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:

$$\mathbf{pn}(u) = e^{\mathbf{x}}, \quad \mathbf{mn}(u) = e^{-\mathbf{x}}, \quad \mathbf{fn}(u) = a \ \mathbf{e}^{\mathbf{x}} + b \ \mathbf{e}^{-\mathbf{x}}, \tag{101}$$

$$\operatorname{sgn}(u) = a \begin{array}{c} e^x \ -b \ \ e^{-x} \end{array}, \quad \operatorname{rn}(u) = \sqrt{1 + m(a \ \ e^x - b \ \ e^{-x})^2}, \tag{102}$$

$$\text{nrf}(u) = \frac{1}{\text{fn}(u)}, \quad \text{ng}(u) = \frac{1}{\text{gn}(u)}, \quad \text{nr}(u) = \frac{1}{\text{rn}(u)}, \tag{103}$$

with u and x related as

The derivatives of the inverse functions are

Nonlinear Optics ‐ Novel Results in Theory and Applications

d dy sna�<sup>1</sup>

d dy cna�<sup>1</sup>

d dy dna�<sup>1</sup>

d dy osa�<sup>1</sup>

d dy oca�<sup>1</sup>

d dy oda�<sup>1</sup>

that Jacobi's functions invert.

Figure 6.

44

Plot of Ea ð Þ u for A ¼ 0:1, B ¼ 0:9, and α ¼ 1:2.

We also introduce the integral

ð Þ¼ y

ð Þ¼ y

ð Þ¼ y

ð Þ¼ y

ð Þ¼ y

ð Þ¼ y

Ea uð Þ¼

¼ n<sup>5</sup> u � α

¼ n<sup>4</sup> u þ α

shown in Figure 6, for a set of values of the parameters.

still introduce another set of nonlinear functions.

�<sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�<sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�<sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n<sup>5</sup> � y<sup>2</sup> ð Þ yð Þ <sup>2</sup> � n<sup>4</sup>

�<sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�<sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�<sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Then, as expected, we can see that these functions also invert the same integrals

dv dna<sup>2</sup>

dv sna2

dv cna<sup>2</sup>

ðu 0

> ðu 0

ðu 0

which resembles Jacobi's elliptic integral of the second kind. This function is

This is the minimum set of properties for these functions. Fortunately, we can

<sup>n</sup><sup>0</sup> � <sup>y</sup><sup>2</sup> ð Þ <sup>n</sup><sup>5</sup> � <sup>α</sup>y<sup>2</sup> ð Þ <sup>p</sup> , (92)

<sup>n</sup><sup>0</sup> � <sup>y</sup><sup>2</sup> ð Þ <sup>n</sup><sup>4</sup> <sup>þ</sup> <sup>α</sup>y<sup>2</sup> ð Þ <sup>p</sup> , (93)

<sup>p</sup> , (94)

<sup>n</sup><sup>0</sup> <sup>y</sup> ð Þ <sup>2</sup> � <sup>1</sup> <sup>n</sup><sup>5</sup> <sup>y</sup> ð Þ <sup>2</sup> � <sup>α</sup> <sup>p</sup> , (95)

<sup>n</sup><sup>0</sup> <sup>y</sup> ð Þ <sup>2</sup> � <sup>1</sup> <sup>n</sup><sup>4</sup> <sup>y</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>α</sup> <sup>p</sup> , (96)

<sup>n</sup><sup>5</sup> <sup>y</sup> ð Þ <sup>2</sup> � <sup>1</sup> <sup>1</sup> � <sup>n</sup><sup>4</sup> <sup>y</sup><sup>2</sup> ð Þ <sup>p</sup> : (97)

ð Þv (98)

ð Þv (99)

ð Þv , (100)

$$u = \int\_0^\infty \frac{dt}{\sqrt{1 + m(a \cdot e^t - b \cdot e^{-t})^2}},\tag{104}$$

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

Note that rn ð Þ u 6¼, rn ð Þ �u , and, then, mn uð Þ is not the mirror image of pn uð Þ, i.e., mn uð Þ 6¼ pnð Þ �u unless a ¼ 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 a and b are related to the mirror symmetry between the functions pn ð Þ u and mn ð Þ u , being b ¼ a the more symmetric case (which would be the case of Jacobi's elliptic functions with complex arguments). The value of m causes that these functions decay or increase more rapidly with respect to the regular exponential functions. The domain of these functions is finite unless m ¼ 0; in fact, increasing the magnitude of x beyond, for instance, ln 10<sup>4</sup>=2a ffiffiffiffi m � � p , does not increase the magnitude of u significantly. One can extend the domain of these functions by setting the value of the function to zero

Figure 7. Nonlinear exponential-like functions for m ¼ 1, a ¼ 0:1, and b ¼ 0:9.

or infinity for larger ∣u∣, making them nonperiodic functions on the real axes. We also note that some of these functions are actually bounded.

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

$$4ab = \text{fr}^2(u) - \text{gn}^2(u),\tag{105}$$

The second derivatives are

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

where

pn00ð Þ� <sup>u</sup> pnð Þ <sup>u</sup> <sup>c</sup><sup>3</sup> <sup>þ</sup> <sup>2</sup>ma<sup>2</sup> pn<sup>2</sup>

Three Solutions to the Nonlinear Schrödinger Equation for a Constant Potential

mn00ð Þ� <sup>u</sup> mnð Þ <sup>u</sup> <sup>c</sup><sup>3</sup> <sup>þ</sup> <sup>2</sup>mb<sup>2</sup> mn2

fn00ð Þ� <sup>u</sup> fnð Þ <sup>u</sup> <sup>c</sup><sup>1</sup> <sup>þ</sup> <sup>2</sup><sup>m</sup> fn<sup>2</sup>

gn00ð Þ� <sup>u</sup> gnð Þ <sup>u</sup> <sup>c</sup><sup>4</sup> <sup>þ</sup> <sup>2</sup><sup>m</sup> gn<sup>2</sup>

rn00ð Þþ <sup>u</sup> 2rnð Þ <sup>u</sup> <sup>c</sup><sup>3</sup> � rn<sup>2</sup>

nf00ð Þ� <sup>u</sup> nfð Þ <sup>u</sup> <sup>c</sup><sup>1</sup> � <sup>8</sup>abc2nf<sup>2</sup>

ng00ð Þ� <sup>u</sup> ngð Þ <sup>u</sup> <sup>c</sup><sup>4</sup> <sup>þ</sup> <sup>8</sup>ab ng<sup>2</sup>

Then, the functions that we have just introduced are solutions of nonlinear second-order differential equations with the one-dimensional Gross-Pitaevskii

<sup>c</sup><sup>3</sup> <sup>þ</sup> ma<sup>2</sup> pn<sup>2</sup>

<sup>c</sup><sup>3</sup> <sup>þ</sup> mb<sup>2</sup> mn2

<sup>c</sup><sup>4</sup> <sup>þ</sup> <sup>m</sup> gn2

<sup>c</sup><sup>3</sup> � rn<sup>2</sup>

<sup>c</sup><sup>1</sup> � <sup>4</sup>abc2nf <sup>2</sup>

<sup>c</sup><sup>4</sup> <sup>þ</sup> <sup>4</sup>ab ng2

<sup>2</sup> �2c<sup>3</sup> <sup>þ</sup> <sup>c</sup><sup>2</sup> nr<sup>2</sup>

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

Some particular cases are the following. When 4mab ¼ 1 or 2mab ¼ 1, we can write down explicit expressions of u in terms of trigonometric, hypergeometric, and

> ffiffiffi a b r e x � �

� tan �<sup>1</sup>

� � � � r

<sup>c</sup><sup>1</sup> <sup>þ</sup> <sup>m</sup> fn<sup>2</sup>

2

ð Þ <sup>u</sup> � � <sup>¼</sup> mb<sup>2</sup>

ð Þ <sup>u</sup> � � <sup>¼</sup> ma<sup>2</sup>

2 ð Þ� u c<sup>3</sup>

c<sup>1</sup> ¼ 1 � 8mab, c<sup>2</sup> ¼ 1 � 4mab, c<sup>3</sup> ¼ 1 � 2mab, (129)

00ð Þ� u 2nr uð Þ c<sup>2</sup> nr

nr

equation form, for a constant potential and real functions.

pn<sup>0</sup> ð Þ u

mn<sup>0</sup> ð Þ u

> fn<sup>0</sup> ð Þ u

> > gn<sup>0</sup> ð Þ u

nf<sup>0</sup> ð Þ u

ng<sup>0</sup> ð Þ u

nr<sup>0</sup> ð Þ u

exponential functions of x. When 4mab ¼ 1, we get

a e<sup>2</sup><sup>t</sup> <sup>þ</sup> b e�2<sup>t</sup> ð Þ <sup>¼</sup> 2 tan �<sup>1</sup>

ffiffiffiffiffiffiffiffi <sup>4</sup>ab <sup>p</sup> dx

and when 2mab ¼ 1, we obtain

u ¼ ðx 0

47

rn<sup>0</sup> ð Þ u

Additionally, the energy or Liapunov functions are given by

2

2

2

2

2

2

<sup>2</sup> <sup>þ</sup> 2rnð Þ <sup>u</sup>

<sup>2</sup> � pnð Þ <sup>u</sup>

<sup>2</sup> � mnð Þ <sup>u</sup>

<sup>2</sup> � fnð Þ <sup>u</sup>

<sup>2</sup> � gnð Þ <sup>u</sup>

<sup>2</sup> � nfð Þ <sup>u</sup>

<sup>2</sup> � ngð Þ <sup>u</sup>

<sup>2</sup> � nrð Þ <sup>u</sup>

functions pn ð Þ u and mn ð Þ u would have the same energy if b ¼ a.

ð Þ <sup>u</sup> � � <sup>¼</sup> <sup>0</sup>, (121)

ð Þ <sup>u</sup> � � <sup>¼</sup> <sup>0</sup>, (122)

ð Þ <sup>u</sup> � � <sup>¼</sup> <sup>0</sup>, (123)

ð Þ <sup>u</sup> � � <sup>¼</sup> <sup>0</sup>, (124)

ð Þ <sup>u</sup> � � <sup>¼</sup> <sup>0</sup>, (126)

ð Þ <sup>u</sup> � � <sup>¼</sup> <sup>0</sup>, (127)

� � <sup>¼</sup> <sup>0</sup>: (128)

c<sup>4</sup> ¼ 1 þ 4mab: (130)

ð Þ <sup>u</sup> � � ¼ �4abc2, (133)

ð Þ <sup>u</sup> � � <sup>¼</sup> <sup>4</sup>ab, (134)

ð Þ <sup>u</sup> � � <sup>¼</sup> m, (136)

ð Þ <sup>u</sup> � � <sup>¼</sup> m, (137)

ð Þ <sup>u</sup> � � <sup>¼</sup> <sup>1</sup>, (138)

ffiffiffi a b

, (139)

ð Þ <sup>u</sup> � � <sup>¼</sup> <sup>c</sup>2, (135)

, (131)

, (132)

ð Þ <sup>u</sup> � � <sup>¼</sup> <sup>0</sup>, (125)

$$\text{tr}\mathbf{n}^2(u) - \mathbf{1} = m \text{ } \text{gn}^2(u) = m\left[\text{fn}^2(u) - 4ab\right] \tag{106}$$

$$\mathbf{f}\mathbf{n}(u)\mathbf{g}\mathbf{n}(u) = a^2 \mathbf{p}\mathbf{n}^2(u) - b^2 \mathbf{m}\mathbf{n}^2(u),\tag{107}$$

$$\text{f}\mathbf{n}^2(\boldsymbol{u}) + \text{gn}^2(\boldsymbol{u}) = 2[\boldsymbol{b}^2 \cdot \text{mn}^2(\boldsymbol{u}) + \boldsymbol{a}^2 \cdot \text{pn}^2(\boldsymbol{u})],\tag{108}$$

whereas the derivatives of them are

$$\text{pm}'(u) = \text{pm}(u) \text{ } \text{rm}(u), \qquad \text{mm}'(u) = -\text{mn}(u) \text{ } \text{rm}(u), \tag{109}$$

$$\text{fn}'(u) = \text{gn}(u) \text{ rn}(u), \qquad \text{gn}'(u) = \text{fn}(u) \text{ rn}(u), \tag{110}$$

$$\text{rn}'(u) = m \text{ \(fn\)} \text{ \(gn\)}, \qquad \text{nf}'(u) = -\text{gn}(u) \text{ \(nf\)} \text{ \(n\)}, \tag{111}$$

$$\text{mg}'(u) = -\text{fn}(u) \text{ } \text{ng}^2(u) \text{ } \text{rn}(u), \qquad \text{nr}'(u) = -m \text{ } \text{fn}(u) \text{ } \text{gn}(u) \text{ } \text{nr}^2(u). \tag{112}$$

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 ¼ 0.

We also have that

$$\frac{d \text{ pm}^{-1}(y)}{dy} = \frac{1}{\sqrt{y^2 + m(a)y^2 - b})},\tag{113}$$

$$\frac{d \text{ mm}^{-1}(y)}{dy} = -\frac{1}{\sqrt{y^2 + m(a - b\_- y^2)^2}},\tag{114}$$

$$\frac{d \text{ fm}^{-1}(\text{y})}{d\text{y}} = \pm \frac{1}{\sqrt{(\text{y}^2 - 4ab)(\text{c}\_2 + m\_\text{ }\text{y}^2)}},\tag{115}$$

$$\frac{d \text{ gen}^{-1}(y)}{dy} = \frac{1}{\sqrt{(y^2 + 4ab)(1 + m\_\perp y^2)}},\tag{116}$$

$$\frac{d \text{ } \text{rn}^{-1}(y)}{dy} = \pm \frac{1}{\sqrt{(1 - y^2)(c\_2 - y^2)}},\tag{117}$$

$$\frac{d \text{ nf}^{-1}(\text{y})}{d\text{y}} = -\frac{1}{\sqrt{(1 - 4ab \text{ } \text{y}^2)[c\text{}y^2 + m]}},\tag{118}$$

$$\frac{d \text{ ng}^{-1}(\mathbf{y})}{d \mathbf{y}} = -\frac{1}{\sqrt{(1 + 4ab \text{ } \mathbf{y}^2)(\mathbf{y}^2 + m)}},\tag{119}$$

$$\frac{d \text{ } \text{nr}^{-1}(y)}{dy} = -\frac{1}{\sqrt{(1-y^2)(1-c\_2y^2)}}.\tag{120}$$

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

Three Solutions to the Nonlinear Schrödinger Equation for a Constant Potential DOI: http://dx.doi.org/10.5772/intechopen.80938

The second derivatives are

$$\left[\mathbf{p}\mathbf{n}''(u) - \mathbf{p}\mathbf{n}(u)\right]c\_3 + 2ma^2 \left[\mathbf{p}\mathbf{n}^2(u)\right] = \mathbf{0},\tag{121}$$

$$\text{mn}''(u) - \text{mn}(u) \left[ c\_3 + 2mb^2 \text{ } \text{mn}^2(u) \right] = \mathbf{0},\tag{122}$$

$$\left[\text{fn}''(u) - \text{fn}(u)\left[c\_1 + 2m\right.\text{fn}^2(u)\right] = \mathbf{0},\tag{123}$$

$$\left[\text{gn}''(u) - \text{gn}(u)\left[c\_4 + 2m\right.\text{gn}^2(u)\right] = 0,\tag{124}$$

$$\text{tr}\mathbf{n}''(u) + 2\text{rn}(u)\left[c\_3 - \text{rn}^2(u)\right] = \mathbf{0},\tag{125}$$

$$\mathbf{n} \mathbf{f}''(u) - \mathbf{n} \mathbf{f}(u) \left[ c\_1 - 8abc\_2 \mathbf{n} \mathbf{f}^2(u) \right] = \mathbf{0},\tag{126}$$

$$\left[\text{ng}''(u) - \text{ng}(u)\left[c\_4 + 8ab \cdot \text{ng}^2(u)\right] = 0,\tag{127}$$

$$\mathbf{n}r''(u) - \mathbf{2}n r(u) \begin{bmatrix} c\_2 & \mathbf{n}r^2(u) - c\_3 \end{bmatrix} = \mathbf{0}.\tag{128}$$

where

or infinity for larger ∣u∣, making them nonperiodic functions on the real axes. We

We can verify easily the following properties which are similar to those for the

pn<sup>2</sup>

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

dy <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dy ¼ � <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dy <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dy ¼ � <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dy ¼ � <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dy ¼ � <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

As expected, from these derivatives, we can see that these functions also invert

ð Þ¼ <sup>u</sup> <sup>2</sup> <sup>b</sup><sup>2</sup> mn<sup>2</sup>

ð Þ� <sup>u</sup> <sup>g</sup>n<sup>2</sup>

ð Þ¼ <sup>u</sup> <sup>m</sup> fn2

ð Þ� <sup>u</sup> <sup>b</sup><sup>2</sup>

ð Þ¼� <sup>u</sup> gnð Þ <sup>u</sup> nf <sup>2</sup>

<sup>y</sup><sup>2</sup> <sup>þ</sup> ma y ð Þ <sup>2</sup> � <sup>b</sup> <sup>2</sup>

<sup>y</sup><sup>2</sup> <sup>þ</sup> m a � b y<sup>2</sup> ð Þ<sup>2</sup>

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

> 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

mn<sup>2</sup>

ð Þþ <sup>u</sup> <sup>a</sup><sup>2</sup> pn2

ð Þ¼� <sup>u</sup> <sup>m</sup> fnð Þ <sup>u</sup> gnð Þ <sup>u</sup> nr<sup>2</sup>

<sup>q</sup> , (113)

<sup>q</sup> , (114)

<sup>y</sup>ð Þ <sup>2</sup> � <sup>4</sup>ab <sup>c</sup><sup>2</sup> <sup>þ</sup> m y<sup>2</sup> ð Þ <sup>p</sup> , (115)

<sup>y</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>4</sup>ab <sup>1</sup> <sup>þ</sup> m y<sup>2</sup> ð Þ <sup>p</sup> , (116)

<sup>1</sup> � <sup>y</sup><sup>2</sup> ð Þ <sup>c</sup><sup>2</sup> � <sup>y</sup><sup>2</sup> ð Þ <sup>p</sup> , (117)

<sup>1</sup> � <sup>4</sup>ab y<sup>2</sup> ð Þ <sup>c</sup>2<sup>y</sup> ½ � <sup>2</sup> <sup>þ</sup> <sup>m</sup> <sup>p</sup> , (118)

<sup>1</sup> <sup>þ</sup> <sup>4</sup>ab y<sup>2</sup> ð Þ <sup>y</sup>ð Þ <sup>2</sup> <sup>þ</sup> <sup>m</sup> <sup>p</sup> , (119)

<sup>1</sup> � <sup>y</sup><sup>2</sup> ð Þ <sup>1</sup> � <sup>c</sup>2y<sup>2</sup> ð Þ <sup>p</sup> : (120)

ð Þ <sup>u</sup> � �, (108)

ð Þ¼� u mnð Þ u rnð Þ u , (109)

ð Þ¼ u fnð Þ u rnð Þ u , (110)

ð Þ u , (105)

ð Þ� <sup>u</sup> <sup>4</sup>ab � � (106)

ð Þ u , (107)

ð Þ u rnð Þ u , (111)

ð Þ u : (112)

also note that some of these functions are actually bounded.

Nonlinear Optics ‐ Novel Results in Theory and Applications

elliptic functions. The square of these functions are related as

ð Þ� <sup>u</sup> <sup>1</sup> <sup>¼</sup> <sup>m</sup> gn<sup>2</sup>

fnð Þ <sup>u</sup> gnð Þ¼ <sup>u</sup> <sup>a</sup><sup>2</sup>

ð Þ¼ u gnð Þ u rnð Þ u , gn<sup>0</sup>

ð Þ u rnð Þ u , nr<sup>0</sup>

<sup>d</sup> pn�<sup>1</sup>ð Þ<sup>y</sup>

<sup>d</sup> mn�<sup>1</sup>ð Þ<sup>y</sup>

ð Þy dy ¼ �

<sup>d</sup> gn�<sup>1</sup>ð Þ<sup>y</sup>

<sup>d</sup> rn�<sup>1</sup>ð Þ<sup>y</sup>

ð Þy

dy ¼ �

d fn�<sup>1</sup>

d nf�<sup>1</sup>

<sup>d</sup> ng�<sup>1</sup>ð Þ<sup>y</sup>

<sup>d</sup> nr�<sup>1</sup>ð Þ<sup>y</sup>

the same integral functions that Jacobi was interested on [1, 4].

ð Þþ <sup>u</sup> gn<sup>2</sup>

ð Þ¼ u pnð Þ u rnð Þ u , mn<sup>0</sup>

ð Þ¼ u m fnð Þ u gnð Þ u , nf<sup>0</sup>

rn<sup>2</sup>

fn<sup>2</sup>

whereas the derivatives of them are

fn<sup>0</sup>

exponential functions for the case m ¼ 0.

pn<sup>0</sup>

ð Þ¼� <sup>u</sup> fnð Þ <sup>u</sup> ng2

We also have that

rn<sup>0</sup>

ng<sup>0</sup>

46

<sup>4</sup>ab <sup>¼</sup> <sup>f</sup>n<sup>2</sup>

$$c\_1 = 1 - 8mab, \quad c\_2 = 1 - 4mab, \quad c\_3 = 1 - 2mab,\tag{129}$$

$$x\_4 = 1 + 4mab.\tag{130}$$

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

$$\left[\mathbf{p}\mathbf{n}'(\boldsymbol{\mu})^2 - \mathbf{p}\mathbf{n}(\boldsymbol{\mu})^2\right]\mathbf{c}\_3 + m\boldsymbol{a}^2 \cdot \mathbf{p}\mathbf{n}^2(\boldsymbol{\mu})\,\mathrm{[}=\boldsymbol{m}\boldsymbol{b}^2,\tag{131}$$

$$\text{mm}'(u)^2 - \text{mn}(u)^2 \left[ c\_3 + mb^2 \text{ } \text{mn}^2(u) \right] = ma^2,\tag{132}$$

$$\left[\text{fn}'(u)^2 - \text{fn}(u)^2\left[c\_1 + m\right.\text{fn}^2(u)\right] = -4abc\_2\tag{133}$$

$$\left[\text{gn}'(u)^2 - \text{gn}(u)^2\Big|c\_4 + m\text{ }\text{gn}^2(u)\right] = 4ab,\tag{134}$$

$$\text{rm}'(u)^2 + 2\text{rn}(u)^2 \left[c\_3 - \text{rm}^2(u)\right] = c\_2. \tag{135}$$

$$\left[\text{nf}'(u)^2 - \text{nf}(u)^2\left[c\_1 - 4abc\_2\text{nf}^2(u)\right] = m,\tag{136}$$

$$\left[\text{ng}'(u)^2 - \text{ng}(u)^2\left[c\_4 + 4ab \cdot \text{ng}^2(u)\right] = m,\tag{137}$$

$$\left[\mathbf{n}\mathbf{r}'(u)^2 - \mathbf{n}\mathbf{r}(u)^2\right] \left[-2c\_3 + c\_2 \cdot \mathbf{n}\mathbf{r}^2(u)\right] = \mathbf{1},\tag{138}$$

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 ð Þ u and mn ð Þ u would have the same energy if b ¼ a.

Some particular cases are the following. When 4mab ¼ 1 or 2mab ¼ 1, we can write down explicit expressions of u in terms of trigonometric, hypergeometric, and exponential functions of x. When 4mab ¼ 1, we get

$$u = \int\_0^\mathbf{x} \frac{\sqrt{4ab} \, d\mathbf{x}}{(ae^{2t} + be^{-2t})} = 2 \left[ \tan^{-1} \left( \sqrt{\frac{a}{b}} e^\mathbf{x} \right) - \tan^{-1} \left( \sqrt{\frac{a}{b}} \right) \right],\tag{139}$$

and when 2mab ¼ 1, we obtain

$$\begin{split} u &= \sqrt{2ab} \int\_0^\infty \frac{dt}{\sqrt{a^2 e^{2t} + b^2 e^{-2t}}} \\ &= \frac{1}{\sqrt{2ab} \, ab} \\ &\times \left\{ e^{-2\kappa} \sqrt{b^2 e^{-2\kappa} + a^2 e^{2\kappa}} \left[ b^2 - e^{2\kappa} (b^2 e^{-2\kappa} + a^2 e^{2\kappa})\_2 F\_1 \left( \frac{3}{4}, 1; \frac{1}{4}; -\frac{a^2 e^{4\kappa}}{b^2} \right) \right] \right\} \\ &\quad - \sqrt{a^2 + b^2} \left[ b^2 - \left( a^2 + b^2 \right)\_2 F\_1 \left( \frac{3}{4}, 1; \frac{1}{4}; -\frac{a^2}{b^2} \right) \right] \end{split} \tag{140}$$

where <sup>2</sup>F<sup>1</sup> is the hypergeometric function.

When a ¼ b ¼ 1, the nonlinear functions reduce to Jacobi's elliptic functions with complex argument:

$$u = \int\_0^\infty \frac{1}{\sqrt{1 + 4m \text{ \sin \text{h}^2(t)}}} dt = -iF(i\infty|4m),\tag{141}$$

where v ¼ �2ak is the velocity, k is the soliton frequency, ω is the soliton wave number, θ is the phase constant, and 0 < m < 1 is the modulus of Jacobi's elliptic

Ψð Þ¼ x; t Acn½ � B xð Þ � vt ; l e

<sup>ω</sup> <sup>¼</sup> <sup>B</sup><sup>2</sup> <sup>2</sup>dA<sup>2</sup> � <sup>a</sup> � ak<sup>2</sup>

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

<sup>B</sup> <sup>¼</sup> <sup>b</sup> 4d <sup>1</sup>=<sup>2</sup>

Three Solutions to the Nonlinear Schrödinger Equation for a Constant Potential

<sup>i</sup><sup>ϕ</sup>, (146)

: (148)

, (147)

function.

sets of soliton waves.

Author details

49

Gabino Torres Vega

Physics Department, Cinvestav, México City, México

provided the original work is properly cited.

\*Address all correspondence to: gabino@fis.cinvestav.mx

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

A second solution was given as

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

where F is elliptic integral of the first kind.

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