The Dispersion Relation of Internal Wave Extended-Korteweg-de Vries Equation in a Two-Layer Fluid

To understand the characteristics of ocean internal waves better, we study the dispersion relation of extended-Korteweg-de Vries (EKdV) equation with quadratic and cubic nonlinear terms in a two-layer fluid by using the Poincaré-Lighthill-Kuo (PLK) method which is one of the perturbation methods. Starting from the partial differential equation, the PLK method can be used to solve the dispersion relation of the equation. In this paper, we use PLK method to solve the equation and derive the dispersion relation of EKdV equation which is related to wave number and amplitude. Based on the dispersion relation obtained in this paper, the expressions of group velocity and phase velocity of the equation are obtained. Under the actual hydrological data, the influence of hydrological parameters on the dispersion relation for descending internal wave is discussed. It is hope that the obtained results will be helpful to the study of energy transfer and other internal wave parameters in the future.


Introduction
Ocean internal solitary wave is a kind of internal waves, which is the result of dispersion effect and nonlinear effect [1]. Under the balance of nonlinear effect and dispersion effect, the waveform can keep constant for hundreds of kilometers in the process of propagation [2] [3]. The dispersion relation is the basis of studying ocean internal solitary waves [4]. The dispersion relation describes the relationship between wave frequency and wave number. The expressions of the How to cite this paper: Feng, P.P. and Meng, X.H. (2021) The Dispersion Relation of Internal Wave Extended-Korteweg-de Vries Equation in a Two-Layer Fluid. Journal of Applied Mathematics and Physics, 9, phase velocity and the group velocity obtained from dispersion relation which can reflect the propagation of internal wave signal and energy can be calculated respectively [5]. The energy exchange in the ocean caused by the occurrence and evolution of internal solitary waves provides abundant nutrients and living space for marine organisms, especially ephemeroptera plants [6] and the shear flow caused by energy exchange seriously threatens the marine operations and military activities. Therefore, the investigation of dispersion relation of internal waves is of great significance to scientific research, marine engineering security, national defense security and marine biological transportation [7]. Hence, this is the main reason why the theoretical dispersion relation of the internal solitary waves equation is widely studied [8]- [13].
Starting from modal equation, Fliegel and Haskell used Thomson-Haskell method to calculate the dispersion relation of internal waves [8]; Munk derived the wave function in the form of Airy function and the corresponding dispersion relation in integral form when the frequency of internal wave was close to Brunt-Väisälä frequency N [14]; Wang and Shang used WKB method to study the dispersion relation of internal wave when the floating frequency was a slow varying function of water depth and internal wave frequency was close to Brunt-Väisälä frequency N [15]; Zhang and Gao solved the vertical structure and dispersion relation of internal waves by using the transformation method of Russian scholars [16]. In reference [17], the dispersion relation of internal solitary waves of the KdV equation was obtained from nonlinear partial differential equations. In this paper, we study the extended-Korteweg-de Vries (EKdV) equation with quadratic and cubic nonlinear terms proposed by T. Sakai and L. G. Redekopp which can better describe large amplitude waves propagation problem [19]. The dispersion relation with the perturbation solution of EKdV equation is obtained by using PLK method. Based on the dispersion relation, the expressions of group velocity and phase velocity are obtained. The effect of wave depth and density difference ratio on the dispersion relation of the EKdV equation is discussed for descending ocean internal waves.

The Dispersion Relation of the EKdV Equation
Grimshaw first describes the weakly nonlinear evolution of interfacial gravity waves on two shallow boundaries with KdV equation [20] [21]. When an extended KdV equation is used, the agreement between the theoretical and experimental data is greatly improved [20] [21] [22]. In view of this, T. Sakai and L.
G. redekopp proposed the EKdV equation which can describe the internal wave packet with large amplitude. The two-layer EKdV equation form is as follows The coefficients of Equation (1) are as follows: where 0 c is the linear velocity, 1 α is the quadratic nonlinear term, 2 α is the cubic nonlinear term, 0 β is the dispersion coefficient and gravity on the first and second order terms of wave amplitude [19]. The dispersion relation of Equation (1) will be derived. Introducing dimen- (1) can be transformed into Introducing phase function where k and ω are wave number and circular frequency respectively, Equation (6) becomes the following nonlinear ordinary differential equation Using the PLK method for Equation (7), the dispersion relation of Equation (1) is obtained. Let 0 η be the amplitude of ζ , and select ε as a small para- Both A and ω are expanded to power series of ε . Because the dimensionless numbers A and ε are small quantities of the same order, A is expanded from the first order of ε , and ω is expanded from the zero order. The perturbation expansion of A with respect to ε is written as and the perturbation expansion of the circular frequency ω is written as Substituting Equation (9) and Equation (10) into Equation (7), we can get The first-order to the third-order approximation is respectively ( ) The first-order approximation Equation (12) is the second-order oscillation equation for The zero order approximation of circular frequency ω is At the same time, the solution of the first-order approximation Equation (12) is obtained Taking Equation (15) and Equation (16)  It can be seen from the above formula that the non-duration condition here is Then Equation (17) is reduced to Its special solution is Taking Equations (15), (16), (18) and (20) into third-order approximation Equation (14), we can obtain It can be seen from the above formula that the non-duration condition here is ( ) The circular frequency is ( ) ( )

Dispersion Relation Diagram Based on Actual Data
Any wave equation has its specific dispersion relation, so we can determine the Combining the expression of phase velocity [23] ( ) the value of wave number k can be deduced by formula (29) and (31) using the measured data.
As seen from Figure 1, for the descending internal solitary wave, within a certain range of the water depth, when the lower layer water depth 2 h is fixed, the value of ω decreases with the increase of the upper layer water depth 1 h in Figure 1(a). When the upper layer water depth 1 h is fixed, the value of ω decreases with the increase of the lower layer water depth 2 h in Figure 1(b).
As seen from Figure 2, for the descending internal solitary wave, within a certain range of density difference ratio, the value of ω increases with the increase of density difference ratio.

Conclusion
In this paper, the dispersion relation with the perturbation solution of EKdV equation is obtained by using PLK method. The dispersion relation derived in this paper is related to water number and amplitude. The expressions of phase  velocity and group velocity are obtained by using the dispersion relation, which can be used to study the propagation characteristics and energy transmission of ocean internal waves. Under the actual hydrological data, the influence of water depth and density difference ratio on the descending internal solitary waves is discussed. The value of ω decreases with the increase of the upper and lower water depth, but it increases with the increase of density difference ratio. We hope to provide a better theoretical basis for solving internal wave parameters by using dispersion relation.