Analytical insights into three models: Exact solutions and nonlinear vibrations

Abstract

Three models are investigated in this paper, including the generalized nonlinear Schrödinger equation with distributed coefficients, the time-dependent-coefficient nonlinear Whitham–Broer–Kaup system and the fractional nonlinear vibration governing equation of an embedded single-wall carbon nanotube. With an analytical method, the aim of this paper is to exactly solve these models. As a result, some explicit and exact solutions which include hyperbolic function solutions, trigonometric function solutions and rational solutions are obtained. To gain more insights into the obtained exact solutions, dynamical evolutions with nonlinear vibrations of the amplitudes are simulated by selecting oscillation functions, noises, coefficient functions and fractional orders. It is graphically shown in the dynamical evolutions that the nonlinear vibrations of the amplitudes are influenced not only by the coefficient functions but also by the oscillation functions, noises and fractional orders.

Introduction

As pointed out by Christov, ¹ a wave in classical continuum physics is a mechanical disturbance caused maybe by the motion of atoms and molecules or the vibration of a lattice structure. From a mathematical point of view, we can understand a wave as a specific type of solution of an appropriate partial differential equation (PDE) modeling a general problem arising in practical applications. It is the hyperbolic property of a given PDE that leads to the possibility of wave solutions. Generally, we associate the solitary solution with such a solution which not only represents a wave of permanent form but also is localized so that it decays or approaches a constant at infinity. If the solution is a periodic function, then we call it periodic solution which reflects the regularity of the system.

It is well known that nonlinear phenomena are everywhere in the real world, and nonlinear PDEs are often used to describe these nonlinear phenomena. Usually, people restore the dynamical evolutions of exact solutions of nonlinear PDEs to gain more insights into the essence behind the nonlinear phenomena for further applications. In the past several decades, many effective methods for exactly solving nonlinear PDEs have been presented.^2–13 Among them, the exp-function method ⁷ proposed by He and Wu is an effective mathematical tool for constructing wave solutions, solitary solutions and periodic solutions. Recently, the analytical investigation of nonlinear vibrations has received considerable attention.^14–17 However, exact analytical solutions of nonlinear vibration equations are hard to find out in most cases. The present paper is motivated by the desire to show the analytical method ⁸ can be used for some special types of nonlinear vibration equations. For this purpose, we shall consider three models in this paper, they are the generalized nonlinear Schrödinger equation with distributed coefficients,^9,18 the time-dependent-coefficient nonlinear Whitham–Broer–Kaup system^19,20 and the fractional nonlinear vibration governing equation which can be thought of as a generalized form in the fractal space of the model for an embedded single-wall carbon nanotube. ¹⁶ Since carbon nanotube is discontinuous and to be more exactly the classical continuum mechanics becomes invalid, then the fractal calculus becomes optional (see those in literature^21–25 for examples). It is because that the fractal calculus is extremely effective to deal with phenomena in hierarchical or porous media. ²³

The first model to consider is the nonlinear generalized Schrödinger equation with gain in the form used in nonlinear fiber optics^9,18

$$i{\psi }_x+\frac{1}{2}\alpha (x){\psi }_{tt}+\beta (x){\left|\psi \right|}^2\psi =i\gamma (x)\psi$$

(1)

where $\psi =\psi (x,t)$ is a complex-valued function of the propagation distance $x$ and the retarded time $t$ , while $\alpha (x)$ , $\beta (x)$ and $\gamma (x)$ are all differentiable functions of $x$ . If $\gamma (x)$ is negative, then equation (1) describes the amplification or attenuation of pulses which propagate nonlinearly in a single mode optical fiber. In this case, $\psi$ represents the complex envelope of the electrical field in a comoving frame, $\alpha (x)$ denotes the group velocity dispersion, $\beta (x)$ denotes the nonlinearity, and $\gamma (x)$ denotes the distributed gain. Particularly, when $\alpha (x)=2$ , $\beta (x)=\pm 2$ and $\gamma (x)=0$ , equation (1) can be reduced to the well-known nonlinear Schrödinger equation $i{\psi }_z+{\psi }_{\tau \tau }\pm 2{\left|\psi \right|}^2\psi =0$ . One of the reasons for the popularity of the reduced Schrödinger equation is that it possesses n-soliton solutions.

The second model to consider is the time-dependent-coefficient nonlinear Whitham–Broer–Kaup system^19,20

$$u_t+{\gamma }_{1}u{u}_x+c{\gamma }_1{v}_x+{\gamma }_2{u}_{xx}=0$$

(2)

$$v_t+{\gamma }_1{(uv)}_x-{\gamma }_2{v}_{xx}+{\gamma }_3{u}_{xxx}=0$$

(3)

where ${\gamma }_1$ , ${\gamma }_2$ and ${\gamma }_3$ representing different dispersion and dissipation forces are all differentiable functions of the time variable $t$ , and $c$ is an arbitrary constant. It is worth mentioning that equations (2) and (3) include some existing important models as special cases, such as the approximate equations for long water waves, ²⁶ and the Boussinesq–Burgers equations. ²⁷

For a single-wall carbon nanotube of length $l$ , Young’s modulus $E$ , density $\rho$ , cross-sectional area $A$ , and cross-sectional inertia moment $I$ , embedded in elastic medium, the third model to consider is the fractional nonlinear vibration governing equation

$$\frac{{\text{d}}^{2\alpha }W}{\text{d}{t}^{2\alpha }}+\left(\frac{{\pi }^{4}EI}{l^4\rho A}+\frac{k}{\rho A}\right)W+\frac{{\pi }^{4}E}{4l^4\rho }{W}^3=0,(0<\alpha \le 1)$$

(4)

where

$${\frac{{\text{d}}^{\alpha }W(t)}{\text{d}{t}^{\alpha }}|}_{t=t_0}=\underset{t\to t_0}{\lim }\frac{{\Delta }^{\alpha }(W(t)-W(t_0))}{{(t-t_0)}^{\alpha }},{\Delta }^{\alpha }(W(t)-W(t_0))\cong \Gamma (1+\alpha )(W(t)-W(t_0))$$

is the fractal derivative ²⁴ of order $\alpha$ at the point $t=t_0$ , $k$ is a constant determined by the material constants of the surrounding elastic medium.

The rest of the paper is organized as follows. In the next section, we construct explicit and exact solutions of equations (1) to (4) by an analytical method. In the subsequent section, we simulate the dynamical evolutions with nonlinear vibrations of some obtained solutions. The final section concludes this paper.

Exact solutions

In this section, the analytical method ⁸ combined with variable-coefficient functions is first employed to construct exact analytical solutions of equations (1) to (3) and then be generalized for solving equation (4) with fractal derivative.

Let us begin with the nonlinear generalized Schrödinger equation (1). Firstly, we make the transformation

$$\psi =A{e}^{i\theta },A=A\left(x,t\right),\theta =\theta \left(x,t\right)$$

(5)

where $A$ and $\theta$ are the amplitude and phase functions, respectively. With the help of equation (5), we separate the real and imaginary parts of equation (1) as follows

$$-A{\theta }_x+\frac{1}{2}\alpha (x)(A_{tt}-A{\theta }_t^2)+\beta (x)A^3=0$$

(6)

$$A_x+\frac{1}{2}\alpha (x)(2A_t{\theta }_t+A{\theta }_{tt})-\gamma (x)A=0$$

(7)

Secondly, we suppose that

$$A=a_0+a_{1}F(\xi ),\xi =\xi (x,t)$$

(8)

where $a_0=a_0(x,t)$ and $a_1=a_1(x,t)$ are undetermined functions of the indicated variables, while $F(\xi )$ satisfies the Ricatti equation ⁸ with a constant $\sigma$

$$F^{\prime }(\xi )=\sigma +F^2(\xi )$$

(9)

Here and thereafter the prime denotes the derivative with respect to the independent variable.

Substituting equation (8) along with equation (9) into equations (6) and (7) and collecting all terms with the same order of $F^j(\xi )$ $(j=0,1,2,\dots )$ together, we derive a set of nonlinear PDEs for $a_0$ , $a_1$ and $\xi$ , from which we have

$$a_1=\pm {\xi }_t\sqrt{-\frac{\alpha (x)}{\beta (x)}},a_0=\pm \frac{{\xi }_{tt}}{2{\xi }_t}\sqrt{-\frac{\alpha (x)}{\beta (x)}}$$

(10)

$${\theta }_t=-\frac{{\xi }_x}{\alpha (x){\xi }_t},{\theta }_x=\frac{4\sigma {\alpha }^2(x){\xi }_t^4-3{\alpha }^2(x){\xi }_{tt}^2+2{\alpha }^2(x){\xi }_t{\xi }_{ttt}-2{\xi }_x^2}{4\alpha (x){\xi }_t^2}$$

(11)

$${\xi }_{xt}=\frac{2\alpha (x)\beta (x)\gamma (x){\xi }_t^2-{\alpha }^{\prime }(x)\beta (x){\xi }_t^2+\alpha (x){\beta }^{\prime }(x){\xi }_t^2+\alpha (x)\beta (x){\xi }_x{\xi }_{tt}}{\alpha (x)\beta (x){\xi }_t}$$

(12)

where $\xi$ satisfies the following constraint

$$\frac{\alpha (x)(4\sigma {\xi }_t^4{\xi }_{tt}+3{\xi }_{tt}^3-4{\xi }_t{\xi }_{tt}{\xi }_{tt}+{\xi }_t^2{\xi }_{tttt})}{4{\xi }_t^3}\sqrt{-\frac{\alpha (x)}{\beta (x)}}=0$$

(13)

Finally, from equations (5) and (10) we obtain exact solutions of equation (1)

$$\psi =\pm \sqrt{-\frac{\alpha \left(x\right)}{\beta \left(x\right)}}\left(\frac{{\xi }_{tt}}{2{\xi }_t}+{\xi }_{t}F\left(\xi \right)\right){\text{e}}^{i\theta }$$

(14)

where $\theta$ and $\xi$ are determined by equations (11) to (13), and $F(\xi )$ satisfies equation (9).

In what follows, from equation (14) we derive some special solutions of equation (1). Considering a special case of equation (13), we set

$$\xi =q(x)t+p(x)$$

(15)

where $p(x)$ and $q(x)$ are undetermined functions of $x$ . Then we can easily see from the first term ${\theta }_t$ of equations (11), (13) and (15) that ${\theta }_{ttt}=0$ . Thus, we further suppose that

$$\theta =f(x)t^2+g(x)t+h(x)$$

(16)

where $f(x)$ , $g(x)$ and $h(x)$ are undetermined functions of $x$ . Substituting equations (15) and (16) into equation (10), the second term of equations (11) and (12) and then solving the resulting equations we have

$$a_1=\pm q(x)\sqrt{-\frac{\alpha (x)}{\beta (x)}},a_0=0$$

(17)

$$f(x)=\frac{1}{f_0+2{\displaystyle \int \alpha (x)}\text{d}x},g(x)=\frac{g_0}{f_0+2{\displaystyle \int \alpha (x)}\text{d}x}$$

(18)

$$h(x)=h_0+\frac{g_0^2-2q_0^2\sigma }{4\left(f_0+2{\displaystyle \int \alpha (x)}\text{d}x\right)}$$

(19)

$$p(x)=\frac{q_0}{f_0+2{\displaystyle \int \alpha (x)}\text{d}x},q(x)=\frac{g_0{q}_0}{2\left(f_0+2{\displaystyle \int \alpha (x)}\text{d}x\right)}$$

(20)

Under the constraint

$$\gamma (x)=-\frac{\alpha (x)}{f_0+2{\displaystyle \int \alpha (x)}\text{d}x}+\frac{{\alpha }^{\prime }(x)}{2\alpha (x)}-\frac{{\beta }^{\prime }(x)}{2\beta (x)}$$

(21)

where $f_0$ , $h_0$ , $g_0$ and $q_0$ are constants.

Finally, from equations (14) to (21), we obtain five pairs of exact solutions of equation (1)

$$\psi =\pm \frac{g_0{q}_0}{2\left(f_0+2{\displaystyle \int \alpha (x)}\text{d}x\right)}\sqrt{-\frac{\alpha (x)}{\beta (x)}}F(\xi )\exp \left\{i\left[\frac{{(2t+g_0)}^2-2q_0^2\sigma }{4\left(f_0+2{\displaystyle \int \alpha (x)}\text{d}\right)x}+h_0\right]\right\}$$

(22)

where $F(\xi )$ takes the five special solutions ⁸

$$F(\xi )=\{\begin{array}{l}-\sqrt{-\sigma }\tanh \left(\sqrt{-\sigma }\xi \right),\sigma <0\hfill \\ -\sqrt{-\sigma }\coth \left(\sqrt{-\sigma }\xi \right),\sigma <0\hfill \\ \sqrt{\sigma }\tan \left(\sqrt{\sigma }\xi \right),\sigma >0\hfill \\ -\sqrt{\sigma }\cot \left(\sqrt{\sigma }\xi \right),\sigma >0\hfill \\ -\frac{1}{\xi },\sigma =0\hfill \end{array}$$

(23)

and

$$\xi =\frac{g_0{q}_0}{2\left(f_0+2{\displaystyle \int \alpha (x)}\text{d}x\right)}t+\frac{q_0}{f_0+2{\displaystyle \int \alpha (x)}\text{d}x}$$

(24)

We next consider the time-dependent-coefficient nonlinear Whitham–Broer–Kaup system in equations (2) and (3). We suppose that

$$u=a_0+a_{1}F(\xi )$$

(25)

$$v=b_0+b_{1}F(\xi )+b_2{F}^2(\xi )$$

(26)

where $\xi =kx+p$ , $a_0=a_0(x,t)$ , $a_1=a_1(x,t)$ , $b_0=b_0(x,t)$ , $b_1=b_1(x,t)$ , $b_2=b_2(x,t)$ , $k=k(t)$ and $p=p(t)$ are undetermined functions of the indicated variables, while $F(\xi )$ satisfies the Ricatti equation (9).

Substituting equations (25) and (26) along with equation (9) into equations (2) and (3) and collecting all terms with the same order of $F^j(\xi )(j=0,1,2,\dots )$ together, we derive a set of nonlinear PDEs for $a_0$ , $a_1$ , $b_0$ , $b_1$ , $b_2$ , $k$ and $p$ . Solving the set of nonlinear PDEs yields

$$a_1=\pm \frac{2k\sqrt{c{\gamma }_1{\gamma }_3+{\gamma }_2^2}}{{\gamma }_1},a_0=-\frac{k^{\prime }x+p^{\prime }}{k{\gamma }_1}$$

(27)

$$b_2=-\frac{2k^2\left(c{\gamma }_1{\gamma }_3+{\gamma }_2^2\pm {\gamma }_2\sqrt{c{\gamma }_1{\gamma }_3+{\gamma }_2^2}\right)}{c{\gamma }_1^2},b_1=0$$

(28)

$$b_0=\frac{k\prime {\gamma }_1{\gamma }_2+k({\gamma }_1{{\gamma }^{\prime }}_2-{{\gamma }^{\prime }}_1{\gamma }_2)-2\sigma k^3{\gamma }_1(c{\gamma }_1{\gamma }_3+{\gamma }_2^2)+{\gamma }_1(k\prime -2\sigma k^3{\gamma }_2)\sqrt{c{\gamma }_1{\gamma }_3+{\gamma }_2^2}}{ck{\gamma }_1^3}$$

(29)

$$k″=\frac{k\prime (2k\prime {\gamma }_1+k{{\gamma }^{\prime }}_1)}{k{\gamma }_1},p″=\frac{p\prime (2k\prime {\gamma }_1+k{{\gamma }^{\prime }}_1)}{k{\gamma }_1}$$

(30)

where ${\gamma }_1$ , ${\gamma }_2$ and ${\gamma }_3$ satisfy the constraints

$${{\gamma }^{\prime }}_3=\frac{c{\gamma }_1{{\gamma }^{\prime }}_1{\gamma }_3-2{\gamma }_2({\gamma }_1{{\gamma }^{\prime }}_2-{{\gamma }^{\prime }}_1{\gamma }_2)}{c{\gamma }_1^2}$$

(31)

$${{\gamma }^{″}}_2=\frac{{\gamma }_1{{\gamma }^{″}}_1{\gamma }_2-3{{\gamma }^{\prime }}_1({{\gamma }^{\prime }}_1{\gamma }_2-{\gamma }_1{{\gamma }^{\prime }}_2)}{{\gamma }_1^2}$$

(32)

We, therefore, obtain exact solutions of equations (2) and (3)

$$u=-\frac{k^{\prime }x+p\prime }{k{\gamma }_1}\pm \frac{2k\sqrt{c{\gamma }_1{\gamma }_3+{\gamma }_2^2}}{{\gamma }_1}F(\xi )$$

(33)

$$v=\frac{k\prime {\gamma }_1{\gamma }_2+k({\gamma }_1{{\gamma }^{\prime }}_2-{{\gamma }^{\prime }}_1{\gamma }_2)-2\sigma k^3{\gamma }_1(c{\gamma }_1{\gamma }_3+{\gamma }_2^2)+{\gamma }_1(k\prime -2\sigma k^3{\gamma }_2)\sqrt{c{\gamma }_1{\gamma }_3+{\gamma }_2^2}}{ck{\gamma }_1^3}-\frac{2k^2\left(c{\gamma }_1{\gamma }_3+{\gamma }_2^2\pm {\gamma }_2\sqrt{c{\gamma }_1{\gamma }_3+{\gamma }_2^2}\right)}{{\gamma }_1^2}{F}^2(\xi )$$

(34)

where $\xi =kx+p$ , $k$ , $p$ , ${\gamma }_1$ , ${\gamma }_2$ and ${\gamma }_3$ are determined by equations (29) to (32), and $F(\xi )$ satisfies equation (9).

If we suppose that

$$\frac{{{\gamma }^{″}}_1}{{\gamma }_1}-3\frac{{{\gamma }^{\prime }}_1^2}{{\gamma }_1^2}=0$$

(35)

then from equation (32) we have

$$\frac{{{\gamma }^{″}}_2}{{{\gamma }^{\prime }}_2}=3\frac{{{\gamma }^{\prime }}_1}{{\gamma }_1}$$

(36)

It is easy to see that equations (31), (35) and (36) have solutions

$${\gamma }_1=\pm \frac{1}{\sqrt{{\gamma }_0-2t}},{\gamma }_2=\pm \frac{1}{\sqrt{{\gamma }_0-2t}},{\gamma }_3=\pm \frac{1}{\sqrt{{\gamma }_0-2t}}$$

(37)

$$k=\frac{1}{k_0\pm \sqrt{{\gamma }_0-2t}},p=\frac{1}{k_0\pm \sqrt{{\gamma }_0-2t}}$$

(38)

where $k_0$ and ${\gamma }_0$ are arbitrary constants. Thus, we can determine the exact solutions (33) and (34) by equations (23), (29), (37) and (38). Obviously, the solutions (33) and (34) determined in this case depend on a linear relationship ${\gamma }_1=\pm {\gamma }_2=\pm {\gamma }_3$ between the coefficient functions ${\gamma }_1$ , ${\gamma }_2$ and ${\gamma }_3$ . This is due to the presupposition in equation (35), which leads to equation (36) and then equation (37). In fact, these coefficient functions can also be linearly independent. For example, if we let ${\gamma }_1={\text{e}}^t$ , then a direct computation on equations (31) and (32) gives

$${\gamma }_2=c_1{\text{e}}^t+c_2{\text{e}}^{2t},{\gamma }_3=c\text{e}{1}^t-\frac{2c_1{c}_2}{c}{\text{e}}^{2t}-\frac{c_2^2}{c}{\text{e}}^{3t}$$

where $c_1$ and $c_2$ are arbitrary constants.

We finally consider the fractional nonlinear vibration governing equation (4). By the similar way as above, we suppose that

$$W=a_0+a_{1}F(t^{\alpha })$$

(39)

where $a_0$ and $a_1$ are constants to be determined later, $F(t^{\alpha })$ satisfies the fractional Ricatti equation ¹¹ with a constant $\sigma$

$$\frac{{\text{d}}^{\alpha }F(t^{\alpha })}{\text{d}{t}^{\alpha }}=\sigma +F^2(t^{\alpha })$$

(40)

which have five special solutions

$$F(t^{\alpha })=\{\begin{array}{l}-\sqrt{-\sigma }{\tanh }_{\alpha }\left(\sqrt{-\sigma }{t}^{\alpha }\right),\sigma <0\hfill \\ -\sqrt{-\sigma }{\coth }_{\alpha }\left(\sqrt{-\sigma }{t}^{\alpha }\right),\sigma <0\hfill \\ \sqrt{\sigma }{\tan }_{\alpha }\left(\sqrt{\sigma }{t}^{\alpha }\right),\sigma >0\hfill \\ -\sqrt{\sigma }{\cot }_{\alpha }\left(\sqrt{\sigma }{t}^{\alpha }\right),\sigma >0\hfill \\ -\frac{\Gamma (1+\alpha )}{t^{\alpha }},\sigma =0\hfill \end{array}$$

(41)

Substituting equation (39) along with equation (40) into equation (4) and collecting all terms with the same order of $F^j(t^{\alpha })(j=0,1,2,\dots )$ together, we derive a set of nonlinear algebraic equations for $a_0$ and $a_1$ . Solving the set of nonlinear algebraic equations yields

$$a_1=\pm i\frac{2l^2}{{\pi }^2}\sqrt{\frac{2\rho }{E}},a_0=0,\sigma =-\frac{{\pi }^{4}EI}{2\rho A{l}^4}-\frac{k}{2\rho A}$$

(42)

With the help of equations (39), (41) and (42), we obtain the following exact solutions of equation (4)

$$W=\pm i\frac{2l^2}{{\pi }^2}\sqrt{\frac{2\rho }{E}}F\left(t^{\alpha }\right),\sigma =-\frac{{\pi }^{4}EI}{2\rho A{l}^4}-\frac{k}{2\rho A}$$

(43)

where $F(t^{\alpha })$ takes the five special solutions in equation (41). Thus, two pairs of hyperbolic function solutions of equation (4) can be obtained as follows

$$W=\mp 2i\sqrt{\frac{I}{A}+\frac{k{l}^4}{{\pi }^{4}EA}}{\tanh }_{\alpha }\left(\sqrt{\frac{{\pi }^{4}EI}{2\rho A{l}^4}+\frac{k}{2\rho A}}{t}^{\alpha }\right)$$

(44)

$$W=\mp 2i\sqrt{\frac{I}{A}+\frac{k{l}^4}{{\pi }^{4}EA}}{\coth }_{\alpha }\left(\sqrt{\frac{{\pi }^{4}EI}{2\rho A{l}^4}+\frac{k}{2\rho A}}{t}^{\alpha }\right)$$

(45)

Nonlinear vibrations

In this section, we further investigate the dynamical evolutions with nonlinear vibrations of some solutions obtained above.

In Figures 1 to 3, we simulate the real part of solution (22) with “+” branch by selecting the hyperbolic tanh function $F(\xi )=-\sqrt{-\sigma }\tanh \left(\sqrt{-\sigma }\xi \right)$ , the oscillation functions

$$\alpha (x)=x^3\sin \frac{1}{x},\beta (x)=-x\sin \frac{1}{x}$$

(46)

and the parameters $f_0=1$ , $g_0=1$ , $h_0=1$ , $q_0=1$ , $\sigma =-1$ .

Figure 1.

Spatial structure of the real part of solution (22) determined by hyperbolic tanh function and oscillation functions.

Figure 2.

Profile of the real part of solution (22) with nonlinear vibrations exited by hyperbolic tanh function and oscillation functions of at $t=0$ .

Figure 3.

Nonlinear vibrations excited by hyperbolic tanh function and oscillation functions of the real part of solution (22) at $x=1$ .

We can see from Figures 1 to 3 that the amplitude of solution (22) has a great many nonlinear vibrations in the dynamical evolutions. This is due to the selected hyperbolic tanh function and oscillation functions.

If we select the same parameters and hyperbolic tanh function as in Figures 1 to 3, the different functions $\alpha (x)$ and $\beta (x)$ with noise effects are given as

$$\alpha (x)=\frac{x^3}{\sqrt{3.14}}\text{Random(}\cdot \text{)}+\frac{2}{\sqrt{3.14}}{\displaystyle \sum _{k=1}^{150}\left(\frac{\sin kx}{k}\text{Random(}\cdot \text{)}\right)}$$

(47)

$$\beta (x)=-\frac{x}{\sqrt{3.14}}\text{Random(}\cdot \text{)}+\frac{2}{\sqrt{3.14}}{\displaystyle \sum _{k=1}^{150}\left(\frac{\sin kx}{k}\text{Random(}\cdot \text{)}\right)}$$

(48)

where $\text{Random(}\cdot \text{)=Random[NormalDistribution[0,1]]}$ is the built-in random function of Mathematica, $\text{NormalDistribution[0,1]}$ is the built-in normal distribution in the interval $[0,1]$ , then the amplitude of solution (22) has experienced a great many nonlinear vibrations in the dynamical evolutions (see Figures 4 to 6). Such nonlinear vibrations are different from those in Figures 1 to 3, and they are caused not only by the selected hyperbolic tanh function but also by the embedded noise effects.

Figure 4.

Spatial structure of the real part of solution (22) determined by hyperbolic tanh function and noise effects.

Figure 5.

Profile of the real part of solution (22) with nonlinear vibrations exited by hyperbolic tanh function and noise effects of at $t=0$ .

Figure 6.

Nonlinear vibrations excited by hyperbolic tanh function and noise effects of the real part of solution (22) at $x=1$ .

In Figures 7 to 9, we simulate the real part of solution (22) with “+” branch by selecting the rational function $F(\xi )=-1/\xi$ , the functions $\alpha (x)$ and $\beta (x)$ and other parameters are same as those in Figures 1 to 3 except for the different $\sigma =0$ . It is shown from Figures 7 to 9 that the amplitude of solution (22) has a great many nonlinear vibrations in the dynamical evolutions. They are caused by the selected rational function and oscillation functions.

Figure 7.

Spatial structure of the real part of solution (22) determined by rational function and oscillation functions.

Figure 8.

Profile of the real part of solution (22) with nonlinear vibrations exited by rational function and oscillation functions of at $t=0$ .

Figure 9.

Nonlinear vibrations excited by rational function and oscillation functions of the real part of solution (22) at $x=1$ .

Selecting the rational function $F(\xi )=-1/\xi$ , the parameters ${\gamma }_0=100$ , $c=1$ , $k_0=1$ and $\sigma =0$ , we simulate in Figure 10 the solution (33) with “+” branch determined by equations (37) and (38). It is easy to see from Figure 10 that the amplitude of solution (33) has experienced a single nonlinear vibration in the dynamical evolutions. This single nonlinear vibration is caused by the selected rational function. In Figure 11, we select the trigonometric tan function $F(\xi )=\sqrt{\sigma }\tan \left(\sqrt{\sigma }\xi \right)$ ; the parameters are same as those in Figure 10 except for the different $\sigma =1$ and simulate the periodic nonlinear vibrations of the amplitude of solution (33) happened in the dynamical evolutions. These nonlinear vibrations are caused by the selected trigonometric function.

Figure 10.

Profiles of solution (33) with nonlinear vibrations exited by rational function at three different times. (a) $t=-40$ . (b) $t=20$ and (c) $t=49.99$ .

Figure 11.

Profile of solution (33) with nonlinear vibrations exited by trigonometric tan function at $t=0$ .

For the hyperbolic function solution (44), we select $A=0.1$ , $E=1$ , $I=0.1$ , $k=3$ , $l=4$ , $\rho =0.01$ , $\pi =3.14$ and simulate its nonlinear vibrations of the modulus with different fractional orders $\alpha =0.1$ , $\alpha =0.2$ and $\alpha =0.7$ in Figure 12, respectively. We can see from Figure 12 that the fractional order influences the amplitude and shape of the nonlinear vibrations in the dynamical evolutions.

Figure 12.

Nonlinear vibrations excited by hyperbolic tanh function of the modulus of solution (44) with different fractional orders.

Conclusion

In summary, with an analytical method we have solved the generalized nonlinear Schrödinger equation with distributed coefficients, the time-dependent-coefficient nonlinear Whitham–Broer–Kaup system and the fractional nonlinear vibration governing equation of an embedded single-wall nanotube and gained some insights into the obtained exact solutions and their nonlinear vibrations happen in process of dynamical evolutions. To the best of our knowledge, the obtained exact solutions (14), (33) to (35) have not been reported in the literatures. It should be noted that when $\alpha =1$ , solutions (44) and (45) give two pairs of exact solutions of the known nonlinear vibration governing equation with integer derivative ¹⁶

$$W=\mp 2i\sqrt{\frac{I}{A}+\frac{k{l}^4}{{\pi }^{4}EA}}\tanh \left(\sqrt{\frac{{\pi }^{4}EI}{2\rho A{l}^4}+\frac{k}{2\rho A}}t\right)$$

(49)

$$W=\mp 2i\sqrt{\frac{I}{A}+\frac{k{l}^4}{{\pi }^{4}EA}}\coth \left(\sqrt{\frac{{\pi }^{4}EI}{2\rho A{l}^4}+\frac{k}{2\rho A}}t\right)$$

(50)

which have not been obtained in Fu et al. ¹⁶ All the obtained solutions presented in this paper have been checked with Mathematica by putting them back into the original equations (1) to (3).

It is shown from the simulations that the nonlinear vibrations of the amplitudes of the obtained exact solutions in the dynamical evolutions are influenced not only by the coefficient functions as pointed out in Zhang and Hong ²⁸ but also by the oscillation functions, noises and fractional orders. Since the rational ansätz of the exp-function method ⁷ contain hyperbolic function solutions and trigonometric function solutions, it is possible for us to use the method ⁷ to construct these solutions obtained in this paper except for the rational solutions. Comparatively speaking, the computation of the exp-function method ⁷ will become complicated when we construct multiple types of solutions in a uniform way. The higher the requirement of the uniformity is, the more complex will the calculation become. How to extend the existing analytical methods to some other nonlinear vibration equations, especially those differential equations with fractal derivatives, is worthy of study.