## 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:

where

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 (

The quarter period of these functions is defined as

If we call

the squares of the nonlinear functions are written as

Some derivatives of these functions are

where

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

Now, the second derivatives are as follows

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

With these results at hand, we can see that the probability current densities associated with cnc

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

The differential equations for cnc

### 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:

and a chemical potential

where

For

with parameters

From these equations, we obtain

and

This last result for

For

with

i.e.,

By combining the expressions for the

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

The equal flux condition results in

Now, equating the functions and their derivatives at

i.e.,

We show these values in
Figure 2
. We observe a behavior similar to the linear system; when

## 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

where

We also define the nonlinear functions

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

The algebraic relationships between the above functions are

The derivatives of these functions are

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

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

Quarter period of these functions is defined as

A plot of Ka

The derivatives of the inverse functions are

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

We also introduce the integral

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.

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:

with

where

Note that rn

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

whereas the derivatives of them are

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

We also have that

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

where

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

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

Some particular cases are the following. When

and when

where

When

where

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:

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

where

A second solution was given as

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.

## References

- 1.
Cayley A. An Elementary Treatise on Elliptic Functions. 2nd ed. N.Y: Dover Publications, Inc; 1961 - 2.
Abramowitz M, Stegun I. Handbook of Mathematical Functions. New York: Dover; 1964 - 3.
Khare A, Sukhatme U. Linear superposition in nonlinear equations. Physical Review Letters. 2002; 88 :244101 - 4.
Markushevich AI. Introduction to the Classical Theory of Abelian Functions, Series Translations of Mathematical Monographs. Vol. 96. Providence, Rhode Island: American Mathematical Society; 1992 - 5.
Carr LD, Clark CW, Reinhardt WP. Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity. Physical Review A. 2000; 62 :063610 - 6.
Carr LD, Clark C,W, Reinhardt WP. Stationary solutions of the one-dimensional nonlinear Schrödinger equation. II. Case of attractive nonlinearity. Physical Review A. 2000; 62 :063611 - 7.
Mahmud KW, Kutz JN, Reinhardt WP. Bose-Einstein condensates in a one-dimensional double square well: Analytical solutions of the nonlinear Schrödinger equation. Physical Review A. 2002; 66 :063607 - 8.
Zayed EME, Alurrfi KAE. A new Jacobi elliptic function expansion method for solving a nonlinear PDE describing pulse narrowing nonlinear transmission lines. Journal of Differential Equations. 2015; 28 :128 - 9.
D’Agosta R, Malomed BA, Presilla C. Stationary solutions of the gross-Pitaevskii equation with linear counterpart. Physics Letters A. 2000; 275 :424-434 - 10.
Torres-Vega G. Some real, periodic stationary solutions of the one-dimensional nonlinear Schrödinger equation for constant potentials. Journal of Physics: Conference Series. 2017; 792 :012054 - 11.
Torres-Vega G. Decaying steady states of the one-dimensional nonlinear Schrödinger equation with a constant potential. Journal of Physics: Conference Series. 2017; 792 :012031 - 12.
Ma H-C, Zhang Z-P, Deng A-P. A new periodic solution to Jacobi elliptic functions of MKdV equation and BBM equation. Acta Mathematicae Applicatae Sinica (English Series). 2012; 28 :409. DOI: 10.1007/s10255-012-0153-7 - 13.
Zayed EME, Alurrfi KAE. A new Jacobi elliptic function expansion method for solving a nonlinear PDE describing the nonlinear low-pass electrical lines. Chaos, Solitons and Fractals. 2015; 78 :148. DOI: 10.1016/j.chaos.2015.07.018 - 14.
Zayed EME, Al-Nowehy A-G. New extended auxiliary equation method for finding many new Jacobi elliptic function solutions of three nonlinear Schrödinger equations. Waves in Random and Complex Media. 2017; 27 :420. DOI: 10.1080/17455030.2016.1259690 - 15.
Sonmezoglu A. Exact solutions for some fractional differential equations. Advances in Mathematical Physics. 2015; 2015 :567842 - 16.
Zhou Q, Zhu Q, Savescu M, Bhrawy A, Biswas A. Optical solitons with nonlinear dispersion in parabolic law medium. Proceedings of the Romanian Academy, Series A. 2015; 16 :152 - 17.
Feng Q, Meng F. Explicit solutions for space-time fractional partial differential equations in mathematical physics by a new generalized fractional Jacobi elliptic equation-based sub-equation method. Optik. 2016; 127 :7450 - 18.
Feng Q. Jacobi elliptic function solutions for fractional partial differential equations. International Journal of Applied Mathematics. 2016; 46 :121 - 19.
Kudryashov NA, Sinelshchikov DI. On the connection of the quadratic Lienard equation with an equation for the elliptic functions. Regular and Chaotic Dynamics. 2015; 20 :486. DOI: 10.1134/S1560354715040073 - 20.
Zhang Z-Y. Jacobi elliptic function expansion method for the modified Korteweg-de Vries-Zakharov-Kuznetsov and the Hirota equations. Romanian Journal of Physics. 2015; 60 :1384 - 21.
Aslan EC, Tchier F, Inc M. On optical solitons of the Schrödinger-Hirota equation with power law nonlinearity in optical fibers. Superlattices and Microstructures. 2017; 105 :48. DOI: 10.1016/j.spmi.2017.03.014 - 22.
Aslan EC, Inc M, Baleanu D. Optical solitons and stability analysis of the NLSE with anti-cubic nonlinearity. Superlattices and Microstructures. 2017; 109 :784. DOI: 10.1016/j.spmi.2017.06.003 - 23.
Tasbozan O, Çenesiz Y, Kurt A. New solutions for conformable fractional Boussinesq and combined KdV-mKdV equations using Jacobi elliptic function expansion method. European Physical Journal Plus. 2016; 131 :244. DOI: 10.1140/epjp/i2016-16244-x - 24.
Tchier F, Aslan EC, Inc M. Optical solitons in parabolic law medium: Jacobi elliptic function solution. Nonlinear Dynamics. 2016; 85 :2577. DOI: 10.1007/s11071-016-2846-6 - 25.
Triki H, Wazwaz A-M. New solitons and periodic wave solutions for the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation. Journal of Electromagnetic Waves and Applications. 2016; 30 :788. DOI: 10.1080/09205071.2016.1153986