3. Nonlinear superposition of trigonometric functions

with

i.e.,

<sup>1</sup> <sup>þ</sup> <sup>α</sup>IIT<sup>2</sup> <sup>¼</sup> <sup>2</sup>ML<sup>2</sup>

Nonlinear Optics ‐ Novel Results in Theory and Applications

ћ2 k2 II <sup>2</sup>ML<sup>2</sup> <sup>þ</sup>

<sup>V</sup><sup>0</sup> <sup>¼</sup> <sup>ћ</sup><sup>2</sup> <sup>k</sup><sup>2</sup>

μ ¼ V<sup>0</sup> þ

The equal flux condition results in

ð Þ a � b kI

kII kI

small perturbation on the evolution of the wave.

q

<sup>¼</sup> ð Þ <sup>a</sup> � <sup>b</sup> ð Þ a þ b

A three-dimensional plot of the values of kII=kI for the potential step. Dimensionless units.

for the parameters:

i.e.,

Figure 2.

40

ћ2 k2 II

> NU<sup>0</sup> <sup>2</sup>A<sup>2</sup> <sup>T</sup><sup>2</sup>

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

αIk<sup>2</sup>

<sup>I</sup> � <sup>k</sup><sup>2</sup> II � � <sup>2</sup>ML<sup>2</sup> <sup>þ</sup>

ð Þ <sup>μ</sup> � <sup>V</sup><sup>0</sup> , <sup>2</sup>ML<sup>2</sup>

, k<sup>2</sup>

<sup>I</sup> <sup>¼</sup> <sup>α</sup>IIk<sup>2</sup>

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

NU<sup>0</sup>

kI <sup>a</sup><sup>2</sup> � <sup>b</sup><sup>2</sup> � � <sup>¼</sup> kIIT<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>α</sup>Ið Þ <sup>a</sup> <sup>þ</sup> <sup>b</sup> <sup>2</sup>

s

We show these values in Figure 2. We observe a behavior similar to the linear system; when μ≫V<sup>0</sup> (kII ! kI), which means very high energies, the step is just a

Now, equating the functions and their derivatives at u ¼ 0, we find two relations

¼ TkII

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>α</sup>Ið Þ <sup>a</sup> <sup>þ</sup> <sup>b</sup> <sup>2</sup> <sup>1</sup> � <sup>α</sup>Ið Þ <sup>a</sup> <sup>þ</sup> <sup>b</sup> <sup>2</sup>

q

k2 <sup>I</sup> <sup>=</sup>k<sup>2</sup> II

NU<sup>0</sup>

ð Þ μ � V<sup>0</sup>

II, (49)

<sup>2</sup>A<sup>2</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � <sup>T</sup><sup>2</sup> � �: (50)

a þ b ¼ T, (52)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>α</sup>IIT<sup>2</sup>

: (51)

, (53)

: (54)

<sup>ћ</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>α</sup>IIT<sup>2</sup> � � : (48)

¼ 2αII, (47)

ћ2 A2 k2 II

II <sup>¼</sup> <sup>2</sup>ML<sup>2</sup>

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 θ to u defined by the Jacobian

$$\text{dna}(u) = \frac{d\theta}{du} = \sqrt{1 + \frac{a}{2} \left(a^2 - b^2\right) \cos\left(2\theta\right) + aab\sin\left(2\theta\right)},\tag{55}$$

where <sup>α</sup>:a, b∈R, and <sup>∣</sup>α<sup>∣</sup> < 4∣ab∣<sup>=</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � �<sup>2</sup> , a plot of 4∣ab∣<sup>=</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � �<sup>2</sup> , is shown in Figure 3. Thus, the relationship between θ and u is

$$u = \int\_0^{\theta} \frac{d\theta}{\sqrt{1 + \frac{a}{2} \left(a^2 - b^2\right) \cos\left(2\theta\right) + aab \sin\left(2\theta\right)}}.\tag{56}$$

We also define the nonlinear functions

$$\sin(u) \coloneqq a\sin\left(\theta\right) - b\cos\left(\theta\right),\tag{57}$$

$$\mathsf{cma}(u) \coloneqq a\,\cos\left(\theta\right) + b\,\sin\left(\theta\right),\tag{58}$$

$$\cos \mathbf{a}(u) \coloneqq \frac{1}{\mathrm{sna}(u)}, \quad \mathrm{oca}(u) \coloneqq \frac{1}{\mathrm{cna}(u)}, \quad \mathrm{oda}(u) \coloneqq \frac{1}{\mathrm{dna}(u)},\tag{59}$$

$$\csc(u) \coloneqq \frac{\text{cna}(u)}{\text{sna}(u)}, \quad \text{sca}(u) \coloneqq \frac{\text{sna}(u)}{\text{cna}(u)}, \quad \text{dsa}(u) \coloneqq \frac{\text{dna}(u)}{\text{sna}(u)}, \tag{60}$$

$$\mathsf{cda}(u) \coloneqq \frac{\mathsf{dna}(u)}{\mathsf{cna}(u)}, \quad \mathsf{sda}(u) \coloneqq \frac{\mathsf{sna}(u)}{\mathsf{dna}(u)}, \quad \mathsf{cda}(u) \coloneqq \frac{\mathsf{cna}(u)}{\mathsf{dna}(u)}.\tag{61}$$

A plot of these functions can be found in Figure 4, for a set of values of α, a, b. The algebraic relationships between the above functions are

.

Figure 3. Three-dimensional plot of <sup>4</sup>∣ab∣<sup>=</sup> <sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> � �<sup>2</sup>

$$a^2 + b^2 = \text{sn}\mathbf{a}^2(u) + \text{cn}\mathbf{a}^2(u),\tag{62}$$

$$\text{d}\text{na}^2(u) = \text{1} - \frac{a}{2} \left(\text{sna}^2(u) - \text{cna}^2(u)\right) \tag{63}$$

$$= n\_4 + a \text{ cna}^2(u) \tag{64}$$

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

ð Þþ <sup>u</sup> dna<sup>4</sup>

ð Þþ <sup>u</sup> <sup>n</sup>4n<sup>5</sup> oda<sup>4</sup>

dt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> <sup>α</sup> <sup>a</sup><sup>2</sup> � <sup>b</sup><sup>2</sup> � � cos 2ð Þ<sup>t</sup> <sup>=</sup><sup>2</sup> <sup>þ</sup> <sup>α</sup>ab sin 2ð Þ<sup>t</sup>

<sup>q</sup> : (91)

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

sna00ð Þþ <sup>u</sup> <sup>n</sup><sup>7</sup> snað Þ� <sup>u</sup> <sup>2</sup><sup>α</sup> sna<sup>3</sup>

cna00ð Þþ <sup>u</sup> <sup>n</sup><sup>2</sup> cnað Þþ <sup>u</sup> <sup>2</sup><sup>α</sup> cna<sup>3</sup>

osa00ð Þþ <sup>u</sup> <sup>n</sup><sup>7</sup> osað Þ� <sup>u</sup> <sup>2</sup>n0n<sup>5</sup> osa<sup>3</sup>

oca00ð Þþ <sup>u</sup> <sup>n</sup><sup>2</sup> ocað Þ� <sup>u</sup> <sup>2</sup>n0n<sup>4</sup> oca3

oda00ð Þ� <sup>u</sup> 2 odað Þþ <sup>u</sup> <sup>2</sup>n4n<sup>5</sup> oda<sup>3</sup>

dna00ð Þþ <sup>u</sup> 2 dnað Þ <sup>u</sup> dna<sup>2</sup>

ð Þ� <sup>u</sup> <sup>α</sup> sna4ð Þ¼ <sup>u</sup> <sup>n</sup>0n5, (79)

ð Þþ <sup>u</sup> <sup>α</sup> cna4ð Þ¼ <sup>u</sup> <sup>n</sup>0n4, (80)

ð Þ� <sup>u</sup> <sup>n</sup>0n5osa<sup>4</sup>ð Þ¼ <sup>u</sup> <sup>α</sup>, (82)

ð Þ� <sup>u</sup> <sup>n</sup>0n<sup>4</sup> oca4ð Þ¼� <sup>u</sup> <sup>α</sup>, (83)

ð Þ¼� u n4n5, (81)

ð Þ¼� u 1: (84)

ð Þ¼ u 0, (85)

ð Þ¼ u 0, (86)

ð Þ¼ u 0, (88)

ð Þ¼ u 0, (89)

ð Þ¼ u 0: (90)

ð Þ� <sup>u</sup> <sup>1</sup> � � <sup>¼</sup> <sup>0</sup>, (87)

sna<sup>0</sup> ½ � ð Þ u

½cna<sup>0</sup>

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

½dna<sup>0</sup>

½osa<sup>0</sup>

½oca<sup>0</sup>

½oda<sup>0</sup>

Quarter period of these functions is defined as

Some of the values of nonlinear quarter period Ka ð Þ α; a; b , for α ¼ 1:2.

A plot of Ka ð Þ α; a; b can be found in Figure 5 for α ¼ 1:2.

ð<sup>π</sup>=<sup>2</sup> 0

Kað Þ¼ α; a; b

Figure 5.

43

<sup>2</sup> <sup>þ</sup> <sup>n</sup><sup>7</sup> sna<sup>2</sup>

Three Solutions to the Nonlinear Schrödinger Equation for a Constant Potential

ð Þ� <sup>u</sup> <sup>2</sup> <sup>þ</sup> <sup>n</sup><sup>2</sup> cna<sup>2</sup>

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

ð Þ� <sup>u</sup> <sup>2</sup> <sup>þ</sup> <sup>n</sup><sup>7</sup> osa2

ð Þ� <sup>u</sup> <sup>2</sup> <sup>þ</sup> <sup>n</sup><sup>2</sup> oca<sup>2</sup>

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

function,

$$
\mathfrak{h} = n\_{\mathfrak{F}} - a \text{ } \mathfrak{sn}^2(u),
\tag{65}
$$

$$\operatorname{sda}^2(u) = n\_0 \operatorname{oda}^2(u) - \operatorname{cda}^2(u),\tag{66}$$

$$\mathbf{1} - \text{oda}^2(u) = \frac{a}{2} \left[ \text{cda}^2(u) - \text{sda}^2(u) \right],\tag{67}$$

$$\mathbf{1} + a \operatorname{sda}^2(\boldsymbol{u}) = n\_5 \operatorname{oda}^2(\boldsymbol{u}),\tag{68}$$

$$\mathbf{1} - a \operatorname{cda}^2(\boldsymbol{\mu}) = n\_4 \operatorname{cda}^2(\boldsymbol{\mu}),\tag{69}$$

$$\operatorname{sca}^2(u) = n\_0 \operatorname{oca}^2(u) - \mathbf{1},\tag{70}$$

$$\mathsf{dca}^2(u) = n\_4 \mathsf{oca}^2(u) + a,\tag{71}$$

$$\csc^2(u) = n\_0 \cos^2(u) - \mathbf{1}.\tag{72}$$

The derivatives of these functions are

$$\mathsf{sna}'(u) = \mathsf{cna}(u) \,\,\mathsf{dna}(u),\tag{73}$$

$$\mathsf{cna}'(u) = -\mathsf{sna}(u) \cdot \mathsf{dna}(u),\tag{74}$$

$$\mathsf{dna}'(u) = -a\mathsf{sna}(u)\mathsf{cna}(u),\tag{75}$$

$$\cos^{\prime}(u) = -\text{cn}\mathbf{a}(u) \text{ dna}(u) \text{os}\mathbf{a}^{\prime}(u),\tag{76}$$

$$\mathsf{acc}'(u) = \mathsf{sna}(u) \cdot \mathsf{dna}(u) \mathsf{oca}^2(u),\tag{77}$$

$$\mathsf{mod}'(u) = a \,\, \mathsf{cna}(u) \,\, \mathsf{sna}(u) \mathsf{od} \,\, ^2(u) \,. \tag{78}$$

## Figure 4.

Plots of the nonlinear functions for a ¼ 0:1, b ¼ 0:9, and α ¼ 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.

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

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

$$\left[\text{sn}\!\!\!\!\!\left(u\right)\right]^2 + n\_7 \text{sn}\!\!\!\left(u\right) - a \text{ sn}\!\!\!\left(u\right) = n\_0 n\_5 \tag{79}$$

$$[\text{cna}'(u)]^2 + n\_2 \text{cna}^2(u) + a \text{ cna}^4(u) = n\_0 n\_4,\tag{80}$$

$$[\text{dna}'(u)]^2 - 2\text{dna}^2(u) + \text{dna}^4(u) = -n\_4 n\_5 \tag{81}$$

$$[\text{osa}'(u)]^2 + n\_{7}\text{osa}^2(u) - n\_{0}n\_{5}\text{osa}^4(u) = a,\tag{82}$$

$$[\mathsf{occ}'(u)]^2 + n\_2 \mathsf{occ}^2(u) - n\_0 n\_4 \mathsf{occ}^4(u) = -a,\tag{83}$$

$$
\frac{1}{2} \left[ \text{oda}'(u) \right]^2 - 2 \text{oda}^2(u) + n\_4 n\_5 \text{oda}^4(u) = -1. \tag{84}
$$

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

$$\text{sna}''(u) + \text{ $n\_7$ sna}(u) - 2a \text{ } \text{sna}^3(u) = \text{0},\tag{85}$$

$$\mathsf{cna}''(u) + n\_2 \mathsf{cna}(u) + 2a \,\, \mathsf{cna}^3(u) = \mathbf{0},\tag{86}$$

$$\mathsf{Inda}''(\mathsf{u}) + \mathsf{Z}\ \mathsf{d}\mathsf{na}(\mathsf{u}) \Big[\mathsf{d}\mathsf{na}^2(\mathsf{u}) - \mathsf{1}\Big] = \mathsf{0},\tag{87}$$

$$
\cos^{\prime\prime}(u) + n\_{7}\cos(u) - 2n\_{0}n\_{5}\cos^{3}(u) = 0,\tag{88}
$$

$$\mathbf{occ}''(u) + n\_2 \mathbf{occ}(u) - 2n\_0 n\_4 \mathbf{occ}^3(u) = \mathbf{0},\tag{89}$$

$$\operatorname{oda}''(u) - 2 \operatorname{ oda}(u) + 2n\_4 n\_5 \operatorname{oda}^3(u) = \mathbf{0}.\tag{90}$$

Quarter period of these functions is defined as

$$\mathbf{K}a(a,a,b) = \int\_0^{\pi/2} \frac{dt}{\sqrt{1 + a(a^2 - b^2)\cos(2t)/2 + aab\sin(2t)}}.\tag{91}$$

A plot of Ka ð Þ α; a; b can be found in Figure 5 for α ¼ 1:2.

Figure 5. Some of the values of nonlinear quarter period Ka ð Þ α; a; b , for α ¼ 1:2.

<sup>a</sup><sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> <sup>¼</sup> sna<sup>2</sup>

ð Þ¼ <sup>u</sup> <sup>1</sup> � <sup>α</sup>

dna<sup>2</sup>

Nonlinear Optics ‐ Novel Results in Theory and Applications

sda<sup>2</sup>

<sup>1</sup> � oda<sup>2</sup>

ð Þþ <sup>u</sup> cna2

ð Þ� <sup>u</sup> cna<sup>2</sup>

ð Þ� <sup>u</sup> cda2

ð Þ� <sup>u</sup> sda2

<sup>2</sup> sna2

<sup>¼</sup> <sup>n</sup><sup>4</sup> <sup>þ</sup> <sup>α</sup> cna<sup>2</sup>

<sup>¼</sup> <sup>n</sup><sup>5</sup> � <sup>α</sup> sna<sup>2</sup>

<sup>2</sup> cda<sup>2</sup>

ð Þ¼ <sup>u</sup> <sup>n</sup><sup>0</sup> oca2

ð Þ¼ <sup>u</sup> <sup>n</sup><sup>4</sup> oca<sup>2</sup>

ð Þ¼ <sup>u</sup> <sup>n</sup><sup>0</sup> osa2

ð Þ¼� <sup>u</sup> cnað Þ <sup>u</sup> dnað Þ <sup>u</sup> osa2

ð Þ¼ <sup>u</sup> snað Þ <sup>u</sup> dnað Þ <sup>u</sup> oca<sup>2</sup>

ð Þ¼ <sup>u</sup> <sup>α</sup> cnað Þ <sup>u</sup> snað Þ <sup>u</sup> oda<sup>2</sup>

Plots of the nonlinear functions for a ¼ 0:1, b ¼ 0:9, and α ¼ 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.

ð Þ¼ <sup>u</sup> <sup>n</sup><sup>5</sup> oda<sup>2</sup>

ð Þ¼ <sup>u</sup> <sup>n</sup><sup>4</sup> oda2

ð Þ¼ <sup>u</sup> <sup>n</sup><sup>0</sup> oda<sup>2</sup>

ð Þ¼ <sup>u</sup> <sup>α</sup>

<sup>1</sup> <sup>þ</sup> <sup>α</sup>sda2

<sup>1</sup> � <sup>α</sup>cda<sup>2</sup>

sca<sup>2</sup>

dca<sup>2</sup>

csa<sup>2</sup>

sna<sup>0</sup>

cna<sup>0</sup>

dna<sup>0</sup>

osa<sup>0</sup>

oda<sup>0</sup>

Figure 4.

42

oca<sup>0</sup>

The derivatives of these functions are

ð Þ u , (62)

ð Þ u (64)

ð Þ u , (65)

ð Þ u , (66)

ð Þ u , (68)

ð Þ u , (69)

ð Þ� u 1, (70)

ð Þþ u α, (71)

ð Þ� u 1: (72)

ð Þ u , (76)

ð Þ u , (77)

ð Þ u : (78)

ð Þ¼ u cnað Þ u dnað Þ u , (73)

ð Þ¼� u snað Þ u dnað Þ u , (74)

ð Þ¼� u αsnað Þ u cnað Þ u , (75)

ð Þ <sup>u</sup> (63)

ð Þ <sup>u</sup> , (67)

The derivatives of the inverse functions are

$$\frac{d}{dy}\text{sna}^{-1}(y) = \frac{\pm 1}{\sqrt{(n\_0 - y^2)(n\_5 - ay^2)}},\tag{92}$$

4. Nonlinear exponential-like functions

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

pnð Þ¼ u e

nfð Þ¼ u

with u and x related as

instance, ln 10<sup>4</sup>=2a ffiffiffiffi

Figure 7.

45

gnð Þ¼ <sup>u</sup> a e<sup>x</sup> � b e�x, rnð Þ¼ <sup>u</sup>

1

u ¼ ðx 0

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

positive and then there is no need to consider branching points.

x, mnð Þ¼ <sup>u</sup> <sup>e</sup>

Three Solutions to the Nonlinear Schrödinger Equation for a Constant Potential

fnð Þ <sup>u</sup> , ngð Þ¼ <sup>u</sup>

exponential type:

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

q

1

dt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> ma et � b e�<sup>t</sup> ð Þ<sup>2</sup>

where a, b∈R and m>0. The required values of a, b, m causes that the radical is

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

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

�x, fnð Þ¼ <sup>u</sup> a e<sup>x</sup> <sup>þ</sup> b e�x, (101)

1

, (102)

rnð Þ <sup>u</sup> , (103)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> ma ex � b e�<sup>x</sup> ð Þ<sup>2</sup>

<sup>q</sup> , (104)

gnð Þ <sup>u</sup> , nrð Þ¼ <sup>u</sup>

$$\frac{d}{dy}\text{cna}^{-1}(y) = \frac{\pm 1}{\sqrt{(n\_0 - y^2)(n\_4 + ay^2)}},\tag{93}$$

$$\frac{d}{dy}\text{dna}^{-1}(y) = \frac{\pm 1}{\sqrt{(n\_5 - y^2)(y^2 - n\_4)}},\tag{94}$$

$$\frac{d}{dy}\text{osa}^{-1}(y) = \frac{\pm 1}{\sqrt{(n\_0 y^2 - 1)(n\_5 y^2 - a)}},\tag{95}$$

$$\frac{d}{dy}\text{oca}^{-1}(y) = \frac{\pm 1}{\sqrt{(n\_0 y^2 - 1)(n\_4 y^2 + a)}},\tag{96}$$

$$\frac{d}{dy}\text{oda}^{-1}(y) = \frac{\pm 1}{\sqrt{(n\_5 y^2 - 1)(1 - n\_4 y^2)}}.\tag{97}$$

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

We also introduce the integral

$$\mathsf{E}a(u) = \int\_{0}^{u} dv \, \mathsf{d}n \mathbf{a}^{2}(v) \tag{98}$$

$$
\delta = n\_5 u - a \int\_0^u dv \, \text{ sna}^2(v) \tag{99}
$$

$$
\dot{u} = n\_4 u + a \int\_0^u dv \, \text{ cna}^2(v), \tag{100}
$$

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.

Figure 6. Plot of Ea ð Þ u for A ¼ 0:1, B ¼ 0:9, and α ¼ 1:2.

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