Variational Principle of the Unstable Nonlinear Schrödinger Equation with Fractal Derivatives

Abstract

The well-known nonlinear Schrödinger equation (NLSE) plays a crucial role in describing the temporal evolution of disturbances in marginally stable or unstable media. However, when the media is a fractal form, it becomes ineffective. Thus, the fractal modification to the NLSE is presented based on the fractal derivative in this work for the first time. The semi-inverse method is employed to establish the fractal variational principle. The entire process of deriving the fractal variational principle is presented in detail. To our knowledge, the fractal variational principle mentioned in this article is the first exploration and report to date. The fractal variational principle established in this paper is expected to deepen our understanding of the essence of physical phenomena in the fractal space and offer new ideas for the application and exploration of the variational approaches.

1. Introduction

The importance of nonlinear partial differential equations (NPDEs) lies in their status as a fundamental discipline in modern mathematics, serving as a bridge between mathematics and other scientific fields involving in optics [1,2], thermodynamics [3,4,5], biomedical science [6,7], plasma physics [8,9] and others [10,11,12,13,14], and capable of describing various important and complex natural phenomena in the objective world. To better reflect the real life and some phenomena that occur in nature, solving NPDEs is particularly important. Therefore, many researchers always focus on researching how to obtain precise solutions. Many mathematicians and physicists have conducted in-depth research in this field, creating many special methods for solving the NPDEs, and these solutions have formed their own theories, for example the Kudryashov’s approach [15,16], tanh-function technique [17,18], generalized auxiliary equation [19,20], Darboux transformation approach [21,22], Hirota bilinear method [23,24], and so on. In addition, the research on the variational principles of the NPDEs is also a hot topic, as the variational principles can help us gain a more intuitive and in-depth understanding of the solutions’ physical properties. However, generally speaking, it is relatively difficult to find the variational principles of the NPDEs, and not all NPDEs can establish their variational principles. In this study, we first consider the unstable NLSE as follows [25]:

$$i\frac{\partial \psi }{\partial t}+\frac{{\partial }^2\psi }{\partial x^2}+2\alpha {\left|\psi \right|}^2\psi -2\beta \psi =0$$

(1)

which can be used to describe the temporal evolution of disturbances in marginally stable or unstable media, $i=\sqrt{-1}$, $\alpha$, and $\beta$ are non-zero arbitrary constants. In [25], the extended simple equation technique is adopted to find the different wave structures of Equation (1). In [26], the lie symmetries are explored. In [27], three different methods, the sine-cosine technique, exp(-ϕ(ξ))-expansion approach and the Riccati–Bernoulli sub-ODE approach are used to study the different wave structures. In [28], the modified Kudraysov method is employed to seek the abundant wave solutions. In [29], the new Jacobi elliptic function rational expansion method is adopted to construct the exact wave solutions. In [30], the improved modified Sardar sub-equation method is used to explore the diverse wave solutions. In [31], the exp_a and hyperbolic function methods are utilized to investigate the explicit exact solutions. In [32], the modified extended direct algebraic method is manipulated to plumb the abundant soliton solutions. In [33], the modified extended mapping method is utilized to seek for different optical soliton solutions. In [34], a modified mathematical method is proposed to look for the exact wave solutions. In [35], the modified exponential rational function method is consindered to study the exact traveling wave solutions. In [36], a new trial equation method is presented to develop the different kinds of soliton solutions. In [37], the extended rational sine-cosine/sinh-cosh approach is employed to explore the solitary and periodic wave solutions. In [38], the three-wave method, double exponential method, and the positive-quadratic function approach are used to find the abundant wave solutions. In [39], the improved auxiliary equation tool is adopted to develop the diverse wave structures. However, Equation (1) becomes powerless when the medium is the fractal form. Thus, we need to carry out the fractal modification for Equation (1) through the fractal derivative as follows:

$$i\frac{\partial \psi }{\partial t^{\gamma }}+\frac{{\partial }^2\psi }{\partial x^{2\gamma }}+2\alpha {\left|\psi \right|}^2\psi -2\beta \psi =0$$

(2)

where $\gamma$($0<\gamma \le 1$) represents the two-scale fractal dimension, $\frac{\partial }{d{x}^{\gamma }}$, and $\frac{\partial }{d{t}^{\gamma }}$ are the fractal derivatives as follows [40,41]:

$$\frac{\partial }{\partial x^{\gamma }}\psi \left(x_0^{\gamma },t^{\gamma }\right)=\mathsf{\Gamma }\left(1+\gamma \right)\underset{\begin{array}{l}x-x_0=\mathsf{\Delta }x\\ \mathsf{\Delta }x\ne 0\end{array}}{\lim }\frac{\psi \left(x^{\gamma },t^{\gamma }\right)-\psi \left(x_0^{\gamma },t^{\gamma }\right)}{{\left(x-x_0\right)}^{\gamma }}$$

(3)

$$\frac{\partial }{\partial t^{\gamma }}\psi \left(x^{\gamma },t_0^{\gamma }\right)=\mathsf{\Gamma }\left(1+\gamma \right)\underset{\begin{array}{l}t-t_0=\mathsf{\Delta }t\\ \mathsf{\Delta }t\ne 0\end{array}}{\lim }\frac{\psi \left(x^{\gamma },t^{\gamma }\right)-\psi \left(x^{\gamma },t_0^{\gamma }\right)}{{\left(t-t_0\right)}^{\gamma }}$$

(4)

which admits the following chain rules [42]:

$$\frac{{\partial }^2}{\partial x^{2\gamma }}=\frac{\partial }{\partial x^{\gamma }}\frac{\partial }{\partial x^{\gamma }}$$

(5)

It should be noted that Equation (2) reduces into the classic unstable NLSE in Equation (1) for $\gamma =1$. As we all know, the variational principle plays a crucial role in describing the nonlinear phenomena from a physical perspective. In addition, it is also the fundamental theoretical basis of the variational approaches. Thus, hereby in this study, we focus on constructing the fractal variational principle for Equation (2) by applying the semi-inverse method.

2. The Fractal Variational Principle

The function $\psi$ can be written as follows:

$$\psi =\theta +i\phi .$$

(6)

Tanking it into Equation (2) yields the following:

$$i\left(\frac{\partial \theta }{\partial t^{\gamma }}+i\frac{\partial \phi }{\partial t^{\gamma }}\right)+\left(\frac{{\partial }^2\theta }{\partial x^{2\gamma }}+i\frac{{\partial }^2\phi }{\partial x^{2\gamma }}\right)+2\alpha \left({\theta }^2+{\phi }^2\right)\left(\theta +i\phi \right)-2\beta \left(\theta +i\phi \right)=0,$$

(7)

which can be divided into the imaginary imaginary and real components, respectively, as follows:

$$\frac{\partial \theta }{\partial t^{\gamma }}+\frac{\partial \phi }{\partial x^{2\gamma }}+2\alpha \left({\theta }^2+{\phi }^2\right)\phi -2\beta \phi =0,$$

(8)

and

$$-\frac{\partial \phi }{\partial t^{\gamma }}+\frac{\partial \theta }{\partial x^{2\gamma }}+2\alpha \left({\theta }^2+{\phi }^2\right)\theta -2\beta \theta =0,$$

(9)

Then, we will employ the semi-inverse method to find the variational principle of Equations (8) and (9) as follows [43,44,45,46,47,48,49,50]:

$$J\left(\theta ,\phi \right)={\displaystyle \iint L\left(\theta ,{\theta }_t^{\left(\gamma \right)},{\theta }_x^{\left(\gamma \right)},\phi ,{\phi }_t^{\left(\gamma \right)},{\phi }_x^{\left(\gamma \right)},\dots \right)d{t}^{\gamma }d{x}^{\gamma }},$$

(10)

where ${\theta }_t^{\left(\gamma \right)}=\frac{\partial \theta }{\partial t^{\gamma }}$, ${\theta }_x^{\left(\gamma \right)}=\frac{\partial \theta }{\partial x^{\gamma }}$, ${\phi }_t^{\left(\gamma \right)}=\frac{\partial \phi }{\partial t^{\gamma }}$, ${\phi }_x^{\left(\gamma \right)}=\frac{\partial \phi }{\partial x^{\gamma }}$, and $L$ is called the trial-Lagrange function (TLF) and the Euler–-Lagrange equations of Equation (10) are the following:

$$\frac{\delta L}{\delta \phi }=\frac{\partial L}{\partial \phi }-\frac{\partial }{\partial x}\left(\frac{\partial L}{\partial {\phi }_x^{\left(\gamma \right)}}\right)-\frac{\partial }{\partial t}\left(\frac{\partial L}{\partial {\phi }_t^{\left(\gamma \right)}}\right).$$

(11)

and

$$\frac{\delta L}{\delta \theta }=\frac{\partial L}{\partial \theta }-\frac{\partial }{\partial x}\left(\frac{\partial L}{\partial {\theta }_x^{\left(\gamma \right)}}\right)-\frac{\partial }{\partial t}\left(\frac{\partial L}{\partial {\theta }_t^{\left(\gamma \right)}}\right),$$

(12)

In order to find the variational principles of Equations (8) and (9), the TLF can be assumed as follows:

$$L=\theta \frac{\partial \phi }{\partial t^{\gamma }}+\frac{1}{2}{\left(\frac{\partial \phi }{\partial x^{\gamma }}\right)}^2+F\left(\theta ,\phi ,{\theta }_t^{\left(\gamma \right)},{\theta }_x^{\left(\gamma \right)},{\phi }_t^{\left(\gamma \right)},{\phi }_x^{\left(\gamma \right)}\right).$$

(13)

where $F\left(\theta ,\phi ,{\theta }_t^{\left(\gamma \right)},{\theta }_x^{\left(\gamma \right)},{\phi }_t^{\left(\gamma \right)},{\phi }_x^{\left(\gamma \right)}\right)$ is a undetermined function of $\theta ,\phi ,{\theta }_t^{\left(\gamma \right)},{\theta }_x^{\left(\gamma \right)},{\phi }_t^{\left(\gamma \right)}$, and ${\phi }_x^{\left(\gamma \right)}$. By Equation (11), we have the following:

$$-\frac{\partial \theta }{\partial t^{\gamma }}-\frac{{\partial }^2\phi }{\partial x^{2\gamma }}+\frac{\delta F}{\delta \phi }=0,$$

(14)

where $\frac{\delta F}{\delta \phi }$ is the variational derivative in this study as follows:

$$\frac{\delta F}{\delta \phi }=\frac{\partial F}{\partial \phi }-\frac{\partial }{\partial t}\left(\frac{\partial F}{\partial {\phi }_t^{\left(\gamma \right)}}\right)-\frac{\partial }{\partial x}\left(\frac{\partial F}{\partial {\phi }_x^{\left(\gamma \right)}}\right).$$

(15)

A comparison between Equations (14) and (8) yields the following:

$$\frac{\delta F}{\delta \phi }=\frac{\partial F}{\partial \phi }-\frac{\partial }{\partial t}\left(\frac{\partial F}{\partial {\phi }_t^{\left(\gamma \right)}}\right)-\frac{\partial }{\partial x}\left(\frac{\partial F}{\partial {\phi }_x^{\left(\gamma \right)}}\right)=-2\alpha \left({\theta }^2+{\phi }^2\right)\phi +2\beta \phi ,$$

(16)

To identify $F$, we can set:

$$\frac{\partial F}{\partial \phi }=-2\alpha \left({\theta }^2+{\phi }^2\right)\phi +2\beta \phi ,$$

(17)

$$\frac{\partial F}{\partial {\phi }_t^{\left(\gamma \right)}}=0,$$

(18)

$$\frac{\partial F}{\partial {\phi }_x^{\left(\gamma \right)}}=0,$$

(19)

From Equations (17)–(19), we can easily determine $F$ as follows:

$$F=-\alpha {\theta }^2{\phi }^2-\frac{1}{2}\alpha {\phi }^4+\beta {\phi }^2,$$

(20)

Now we can update the TLF as follows:

$$L=\theta \frac{\partial \phi }{\partial t^{\gamma }}+\frac{1}{2}{\left(\frac{\partial \phi }{\partial x^{\gamma }}\right)}^2-\alpha {\theta }^2{\phi }^2-\frac{1}{2}\alpha {\phi }^4+\beta {\phi }^2+f\left(\theta ,\phi ,{\theta }_t^{\left(\gamma \right)},{\theta }_x^{\left(\gamma \right)},{\phi }_t^{\left(\gamma \right)},{\phi }_x^{\left(\gamma \right)}\right),$$

(21)

where $f\left(\theta ,\phi ,{\theta }_t^{\left(\gamma \right)},{\theta }_x^{\left(\gamma \right)},{\phi }_t^{\left(\gamma \right)},{\phi }_x^{\left(\gamma \right)}\right)$ is an undetermined function of $\theta ,\phi ,{\theta }_t^{\left(\gamma \right)},{\theta }_x^{\left(\gamma \right)},{\phi }_t^{\left(\gamma \right)}$, and ${\phi }_x^{\left(\gamma \right)}$. By Equation (12), we have the following:

$$\frac{\partial \phi }{\partial t^{\gamma }}-2\alpha \theta {\phi }^2+\frac{\delta f}{\delta \theta }=0,$$

(22)

where $\frac{\delta f}{\delta \phi }$ is the variational derivative in this study as follows:

$$\frac{\delta f}{\delta \theta }=\frac{\partial f}{\partial \theta }-\frac{\partial }{\partial t}\left(\frac{\partial f}{\partial {\theta }_t^{\left(\gamma \right)}}\right)-\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial {\theta }_x^{\left(\gamma \right)}}\right).$$

(23)

From Equations (9), (22), and (23), there are the following:

$$\frac{\delta f}{\delta \theta }=\frac{\partial f}{\partial \theta }-\frac{\partial }{\partial t}\left(\frac{\partial f}{\partial {\theta }_t^{\left(\gamma \right)}}\right)-\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial {\theta }_x^{\left(\gamma \right)}}\right)=-\frac{{\partial }^2\theta }{\partial x^{2\gamma }}-2\alpha {\theta }^3+2\beta \theta ,$$

(24)

For identifying $f$, we can set:

$$\frac{\partial f}{\partial \theta }=-2\alpha {\theta }^3+2\beta \theta ,$$

(25)

$$\frac{\partial f}{\partial {\theta }_t^{\left(\gamma \right)}}=0,$$

(26)

$$-\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial {\theta }_x^{\left(\gamma \right)}}\right)=-\frac{{\partial }^2\theta }{\partial x^{2\gamma }},$$

(27)

Based on Equations (25)–(27), we can easily determine $f$ as follows:

$$f=\frac{1}{2}{\left(\frac{\partial \theta }{\partial x^{\gamma }}\right)}^2-\frac{1}{2}\alpha {\theta }^4+\beta {\theta }^2$$

(28)

Then, the TLF $L$ can be found as follows:

$$L=\theta \frac{\partial \phi }{\partial t^{\gamma }}+\frac{1}{2}{\left(\frac{\partial \phi }{\partial x^{\gamma }}\right)}^2+\frac{1}{2}{\left(\frac{\partial \theta }{\partial x^{\gamma }}\right)}^2-\alpha {\theta }^2{\phi }^2-\frac{1}{2}\alpha \left({\phi }^4+{\theta }^4\right)+\beta \left({\phi }^2+{\theta }^2\right).$$

(29)

Thus, the variational principle is established as follows:

$$J\left(\theta ,\phi \right)={\displaystyle \iint \left\{\theta \frac{\partial \phi }{\partial t^{\gamma }}+\frac{1}{2}{\left(\frac{\partial \phi }{\partial x^{\gamma }}\right)}^2+\frac{1}{2}{\left(\frac{\partial \theta }{\partial x^{\gamma }}\right)}^2-\alpha {\theta }^2{\phi }^2-\frac{1}{2}\alpha \left({\phi }^4+{\theta }^4\right)+\beta \left({\phi }^2+{\theta }^2\right)\right\}d{t}^{\gamma }d{x}^{\gamma }}.$$

(30)

Proof. 

Computing the stationary conditions of Equation (30) with respect to $\phi$ and $\theta$, we can obtain the Euler–Lagrange equations as follows:

$$\frac{\delta L}{\delta \phi }:\frac{\partial \theta }{\partial t^{\gamma }}+\frac{{\partial }^2\phi }{\partial x^{2\gamma }}+2\alpha \phi \left({\theta }^2+{\phi }^2\right)-2\beta \phi =0,$$

(31)

and

$$\frac{\delta L}{\delta \theta }:-\frac{\partial \phi }{\partial t^{\gamma }}+\frac{{\partial }^2\theta }{\partial x^{2\gamma }}+2\alpha \theta \left({\phi }^2+{\theta }^2\right)-2\beta \theta =0.$$

(32)

Obviously, we can see Equations (31) and (32) equal to Equations (8) and (9), respectively, which fully confirms the correctness of the obtained variational principle in Equation (30).

Then, by Equation (6), we have the following:

$$\theta =\frac{\psi +{\psi }^{\ast }}{2},$$

(33)

and

$$\phi =-\frac{i\left(\psi -{\psi }^{\ast }\right)}{2}.$$

(34)

Taking them into Equation (30) yields the fractal variational principle to Equation (2) as follows:

$$J\left(\psi ,{\psi }^{\ast }\right)={\displaystyle \iint \left\{-i\left(\psi +{\psi }^{\ast }\right)\frac{\partial }{\partial t^{\gamma }}\left(\psi -{\psi }^{\ast }\right)+2\frac{\partial \psi }{\partial x^{\gamma }}\frac{\partial {\psi }^{\ast }}{\partial x^{\gamma }}-2\alpha {\psi }^2{\left({\psi }^{\ast }\right)}^2+4\beta \psi {\psi }^{\ast }\right\}d{t}^{\gamma }d{x}^{\gamma }}.$$

(35)

□

3. Proof of the Fractal Variational Principle

Proof. 

Calculating the functional stationary conditions of Equation (35) in regard to ${\psi }^{\ast }$ and $\psi$, respectively, yields the following:

$$-i\left[\frac{\partial }{\partial t^{\gamma }}\left(\psi -{\psi }^{\ast }\right)+\frac{\partial }{\partial t^{\gamma }}\left(\psi +{\psi }^{\ast }\right)\right]-2\frac{{\partial }^2\psi }{\partial x^{2\gamma }}-4\alpha {\psi }^2{\psi }^{\ast }+4\beta \psi =0.$$

(36)

and

$$\frac{\delta L}{\delta \psi }:-i\left[\frac{\partial }{\partial t^{\gamma }}\left(\psi -{\psi }^{\ast }\right)-\frac{\partial }{\partial t^{\gamma }}\left(\psi +{\psi }^{\ast }\right)\right]-2\frac{{\partial }^2{\psi }^{\ast }}{\partial x^{2\gamma }}-4\alpha \psi {\left({\psi }^{\ast }\right)}^2+4\beta {\psi }^{\ast }=0,$$

(37)

After simple calculations, we have the following:

$$\frac{\delta L}{\delta {\psi }^{\ast }}:i\frac{\partial \psi }{\partial t^{\gamma }}+\frac{{\partial }^2\psi }{\partial x^{2\gamma }}+2\alpha {\left|\psi \right|}^2\psi -2\beta \psi =0,$$

(38)

and

$$\frac{\delta L}{\delta \psi }:i\frac{\partial {\psi }^{\ast }}{\partial t^{\gamma }}-\frac{{\partial }^2{\psi }^{\ast }}{\partial x^{2\gamma }}-2\alpha {\left|{\psi }^{\ast }\right|}^2{\psi }^{\ast }+2\beta {\psi }^{\ast }=0,$$

(39)

It is obvious that Equation (38) is the same with the fractal unstable NSLE given in Equation (2), thus confirming the correctness of the constructed fractal variational principle in Equation (35). □

4. Conclusions

The fractal unstable NLSE for the fractal media was proposed in this work for the first time. Taking advantage of the semi-inverse method, its fractal variational principle was developed and the entire construction process was displayed in detail. The correctness of the established fractal variational principle was also confirmed through the Euler–Lagrange equations, which were obtained by taking the stationary conditions. To the best of the author’s knowledge, the fractal variational principle is first probed in this work and has never been reported in other literature yet. In addition, the extracted fractal variational principle can describe the nonlinear phenomena from a physical perspective in a fractal space, enabling a better understanding of the essence of physical phenomena and offer some new inspiration for the exploration of the analytical solutions and numerical scheme of the fractal NPDEs arising in physics.