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 [1–4], 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.

it reaches the soliton value, <sup>α</sup> <sup>¼</sup> <sup>1</sup>=max ð Þ <sup>a</sup> � <sup>b</sup> <sup>2</sup> h i. The functions become concen-

<sup>2</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � �, n<sup>5</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>α</sup>

<sup>¼</sup> <sup>n</sup><sup>1</sup> <sup>þ</sup> <sup>α</sup>j j sncð Þ <sup>u</sup>; <sup>α</sup> <sup>2</sup>

ð Þ¼ <sup>u</sup>; <sup>α</sup> <sup>1</sup> � <sup>4</sup>ab ncc2

ð Þ¼ <sup>u</sup>; <sup>α</sup> <sup>1</sup> <sup>þ</sup> <sup>4</sup>abnsc2

dt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>α</sup>j j cncð Þ <sup>t</sup>; <sup>α</sup> <sup>2</sup>

<sup>q</sup> : (6)

<sup>2</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � �, (9)

<sup>α</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � �, (10)

ð Þ¼ u; α 4ab, (11)

, (14)

ð Þ u; α , (15)

ð Þ u; α : (16)

, n<sup>1</sup> <sup>¼</sup> <sup>1</sup> � <sup>2</sup><sup>α</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � �, (7)

<sup>α</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � �, n<sup>3</sup> <sup>¼</sup> <sup>1</sup> � <sup>α</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � �, (8)

3 2

j j cncð Þ <sup>u</sup>; <sup>α</sup> <sup>2</sup> <sup>þ</sup> j j sncð Þ <sup>u</sup>; <sup>α</sup> <sup>2</sup> <sup>¼</sup> <sup>2</sup>n0, (12)

ð Þ¼ <sup>u</sup>; <sup>α</sup> <sup>1</sup> � <sup>α</sup>j j cncð Þ <sup>u</sup>; <sup>α</sup> <sup>2</sup> (13)

ð Þ¼ u; α i sncð Þ u; α dncð Þ u; α , (17)

ð Þ¼ u; α i cncð Þ u; α dncð Þ u; α , (18)

ð Þ <sup>u</sup>; <sup>α</sup> <sup>ℑ</sup> cnc<sup>∗</sup> f g ð Þ <sup>u</sup>; <sup>α</sup> sncð Þ <sup>u</sup>; <sup>α</sup> , (22)

ð Þ <sup>u</sup>; <sup>α</sup> � �dncð Þ <sup>u</sup>; <sup>α</sup> (23)

2 <sup>r</sup> � � , (25)

ð Þ u; α dncð Þ u; α , (24)

ð Þ¼ <sup>u</sup>; <sup>α</sup> <sup>α</sup><sup>ℑ</sup> cnc<sup>∗</sup> f g ð Þ <sup>u</sup>; <sup>α</sup> sncð Þ <sup>u</sup>; <sup>α</sup> , (19)

ð Þ¼� u; α itacð Þ u; α nccð Þ u; α dncð Þ u; α , (20)

ð Þ¼� u; α icocð Þ u; α nscð Þ u; α dncð Þ u; α , (21)

i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi yð Þ <sup>2</sup> � 4ab 1 � αj j y

trated around the origin for the soliton value of α. The quarter period of these functions is defined as

DOI: http://dx.doi.org/10.5772/intechopen.80938

<sup>n</sup><sup>0</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup>

2

the squares of the nonlinear functions are written as

cnc<sup>2</sup>

dnc<sup>2</sup>

tac2

coc2

cnc<sup>0</sup>

snc<sup>0</sup>

ð Þ¼� <sup>u</sup>; <sup>α</sup> <sup>α</sup> ndc<sup>2</sup>

ð Þ¼ <sup>u</sup>; <sup>α</sup> <sup>i</sup> <sup>1</sup> <sup>þ</sup> tac2

ð Þ¼� <sup>u</sup>; <sup>α</sup> <sup>i</sup>4abnsc<sup>2</sup>

where ℑ indicates to take the imaginary part of the quantity. We also have that the derivative of the inverse functions is given by

ð Þ¼� y

tac<sup>0</sup>

coc<sup>0</sup>

d dy cnc�<sup>1</sup>

dnc<sup>0</sup>

ncc<sup>0</sup>

nsc<sup>0</sup>

ndc<sup>0</sup>

37

Some derivatives of these functions are

<sup>n</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> � <sup>3</sup>

<sup>n</sup><sup>4</sup> <sup>¼</sup> <sup>1</sup> � <sup>α</sup>

If we call

Kc ¼

ð<sup>π</sup>=<sup>2</sup> 0

Three Solutions to the Nonlinear Schrödinger Equation for a Constant Potential

<sup>n</sup><sup>6</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>α</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � �, n<sup>7</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup>

ð Þ� <sup>u</sup>; <sup>α</sup> snc2
