A governing equation of Rossby waves and its dynamics evolution by Bilinear neural network method

Fluid mechanics is a basic science, it is widely used in industry, agriculture, astronomy, geosciences, biology, medicine and many other fields [1–3]. The research of fluid mechanics mainly focuses on the application research and the basic research for the purpose of exploring the complex flow law and mechanism of fluid. In many fields of fluid mechanics research, the geophysical fluid dynamics in large scale and the micro and nano fluid mechanics in small scale are very attractive [4, 5]. Both large-scale and small-scale researches follow the following theoretical analysis rules. First of all, mechanical models are established according to the mechanical problems of actual fluids. In the second step, the governing equations are established according to the boundary conditions and initial conditions. The third step is to use mathematical methods to study these equations. The last step is to explain the physical meaning and flow mechanism [6].

Rossby waves is one of the significant waves in geophysical fluid dynamics, which characterizes large-scale motion. Rossby waves evolution research is of great importance in nonlinear atmospheric and oceanic dynamics, such as the evolution of weather systems, the interaction between atmosphere and ocean, blocking situations in atmospheric circulation, the theory of large-scale enveloping soliton in atmosphere, ocean vortices, nonlinear dynamics in equatorial regions, and numerical weather prediction [7–10]. Rossby waves is mainly divided into geostrophic Rossby waves [11, 12] and topographic Rossby waves [13, 14]. Geostrophic Rossby waves is related to β effect, while topographic Rossby wave is related to relief of terrain. In the history of Rossby waves development, scholars have derived several types of nonlinear partial differential equations to reveal the physical mechanism of Rossby waves. The famous KdV model [15] was first derived by Long [16] to model the evolution of nonlinear Rossby waves amplitudes in the β-plane approximation. A stronger nonlinearity mKdV equation is proposed to describe Rossby waves by Wadati [17]. Ono [18] studied the propagation of Rossby waves by applying BDO equation. The main direction of the Benjamin-Ono (BO) equation is the algebraic Rossby solitary wave theory [19]. Luo [20] studied the dynamical characteristics of atmospheric block by applying the nonlinear Schrödinger equation, and the following research proved NLS is suitable to describe the propagation characteristics of Rossby waves [21]. In the following period, more and more interesting extended equations appeared to be used to characterize the dynamics of Rossby waves. But most of these studies are traditional approximation, which ignores the horizontal component of the Coriolis parameter due to the Earth's rotation. However, an increasing number of studies have found that the horizontal component of Coriolis parameter plays an important role in atmospheric and oceanic motions near the equator. So, all the subsequent studies on Rossby waves have been proposed about complete Coriolis parameters. Such as Zhang et al got a mZK governing model by considering the complete Coriolis parameters [22]. Yin et al obtained a nonlinear Schrödinger governing model [23] and a mKdV-Burgers governing model [24] by considering the complete Coriolis parameters. Yang et al derived an integer order mZK governing model by considering the complete Coriolis parameters [25].

Discussing here, this paper focuses on equation (1) in the literature [26], which is an extended KdV equation derived from the fundamental equation of motion for the ocean, the horizontal component of Coriolis force was considered. The equation (1) is obtained by using scalar analysis and perturbation expansions in the equatorial β-plane approximation, which is utilized to imitate the equatorial nonlinear near-inertial wave's evolution process.

$$\begin{eqnarray}&&C\displaystyle \frac{\partial A}{\partial T}+{C}_{1}\displaystyle \frac{\partial A}{\partial \xi }+{C}_{2}A\displaystyle \frac{\partial A}{\partial \xi }+{C}_{3}\displaystyle \frac{{\partial }^{3}A}{\partial {\xi }^{3}}=0,\end{eqnarray} \tag{ 1 }$$

where $C=-\displaystyle {\iint }_{y,z}\displaystyle \frac{{\tilde{q}}_{0}^{* }}{{\bar{v}}_{x}-c}\left({L}_{3}\left({\tilde{t}}_{0}\right)+{L}_{4}\left({\tilde{v}}_{x0}\right)\right){\rm{d}}y{\rm{d}}z,$

$$\begin{eqnarray*}&&{C}_{1}=\gamma \displaystyle {\iint }_{y,z}\displaystyle \frac{{\tilde{q}}_{0}^{* }}{{\bar{v}}_{x}-c}{L}_{2}\left.\left({\tilde{v}}_{x0}\right)\right){\rm{d}}y{\rm{d}}z,\end{eqnarray*}$$

$$\begin{eqnarray*}&&\begin{array}{c}{C}_{2}=-\displaystyle {\iint }_{y,z}\displaystyle \frac{{\tilde{q}}_{0}^{* }}{{\bar{v}}_{x}-c}\left\{{L}_{3}\left[\displaystyle \frac{1}{{d}_{s}}\left({\tilde{v}}_{x0}{\tilde{t}}_{0}+{\tilde{v}}_{y0}\displaystyle \frac{\partial {\tilde{t}}_{0}}{\partial y}\right)\right]\right.\\ \,\left.+\,{L}_{4}\left[\displaystyle \frac{1}{{d}_{s}}\left({\tilde{v}}_{x0}^{2}+{\tilde{v}}_{y0}\displaystyle \frac{\partial {\tilde{v}}_{x0}}{\partial y}+{\tilde{v}}_{z0}\displaystyle \frac{\partial {\tilde{v}}_{x0}}{\partial z}\right)\right]\right\}{\rm{d}}y{\rm{d}}z,\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}&&{C}_{3}=-\displaystyle {\iint }_{y,z}\displaystyle \frac{{\tilde{q}}_{0}^{* }}{{\bar{v}}_{x}-c}\left({L}_{1}\left[\left({\bar{v}}_{x}-c\right){\tilde{v}}_{y0}\right]\right.{\rm{d}}y{\rm{d}}z.\end{eqnarray*}$$

when ${a}_{1}=\displaystyle \frac{{C}_{2}}{C},$ ${a}_{2}=\displaystyle \frac{{C}_{3}}{C},$ ${a}_{3}=\displaystyle \frac{{C}_{1}}{C},$ it can be simplified as follow:

$$\begin{eqnarray}&&\displaystyle \frac{\partial n}{\partial T}+{a}_{1}n\displaystyle \frac{\partial n}{\partial \xi }+{a}_{2}\displaystyle \frac{{\partial }^{3}n}{\partial {\xi }^{3}}+{a}_{3}\displaystyle \frac{\partial n}{\partial \xi }=0.\end{eqnarray} \tag{ 2 }$$

a very critical coefficient ${a}_{3}$ expresses the horizontal component of Coriolis parameters, which has effect by qualifying the velocity characteristic of the near-inertial waves. The coefficient ${a}_{3}$ gives this equation a physical background, which is the main factor that distinguishes it from traditional KdV equation [15]. Although such a meaningful equation is derived in the reference [26], it was solved by using the Jacobian elliptic function expansions law and only got periodic and single solitary wave solutions. Abundant meaningful exact solutions have not been obtained. In fact, exact solutions of nonlinear extended equations play an irreplaceable part for many domains, especially when describing some complex nonlinear phenomena in physics. A growing number of researchers have worked on solving the exact analytic solutions of the nonlinear extended equations, including solitons [27–29], lumps [30, 31], breathers [32, 33], rouge waves [34–37], interaction phenomena [38, 39] and some rational function solutions [40–43].

To derive the exact solutions of nonlinear equations, many methods have been raised, for example, the Hirota bilinear transformation [44–47], symbolic computation approach [48, 49], Darboux transformation [50, 51], Bäcklund transformation [52, 53], and so on. A new method incorporates virtually all the above methods and is a general approach to search the exact solutions of NLPDEs. The most powerful method is the bilinear neural network method (BNNM), which is an effective method combining bilinear and neural network. This approach provides the flexibility to build various test functions for obtaining exact solutions. This approach constructs a number of multi-layer composite functions, unlike the classical solution method which is just a simple linear function. This method was first proposed by Zhang [54] and used to solve nonlinear partial differential equations. In addition, Zhang et al used this method to obtain exact analytical solutions of many NLPDEs, such as the p-gBKP equation [55, 56], the gBS equation [57], the SK-like equation [58], the CDGKS-like equation [59]. And there are scholars interested in this approach, such as Qiao et al and Shen et al used BNNM to obtain three types of periodic solutions [60] and periodic-soliton [61] for BMLP equation. Zeynel et al used BNNM to obtain a rogue, bright and dark wave solutions for a Hirota type equation [62].

In this paper, we obtain the rogue wave, interaction solution under the single-layer neural network and rational function exact solutions under the multi-layer neural network of the extended KdV governing equation by using BNNM, and study the influence of horizontal component of Coriolis parameter ${a}_{3}$ on the rogue wave and interaction phenomenon. This paper mainly studies following points, firstly, the BNNM is introduced. Secondly, the single layer neural network model (2–2–1 and 2–3–1), double layers neural network model (2–2–2–1and 2–2–3–1), three layers neural network model (2–2–2–2–1) are used to solve the exact solutions of the equation. Finally, dynamic graphs of these exact solutions are given by the aid of Mathematica.

2.1. Bilinear neural network

The tensor formula of BNNM is established here:

$$\begin{eqnarray}&&f={w}_{{l}_{n},f}{\sigma }_{{l}_{n}}({\eta }_{{l}_{n}}),\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray*}&&{\eta }_{{l}_{i}}={w}_{{l}_{i-1},{l}_{i}}{\sigma }_{{l}_{i-1}}({\eta }_{{l}_{i-1}})+{b}_{{l}_{i}},i=1,2,\mathrm{..}.,n,\end{eqnarray*}$$

a weight coefficient from neuron $p$ to $q$ is given by ${w}_{p,q},$ $\sigma$ is a activation function, any function can be obtained by using it, but the final layer activation function must satisfy ${\sigma }_{{l}_{n}}(\eta )\geqslant 0.$

$$\begin{eqnarray}&&{l}_{n}=\left\{{m}_{n-1}+1,{m}_{n-1}+2,\mathrm{..}.,n\right\}.\end{eqnarray} \tag{ 4 }$$

represents the neural network model's $n$ th layer space, and

${l}_{0}=\left\{{x}_{1},{x}_{2},\mathrm{..}.,{x}_{n}\right\},$ ${l}_{1}=\left\{1,2,\mathrm{..}.,{m}_{1}\right\},$ ${l}_{i}=\left\{{m}_{i-1}+1,{m}_{i-1}+2,\mathrm{..}.,{m}_{i}\right\},(i=2,3,\mathrm{..}.,n-1),$

$b$ is a constant.

Figure 1 shows the structure of the tensor formula of the neural network in detail. The specific algorithm of BNNM is shown in figure 2.

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2.2. Hirota bilinear form

Under the following integrable constraint and the dependent variable transforma-

tion

$$\begin{eqnarray}&&u=2{(\mathrm{ln}\,n)}_{\xi \xi },\end{eqnarray} \tag{ 5 }$$

in which $u=u(\xi ,T),$ $n=n(\xi ,T),$ ${a}_{1}=6{a}_{2}.$

Substituting (5) into (2), then we obtain the following Hirota bilinear equation

$$\begin{eqnarray}&&\begin{array}{c}{B}_{KdV}\left(n\right):\left({D}_{\xi }{D}_{T}+{a}_{2}{D}_{\xi }^{4}+{a}_{3}{D}_{\xi }^{2}\right)n\cdot n=0\\ =\left({n}_{\xi T}n-{n}_{\xi }{n}_{T}\right)+{a}_{2}\left({n}_{\xi \xi \xi \xi }n-4{n}_{\xi }{n}_{\xi \xi \xi }+3{n}_{\xi \xi }^{2}\right)+{a}_{3}\left({n}_{\xi \xi }n-{n}_{\xi }^{2}\right)=0.\end{array}\end{eqnarray} \tag{ 6 }$$

The rogue wave solution of the governing equation is obtained under single layer network model 2–2–1. Figure 3 shows the structure of the single hidden layer neural network of equation (3). The structure of the single hidden layer nonlinear neural network of equation (2) is called 2–ca1, the first '2' means two variables, the second '2' means two activation functions in the first layer, the last '1' is the desired function.

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We can set 2–2–1 model as following

$$\begin{eqnarray}&&n={w}_{1,n}{\sigma }_{1}({\eta }_{1})+{w}_{2,n}{\sigma }_{2}({\eta }_{2})+{b}_{3},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray*}&&\left\{\begin{array}{c}{\eta }_{1}={w}_{\xi ,1}\xi +{w}_{T,1}T+{b}_{1},\\ {\eta }_{2}={w}_{\xi ,2}\xi +{w}_{T,2}T+{b}_{2}.\end{array}\right.\end{eqnarray*}$$

Take the activation function as ${\sigma }_{1}({\eta }_{1})={{\eta }_{1}}^{2},{\sigma }_{2}({\eta }_{2})={{\eta }_{2}}^{2}$ [58], then exact solution is given as

$$\begin{eqnarray}&&n={w}_{1,n}{\left({w}_{\xi ,1}\xi +{w}_{T,1}T+{b}_{1}\right)}^{2}+{w}_{2,n}{\left({w}_{\xi ,2}\xi +{w}_{T,2}T+{b}_{2}\right)}^{2}+{b}_{3},\end{eqnarray} \tag{ 8 }$$

Where ${b}_{k}$ and ${w}_{i,j}(i=\xi ,T,j=1,2,\cdots )$ are real constants.

A group of equations can be created by substituting (8) into the bilinear form (6), and results into a group of six equations, the solution of the equations can be obtained as follows

$$\begin{eqnarray*}&&{w}_{T,1}=-{a}_{3}{w}_{\xi ,1},\end{eqnarray*}$$

$$\begin{eqnarray*}&&{w}_{T,2}=\displaystyle \frac{1}{2}\left(-{a}_{3}{w}_{\xi ,2}\pm \sqrt{-4{w}_{T,1}^{2}-4{a}_{3}{w}_{T,1}{w}_{\xi ,1}+{a}_{3}^{2}{w}_{\xi ,2}^{2}}\right),\end{eqnarray*}$$

$$\begin{eqnarray}\begin{array}{cc}{b}_{3}\,= & \,\frac{-12{a}_{2}{w}_{\xi ,1}^{4}+2{a}_{3}{b}_{1}{b}_{2}{w}_{\xi ,1}{w}_{\xi ,2}-2{a}_{3}{b}_{1}^{2}{w}_{\xi ,2}^{2}+2{a}_{3}{b}_{2}^{2}{w}_{\xi ,2}^{2}-24{a}_{2}{w}_{\xi ,1}^{2}{w}_{\xi ,2}^{2}-12{a}_{2}{w}_{\xi ,2}^{4}}{2{a}_{3}{w}_{\xi ,2}^{2}}.\end{array}\end{eqnarray} \tag{ 9 }$$

substituting (7) into (4), the first order rogue wave solution is obtained:

$$\begin{eqnarray}&&u=2{\left(-\displaystyle \frac{2{w}_{\xi ,1}A+2{w}_{\xi ,2}{B}^{2}+\left(2{w}_{\xi ,1}^{2}+2{w}_{\xi ,2}^{2}\right)}{{\rm{\Theta }}}\right)}_{\xi \xi },\end{eqnarray} \tag{ 10 }$$

Here $A={b}_{1}-{a}_{3}{w}_{\xi ,1}T+{w}_{\xi ,1}\xi ,B={b}_{2}+{w}_{\xi ,2}\xi ,$

$$\begin{eqnarray*}&&\begin{array}{c}{\rm{\Theta }}=\displaystyle \frac{-12{a}_{2}{w}_{\xi ,1}^{4}+4{a}_{3}{b}_{1}{b}_{2}{w}_{\xi ,1}{w}_{\xi ,2}-2{a}_{3}{b}_{1}^{2}{w}_{\xi ,2}^{2}+2{a}_{3}{b}_{2}^{2}{w}_{\xi ,2}^{2}-24{a}_{2}{w}_{\xi ,1}^{2}{w}_{\xi ,2}^{2}-12{a}_{2}{w}_{\xi ,2}^{4}}{2{a}_{3}{w}_{\xi ,2}^{2}}\\ \,+{A}^{2}+{B}^{2}.\end{array}\end{eqnarray*}$$

Selecting appropriate parameters for equation (10), the dynamic behavior of single hidden layer exact solution can be demonstrated in figure 4 with the parameters ${w}_{1,n}=$ ${w}_{2,n}=$ ${w}_{\xi ,1}=1,{w}_{\xi ,2}=1/2,{b}_{1}=-1,{b}_{2}=-2,{b}_{3}=1,{a}_{2}=1/5,$ (g1) ${a}_{3}=1;$ (g2)${a}_{3}=2;$(g3) ${a}_{3}=3.$ From figures (g1), (g2) and (g3), the values of ${w}_{1,n},{w}_{2,n},{w}_{\xi ,1},{w}_{\xi ,2},{b}_{1},{b}_{2},{b}_{3}$ and ${a}_{2}$ are not changed, while the horizontal Coriolis parameter ${a}_{3}$ takes different values, the propagation direction of the wave is changed with the Coriolis parameter ${a}_{3}$ increasing and the wave rotates clockwise. From contour figures (g4), (g5) and (g6), different values of the horizontal Coriolis parameter ${a}_{3}$ lead to a large change of contour density. From figures (g7) and (g8), when $\xi =0$ and $T=0,$ different values of the horizontal Coriolis parameter ${a}_{3}$ lead to a significant change in their amplitudes and shapes, with the Coriolis parameter ${a}_{3}$ increasing, the Rossby wave amplitude decreases.

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The interaction phenomena of the governing equation is obtained under single layer network model 2–3–1. The structure of the single hidden layer nonlinear neural network of equation (2) is called 2–is1, the first '2' means two variables, the second '3' means two activation functions in the first layer, the last '1' is the desired function.

We can set 2–3–1 model as following

$$\begin{eqnarray}&&n={w}_{1,n}{\sigma }_{1}({\eta }_{1})+{w}_{2,n}{\sigma }_{2}({\eta }_{2})+{w}_{3,n}{\sigma }_{3}({\eta }_{3})+{b}_{4},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray*}&&\left\{\begin{array}{c}{\eta }_{1}={w}_{\xi ,1}\xi +{w}_{T,1}T+{b}_{1},\\ \begin{array}{c}{\eta }_{2}={w}_{\xi ,2}\xi +{w}_{T,2}T+{b}_{2},\\ {\eta }_{3}={w}_{\xi ,3}\xi +{w}_{T,3}T+{b}_{3}.\end{array}\end{array}\right.\end{eqnarray*}$$

Take the activation function as ${\sigma }_{1}({\eta }_{1})={{\eta }_{1}}^{2},{\sigma }_{2}({\eta }_{2})={{\eta }_{2}}^{2},$ ${\sigma }_{3}\left({\eta }_{3}\right)={e}^{{\eta }_{3}}$ [56], then exact solution is given as

$$\begin{eqnarray}&&\begin{array}{c}n={w}_{1,n}{\left({w}_{\xi ,1}\xi +{w}_{T,1}T+{b}_{1}\right)}^{2}\\ \,+\,{w}_{2,n}{\left({w}_{\xi ,2}\xi +{w}_{T,2}T+{b}_{2}\right)}^{2}\\ \,+\,{w}_{3,n}{\left({w}_{\xi ,3}\xi +{w}_{T,3}T+{b}_{3}\right)}^{2}+{b}_{4},\end{array}\end{eqnarray} \tag{ 12 }$$

Where ${b}_{k}$ and ${w}_{i,j}\left(i=\xi ,T,j=1,2,\cdots \right)$ are real constants.

A group of equations can be created by substituting (12) into the bilinear form (6), and results into a group of equations, the solution of the equations can be obtained as follows

$$\begin{eqnarray*}&&{w}_{\xi ,1}={w}_{\xi ,2}=-\displaystyle \frac{{w}_{T,1}}{3{a}_{2}{w}_{\xi ,3}^{2}+{a}_{3}}\end{eqnarray*}$$

$$\begin{eqnarray*}&&{w}_{T,2}={w}_{T,1}\end{eqnarray*}$$

$$\begin{eqnarray}&&{w}_{T,3}=-{a}_{2}{w}_{\xi ,3}^{3}-{a}_{3}{w}_{\xi ,3}\end{eqnarray} \tag{ 13 }$$

substituting (13) into (5), the interaction solution is obtained:

$$\begin{eqnarray}&&u=2{(\mathrm{ln}\,n)}_{\xi \xi },\end{eqnarray} \tag{ 14 }$$

here $n={w}_{1,n}{\left(-\displaystyle \frac{{w}_{T,1}}{3{a}_{2}{w}_{\xi ,3}^{2}+{a}_{3}}\xi +{w}_{T,2}T+{b}_{1}\right)}^{2}$ + ${w}_{2,n}{\left(-\displaystyle \frac{{w}_{T,1}}{3{a}_{2}{w}_{\xi ,3}^{2}+{a}_{3}}\xi +{w}_{T,2}T+{b}_{2}\right)}^{2}$ $+{w}_{3,n}{\left({w}_{\xi ,3}\xi +\left(-{a}_{2}{w}_{\xi ,3}^{3}-{a}_{3}{w}_{\xi ,3}\right)T+{b}_{3}\right)}^{2}+{b}_{4}.$

Selecting appropriate parameters for equation (14), the dynamic behavior of the interaction phenomenon consisting of rogue wave and soliton wave can be demonstrated in figure 5 with the parameters ${w}_{1,n}=$ ${w}_{2,n}={w}_{3,n}=$ ${w}_{T,1}={w}_{\xi ,3}=1,$ ${b}_{1}\,=\,$

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${b}_{2}={b}_{3}={b}_{4}=1,{a}_{2}=1,$

${a}_{3}=-1;$

${a}_{3}=0;$

${a}_{3}=$

$1.$

${w}_{2,n},{w}_{3,n},{w}_{T,1},{w}_{\xi ,3},{b}_{1},{b}_{2},{b}_{3},{b}_{4}$

${a}_{2}$

${a}_{3}$

${a}_{3}$

${a}_{3}$

$\xi =0$

$T=0,$

${a}_{3}$

${a}_{3}$

The exact solutions of the extended governing equation under double layers are obtained, the double hidden layer network model 2–2–2–1 is illustrated in figure 6. We call it 2–2–2–1, the first '2' means two variables, the second '2' means two activation functions in the first layer, the third '2' means two activation functions in the second layer, the last '1' is the desired function.

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We can set 2–2–2–1 model as following

$$\begin{eqnarray}&&n={w}_{3,n}{\sigma }_{3}({\eta }_{3})+{w}_{4,n}{\sigma }_{4}({\eta }_{4})+{b}_{5},\end{eqnarray} \tag{ 15 }$$

Here $\left\{\begin{array}{l}{\eta }_{1}={w}_{\xi ,1}\xi +{w}_{T,1}T+{b}_{1},\\ \begin{array}{c}{\eta }_{2}={w}_{\xi ,2}\xi +{w}_{T,2}T+{b}_{2},\\ {\eta }_{3}={w}_{2,3}{\sigma }_{2}({\eta }_{2})+{w}_{1,3}{\sigma }_{1}({\eta }_{1})+{b}_{3},\\ {\eta }_{4}={w}_{2,4}{\sigma }_{2}({\eta }_{2})+{w}_{1,4}{\sigma }_{1}({\eta }_{1})+{b}_{4}.\end{array}\end{array}\right.$take the activation function as ${\sigma }_{1}({\eta }_{1})=\,\cos ({\eta }_{1}),{\sigma }_{2}({\eta }_{2})={e}^{{\eta }_{2}},{\sigma }_{3}({\eta }_{3})={\eta }_{3}^{2},{\sigma }_{4}({\eta }_{4})={\eta }_{4}^{2}$ [60], then the exact solution is given as

$$\begin{eqnarray}&&\begin{array}{c}n={w}_{3,n}{\left({w}_{2,3}{e}^{{w}_{\xi ,2}\xi +{w}_{T,2}T+{b}_{2}}+{w}_{1,3}\,\cos \left({w}_{\xi ,1}\xi +{w}_{T,1}T+{b}_{1}\right)+{b}_{3}\right)}^{2}+\\ {w}_{4,n}{\left({w}_{2,4}{e}^{{w}_{\xi ,2}\xi +{w}_{T,2}T+{b}_{2}}+{w}_{1,4}\,\cos \left({w}_{\xi ,1}\xi +{w}_{T,1}T+{b}_{1}\right)+{b}_{4}\right)}^{2}+{b}_{5},\end{array}\end{eqnarray} \tag{ 16 }$$

Where ${b}_{k}$ and ${w}_{i,j}$ are real constants.

A group of nonlinear algebraic equation can be created by substituting 2–2–2–1 model (16) into the bilinear form (6), and results into a group of equations, the solution of the equations can be obtained as follows

$$\begin{eqnarray}&&{\rm{Case}}\,1\,{w}_{T,1}=4{w}_{\xi ,1}^{3}{a}_{2}-{w}_{\xi ,1}{a}_{3},{w}_{1,4}=-{w}_{1,3},{w}_{3,n}=-{w}_{4,n}.\end{eqnarray} \tag{ 17 }$$

Substituting (16) (17) into transformation (5) and selecting appropriate parameters, the dynamic behavior of double hidden layer exact solution can be demonstrated in figure 7 with the parameters ${w}_{\xi ,1}=2,{w}_{T,2}=1,{w}_{\xi ,2}=2,$ ${w}_{2,3}=1,{w}_{1,3}=5,{w}_{2,4}=1,$ ${w}_{4,n}=2,{a}_{2}={a}_{3}=1,{b}_{k}=0\left(k=1\ldots 5\right).$ From figure 7, it can be observed that it is a periodic rational function solution.

$$\begin{eqnarray}&&{\rm{Case}}\,2\,{w}_{T,1}=0,{w}_{\xi ,1}=0,{w}_{T,2}=-4{w}_{\xi ,2}^{3}{a}_{2}-{w}_{\xi ,2}{a}_{3},{w}_{1,4}=-\displaystyle \frac{{w}_{1,3}{w}_{3,n}}{{w}_{4,n}}.\end{eqnarray} \tag{ 18 }$$

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Substituting (16) (18) into transformation (5) and selecting appropriate parameters, the dynamic behavior of double hidden layer exact solution can be demonstrated in figure 8 with the parameters: ${w}_{\xi ,2}=1,{w}_{2,3}=1,{w}_{1,3}=2,{w}_{2,4}=1,{w}_{3,n}=1,{w}_{4,n}=1,{a}_{2}=$ ${a}_{3}=1,{b}_{k}=0(k=1,\ldots ,5).$ From figure 8, it can be observed that it is a bright soliton solution, and has general traveling wave properties.

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The exact solutions of the extended governing equation under double layers are obtained, the double hidden layer network model 2–2–3–1 is illustrated in figure 9. We call it 2–2–3–1, the first '2' means two variables, the second '2' means two activation functions in the first layer, the third '3' means three activation functions in the second layer, the last '1' is the desired function.

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We can set 2–2–3–1 model as following

$$\begin{eqnarray}&&n={w}_{3,n}{\sigma }_{3}({\eta }_{3})+{w}_{4,n}{\sigma }_{4}({\eta }_{4})+{w}_{5,n}{\sigma }_{5}({\eta }_{5})+{b}_{6},\end{eqnarray} \tag{ 19 }$$

Here $\left\{\begin{array}{l}{\eta }_{1}={w}_{\xi ,1}\xi +{w}_{T,1}T+{b}_{1},\\ \begin{array}{c}{\eta }_{2}={w}_{\xi ,2}\xi +{w}_{T,2}T+{b}_{2},\\ {\eta }_{3}={w}_{2,3}{\sigma }_{2}({\eta }_{2})+{w}_{1,3}{\sigma }_{1}({\eta }_{1})+{b}_{3},\\ {\eta }_{4}={w}_{2,4}{\sigma }_{2}({\eta }_{2})+{w}_{1,4}{\sigma }_{1}({\eta }_{1})+{b}_{4},\\ {\eta }_{5}={w}_{2,5}{\sigma }_{2}({\eta }_{2})+{w}_{1,5}{\sigma }_{1}({\eta }_{1})+{b}_{5}.\end{array}\end{array}\right.$take the activation function as ${\sigma }_{1}({\eta }_{1})=\,\cos ({\eta }_{1}),{\sigma }_{2}({\eta }_{2})={e}^{{\eta }_{2}},{\sigma }_{3}({\eta }_{3})={\eta }_{3}^{2},{\sigma }_{4}({\eta }_{4})={\eta }_{4}^{2},$ ${\sigma }_{5}({\eta }_{5})={\eta }_{5}^{2},$ then exact solution is given as

$$\begin{eqnarray}&&\begin{array}{c}n={w}_{3,n}{\left({w}_{2,3}{e}^{{w}_{\xi ,2}\xi +{w}_{T,2}T+{b}_{2}}+{w}_{1,3}\,\cos \left({w}_{\xi ,1}\xi +{w}_{T,1}T+{b}_{1}\right)+{b}_{3}\right)}^{2}+\\ {w}_{4,n}{\left({w}_{2,4}{e}^{{w}_{\xi ,2}\xi +{w}_{T,2}T+{b}_{2}}+{w}_{1,4}\,\cos \left({w}_{\xi ,1}\xi +{w}_{T,1}T+{b}_{1}\right)+{b}_{4}\right)}^{2}+\\ {w}_{5,n}{\left({w}_{2,5}{e}^{{w}_{\xi ,2}\xi +{w}_{T,2}T+{b}_{2}}+{w}_{1,1}\,\cos \left({w}_{\xi ,1}\xi +{w}_{T,1}T+{b}_{1}\right)+{b}_{5}\right)}^{2}+{b}_{6},\end{array}\end{eqnarray} \tag{ 20 }$$

where ${b}_{k}$ and ${w}_{i,j}$ are real constants.

A group of nonlinear algebraic equation can be created by substituting 2–2–3–1 model (20) into the bilinear form (6), and results in a group of equations, the solution of the equations can be obtained as follows

Case 1

$$\begin{eqnarray}&&{w}_{T,2}=-2\left(4{w}_{T,1}^{3}{a}_{2}+{w}_{T,1}{a}_{3}\right),{w}_{1,5}=0,{w}_{5,n}=-\frac{{w}_{2,3}^{2}+{w}_{2,5}^{2}}{{w}_{2,5}^{2}}.\end{eqnarray} \tag{ 21 }$$

Substituting (20) (21) into transformation (5) and selecting appropriate parameters, the dynamic behavior of double hidden layer exact solution can be demonstrated in figure 10 with the parameters ${w}_{T,1}=1/2,{w}_{\xi ,1}=-2,{w}_{\xi ,2}=1,{w}_{2,3}=1,{w}_{1,3}=0,{w}_{2,4}=1,$ ${w}_{1,4}=2,{w}_{2,5}=1,{w}_{3,n}=1,{w}_{4,n}=1,{a}_{2}=0,{a}_{3}=1,{b}_{k}=0(k=1,\ldots ,6).$ From figure 10, it can be observed that it is a periodic rational function solution.

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Case 2

$$\begin{eqnarray}&&{w}_{T,2}=-8{w}_{T,1}^{3}{a}_{2}-2{w}_{T,1}{a}_{3},{w}_{2,3}=0,{w}_{2,5}=0,{w}_{1,5}=0.\end{eqnarray} \tag{ 22 }$$

Substituting (20) (22) into transformation (5) and selecting appropriate parameters, the dynamic behavior of double hidden layer exact solution can be demonstrated in figure 11 with the parameters ${w}_{T,1}=-1/2,{w}_{\xi ,1}=8,{w}_{\xi ,2}=-1,{w}_{1,3}=0,{w}_{2,4}=0,{w}_{1,4}=$ $2,{w}_{3,n}={w}_{4,n}={w}_{5,n}=1,{a}_{2}=2,{a}_{3}=2,{b}_{k}=0(k=1,\ldots ,6).$ From figure 11, it can be observed that it is a periodic rational function solution.

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The exact solutions of the extended governing equation under three layers are obtained, the three hidden layers network model 2–2–2–2–1 is illustrated in figure 12. We call it 2–2–2–2–1, the first '2' means two variables, the second '2' means two activation functions in the first layer, the third '2' means two activation functions in the second layer, the fourth '2' means two activation functions in the third layer, the last '1' is the desired function.

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We can set 2–2–2–2–1 model as following

$$\begin{eqnarray}&&n={w}_{5,n}{\sigma }_{5}({\eta }_{5})+{w}_{6,n}{\sigma }_{6}({\eta }_{6})+{b}_{7},\end{eqnarray} \tag{ 23 }$$

Here $\left\{\begin{array}{l}{\eta }_{1}={w}_{\xi ,1}\xi +{w}_{T,1}T+{b}_{1},\\ \begin{array}{c}{\eta }_{2}={w}_{\xi ,2}\xi +{w}_{T,2}T+{b}_{2},\\ {\eta }_{3}={w}_{2,3}{\sigma }_{2}({\eta }_{2})+{w}_{1,3}{\sigma }_{1}({\eta }_{1})+{b}_{3},\\ {\eta }_{4}={w}_{2,4}{\sigma }_{2}({\eta }_{2})+{w}_{1,4}{\sigma }_{1}({\eta }_{1})+{b}_{4},\\ {\eta }_{5}={w}_{3,5}{\sigma }_{3}({\eta }_{3})+{w}_{4,5}{\sigma }_{4}({\eta }_{4})+{b}_{5},\\ {\eta }_{6}={w}_{3,6}{\sigma }_{3}({\eta }_{2})+{w}_{4,6}{\sigma }_{4}({\eta }_{1})+{b}_{6}.\end{array}\end{array}\right.$

Take the activation function as ${\sigma }_{1}({\eta }_{1})=\,\cos ({\eta }_{1}),{\sigma }_{2}({\eta }_{2})={e}^{{\eta }_{2}},{\sigma }_{3}({\eta }_{3})={\eta }_{3}^{2},$

${\sigma }_{4}({\eta }_{4})={\eta }_{4}^{2},{\sigma }_{5}({\eta }_{5})={\eta }_{5}^{2},{\sigma }_{6}({\eta }_{6})={\eta }_{6}^{2}$ [60], then exact solution is given as

$$\begin{eqnarray}&&\begin{array}{c}n={w}_{5,n}{\left({w}_{3,5}\left({w}_{2,3}{e}^{{w}_{\xi ,2}\xi +{w}_{T,2}T+{b}_{2}}+{w}_{1,3}\,\cos \left({w}_{\xi ,1}\xi +{w}_{T,1}T+{b}_{1}\right)+{b}_{3}\right)\right.}^{2}+\\ \,{w}_{4,5}{\left.{\left({w}_{2,4}{e}^{{w}_{\xi ,2}\xi +{w}_{T,2}T+{b}_{2}}+{w}_{1,4}\,\cos \left({w}_{\xi ,1}\xi +{w}_{T,1}T+{b}_{1}\right)+{b}_{4}\right)}^{2}+{b}_{5}\right)}^{2}+\\ \,{w}_{6,n}{\left({w}_{3,6}\left({w}_{2,3}{e}^{{w}_{\xi ,2}\xi +{w}_{T,2}T+{b}_{2}}+{w}_{1,3}\,\cos \left({w}_{\xi ,1}\xi +{w}_{T,1}T+{b}_{1}\right)+{b}_{3}\right)\right.}^{2}+\\ \,{w}_{4,6}{\left.{\left({w}_{2,4}{e}^{{w}_{\xi ,2}\xi +{w}_{T,2}T+{b}_{2}}+{w}_{1,4}\,\cos \left({w}_{\xi ,1}\xi +{w}_{T,1}T+{b}_{1}\right)+{b}_{4}\right)}^{2}+{b}_{6}\right)}^{2}+{b}_{7},\end{array}\end{eqnarray} \tag{ 24 }$$

where ${b}_{k}$ and ${w}_{i,j}$ are real constants.

A group of nonlinear algebraic equation can be created by substituting 2–2–2–2–1 model (24) into the bilinear form (6), and results in a group of equations, the solution of the equations can be obtained as follows

Case 1

$$\begin{eqnarray}&&{w}_{1,3}=\frac{25{w}_{4,6}(-1+{w}_{4,6}^{4})}{4{w}_{T,1}},{w}_{6,n}=-\frac{1}{{w}_{4,6}^{2}}.\end{eqnarray} \tag{ 25 }$$

Substituting (24) (25) into transformation (5) and selecting appropriate parameters, the dynamic behavior of double hidden layer exact solution can be demonstrated in figure 13 with the parameters ${w}_{\xi ,1}={w}_{T,2}={w}_{4,5}={w}_{3,5}={w}_{1,4}={w}_{4,6}=1,{w}_{2,3}=0,{w}_{T,1}={w}_{\xi ,2}={w}_{2,4}={w}_{3,6}={w}_{5,n}=2,{a}_{2}=1,{a}_{3}=-1,$ $\,{b}_{k}=0(k=1,\ldots ,7).$ From figure 13, it can be observed that it is a periodic rational function solution.

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Case 2

$$\begin{eqnarray}&&{w}_{T,1}=0,{w}_{\xi ,1}=0,{w}_{6,n}=-{w}_{5,n},{w}_{T,2}=-4{w}_{\xi ,2}^{3}.\end{eqnarray} \tag{ 26 }$$

Substituting (24) (26) into transformation (5) and selecting appropriate parameters, the dynamic behavior of double hidden layer exact solution can be demonstrated in figure 14 with the parameters ${w}_{2,3}=2,{w}_{1,4}=3,{w}_{\xi ,2}={w}_{1,3}={w}_{2,4}={w}_{3,5}={w}_{4,5}={w}_{3,6}=$ ${w}_{4,6}={w}_{5,n}=1,{a}_{2}={a}_{3}=1,{b}_{k}=0(k=1,\ldots ,7).$ From figure 14, it can be observed that it is a dark soliton solution.

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Case 3

$$\begin{eqnarray}&&{w}_{T,1}=4{w}_{\xi ,1}^{3},{w}_{6,n}=-{w}_{5,n},{w}_{T,2}=0,{w}_{\xi ,2}=0.\end{eqnarray} \tag{ 27 }$$

Substituting (24) (27) into transformation (5) and selecting appropriate parameters, the dynamic behavior of double hidden layer exact solution can be demonstrated in figure 15 with the parameters ${w}_{2,3}=2,{w}_{1,4}=3,{w}_{\xi ,1}={w}_{1,3}={w}_{2,4}={w}_{3,5}={w}_{4,5}={w}_{3,6}\,=$ ${w}_{4,6}={w}_{5,n}=1,{a}_{2}={a}_{3}=1,{b}_{k}=0(k=1,\ldots ,7)..$ From figure 15, it can be observed that it is a periodic rational function solution.

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In summary, based on the existing ocean model, this manuscript studies the governing equation with complete Coriolis parameter through the five steps of theoretical analysis, and obtains the rational function solutions of the equation. We mainly use BNNM framework, which is not limited to single layer network, and try to build deeper and broader network structures (such as ''2–2–2–1'', ''2–2–3–1'' and ''2–2–2–1'' models) to build test functions. And with the assistance of symbolic calculation method, some exact solutions are represented, such as rogue waves solution, interaction between rogue wave and soliton wave, bright and dark soliton solutions and some periodic rational function solutions. These solutions are new solutions. We studied the effect of the horizontal component of Coriolis parameter ${a}_{3}$ on rogue wave and interaction wave, and it is concluded that the change of the horizontal component of Coriolis parameter ${a}_{3}$ will affect the propagation direction and amplitude of Rossby waves. After each set of solutions, we show their 3D stereograms, density planes and 2D planes with the aid of Mathematica.

In the future, we can get more and more meaningful exact solutions by trying to use new single-layer test functions and deeper and wider multi-layer test functions. In this work, we have encountered many problems, the first is to solve the complex equations, the second is to select appropriate parameters to dynamically display these exact solutions, the third is to give more understanding of these new solutions, these will be our follow-up further research work.

This project is supported by the National Natural Science Foundation of China (12102205, 12262025). The Natural Science Foundation of Inner Mongolia Autonomous Region (2022QN01003). The Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (NJYT23099, NMGIRT2208). Inner Mongolia Autonomous Region University research Project (NJZY23116). The Basic Research Operation Funds for Universities directly under Inner Mongolia Autonomous Region (22BR0902). The Program for improving the Scientific Research Ability of Youth Teachers of Inner Mongolia Agricultural University (BR220126).

No new data were created or analysed in this study.

All authors of this article declare no conflict of interest.