=1or 2mab=1, we can write down explicit expressions of uin terms of trigonometric, hypergeometric, and exponential functions of x. When 4mab=1, we get

u=0x4abdxae2t+be2t=2tan1abextan1ab,E139

and when 2mab=1, we obtain

u=2ab0xdta2e2t+b2e2t
=12abab
×e2xb2e2x+a2e2xb2e2xb2e2x+a2e2x2F134114a2e4xb2
a2+b2b2a2+b22F134114a2b2,E140

where 2F1is the hypergeometric function.

When a=b=1, the nonlinear functions reduce to Jacobi’s elliptic functions with complex argument:

u=0x11+4msinh2tdt=iFix4m,E141

where Fis elliptic integral of the first kind.

This is the minimum set of properties of the exponential-type nonlinear functions.

5. Remarks

Thus, we were able to obtain three sets of nonlinear functions which are solutions to the Gross-Pitaevskii equation. With these functions, we have the nonlinear versions of the trigonometric, real, and complex exponential functions and their linear combinations, and a complete set of functions as in the linear counterpart.

Due to the method of solution, which makes use of elliptic functions, these functions will expand the set of solutions that can be given to polynomial nonlinear equations, in general [8, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].

For instance, a well-known optical phenomenon is the nonlinear dispersion in parabolic law medium with Kerr law nonlinearity [24]. This system is described by a nonlinear Schrödinger equation:

iΨt+aΨxx+bΨ2Ψ+cΨ4Ψ+dΨ2xxΨ=0,E142

where a subindex indicates a derivative with respect to that index. The second term of the above equation represents the group velocity dispersion, the third and fourth terms are the parabolic law nonlinearity, and the last term is the nonlinear dispersion. Some solutions of Eq. (142) were found in Ref. [24]. A solution is the traveling wave, with Jacobi’s sn function profile, given by

Ψxt=AsnBxvtme,E143
B=bA2am1+m2dm2+m+2A21/2,E144
ω=B22dA2a1+m.E145

where v=2akis the velocity, kis the soliton frequency, ωis the soliton wave number, θis the phase constant, and 0<m<1is the modulus of Jacobi’s elliptic function.

A second solution was given as

Ψxt=AcnBxvtle,E146
B=b4d1/2,E147
ω=B22dA2aak2.E148

Since the functions that we have introduced in these chapters comply with differential and algebraic equations similar to the ones for Jacobi’s elliptic functions, we can give additional solutions in terms of these new functions, giving rise to new sets of soliton waves.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Gabino Torres Vega (November 5th 2018). Three Solutions to the Nonlinear Schrödinger Equation for a Constant Potential, Nonlinear Optics - Novel Results in Theory and Applications, Boris I. Lembrikov, IntechOpen, DOI: 10.5772/intechopen.80938. Available from:

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