Horizontal stratification of fluids and the behavior of long waves

Abstract

This study employs contemporary and precise computational techniques to obtain innovative solitary wave solutions for the (3 + 1)-dimensional Kadomtsev-Petviashvili (${\mathcal{K}\mathcal{P}}$) equation, which characterizes the behavior of long waves with small amplitudes compared to wavelengths in horizontally stratified fluids. The model accounts for nonlinearity, dispersion, and dissipation, all of which significantly influence wave propagation. The nonlinearity results in wave interactions and soliton formation, dispersion causes wave spreading over time, while dissipation represents energy loss. The generalized exponential function (${\mathcal {GEF}}$) method is utilized to construct solitary wave solutions, which are analyzed for their amplitude, width, and wave speed. The contribution of this study lies in enhancing the comprehension of physical phenomena dynamics, including surface waves in deep water, Langmuir waves in plasma physics, and electromagnetic waves in nonlinear optics. Comparisons with prior studies indicate the proposed technique’s reliability and accuracy in obtaining solitary wave solutions for the ${\mathcal{K}\mathcal{P}}$ equation. Therefore, this study provides valuable insights into nonlinear science and has the potential to inform practical applications across various fields.

Appendix

Consider the general form of the nonlinear partial differential equation in the following form:

$$\begin{aligned} \Psi (E,E_t,E_x,E_{t\,t},E_{x\,x},....)=0, \end{aligned}$$

(29)

such that $\Psi$ is a polynomial function in E(x, t) and its partial derivatives. In the following, we show the basic structure of modified exp $\left( -\phi \left( \varsigma \right) \right)$ expansion function method:

First Step: Implementation of the wave transformation

$$\begin{aligned} E(x,t)=S(\varsigma ), \ \ \ \ \ \ \varsigma =(x-c\,t), \end{aligned}$$

(30)

where (c) is the wave velocity that would turn the partial differential equation into an ordinary differential equation to be in the following form:

$$\begin{aligned} \Omega (S,S',S'',S''',.....)=0, \end{aligned}$$

(31)

where $\Omega$ is a polynomial function in $S(\varsigma )$ and its total derivatives.

Second Step: Assume that the general solution of ODE (31) can be voiced by a polynomial in $\mathrm{{exp}}(-\phi (\varsigma ))$ as follows

$$\begin{aligned} u(\varsigma )=\frac{\sum \nolimits _{0} ^{N} a_{i}\,\mathrm{{exp}}(-i\,\phi (\varsigma )) }{\sum \nolimits _{0} ^{M} b_{j}\,\mathrm{{exp}}(-j\,\phi (\varsigma )) }, \end{aligned}$$

(32)

where $\phi (\varsigma )$ fulfill the following ordinary differential equation:

$$\begin{aligned} \phi '(\varsigma )= \zeta _1+\zeta _2 \,e^{\phi (\varsigma )}+\frac{\zeta _2 }{e^{\phi (\varsigma )}}. \end{aligned}$$

(33)

The solutions of ODE (33) have many cases; these solutions depend on the values of these parameters $(\zeta _1,\, \zeta _2,\, \zeta _2)$, where $\left[ a_{0},...,a_{N},\, b_{1},...,b_{M},\, \zeta _1,\,\zeta _2\right]$ are constants to be determined later. Where $(a_{N},\, b_{M})\ne 0$.

Third Step: Substituting Eq. (32) and its derivatives together Eq. (33) into Eq. (31) and gathering all the terms of the same power $\mathrm{{exp}}\left( -m\varphi (\varsigma )\right)$, $(m= 0,1, 2, 3,....)$ and equating them to zero, we obtain a system of algebraic equations, which can be solved by Maple or Mathematica to get the values of $(a_{i},\, b_{j},\, \zeta _1,\,\zeta _2)$.

Fourth Step: Replacing these values into the general exact traveling solution (32) that suggested by the method, we obtain the exact solutions of Eq. (31).