In the field of the soliton theory, it is vital to find the exact solution of nonlinear evolution equations, including soliton solutions, breathers solution, rogue wave solutions, lump solutions, period line wave solutions, interaction solutions, etc. Among these exact solutions, rogue wave solutions are a class of analytic rational solutions which can reach very high amplitudes in a short time. Because of its great destructive power and unpredictability, the rouge wave has been found in the ocean for a long time. As another non-linear scientific revolution after soliton, the rouge wave has attracted wide attention of researchers, and has gradually risen in the category of social and scientific contexts, such as Bose-Einstein condensates[1-3], oceanic[4, 5], even in finance[6] and hydrodynamics[7]. In 1983, some analytical solutions of nonlinear system (NLS) equations were obtained by D. H. Peregrine[8, 9], then the rouge wave solutions were researched for NLS equation[10, 11]. In recent years, the rouge wave solution of the nonlinear Schrödinger equation [12] and the coupled Hirota systems[13] have been studied. And the abound rogue wave type solutions to the extended (3+1)-dimensional Jimbo–Miwa equation are obtained by Liu, Yang, et al. in[14]. Furthermore, breathers [15] are considered as a special type of soliton, which can periodically occur and propagate in a localized and oscillatory way. Breathers can be divided into three types: generalized breathers, Akhmediev breathers[16] and Kuznetsov–Ma breathers[17, 18]. In 2019, Liu obtained breathers solution and higher order breathers solution of some equations in [19]. Recently, Mandel and Scheider[20] obtained the breathers solution of nonlinear equation by variational method. And Guo and Scheider studied the higher order breathers solutions of Jimbo-Miwa equation and the breathers solutions to the cubic Klein Gordon equation in[21] and[22], respectively.

High rogue waves play an critical role in the study of phase shift, propagation direction, shape and energy distribution, which makes the significance of these solutions a very practical research topic. In addition, the dynamics feature has become a central subject because it can reveal the collision, rebound and absorption characteristics of particles in the process of interaction, as well as the underlying laws of physics.

In this paper, we would like to consider the $ (4+1) $-dimensional nonlinear Fokas equation as follows:

$ \begin{equation} {u_{xt}} - \frac{1}{4}{u_{xxxy}} + \frac{1}{4}{u_{xyyy}} + \frac{3}{2}{({u^2})_{xy}} - \frac{3}{2}{u_{zw}} = 0. \end{equation} $

(1.1)

The nonelasticity of the equation (1.1) and the interaction between elasticity were described[23, 24]. The meaning of the Fokas equation in water waves originates in the physical application of Kadomtsev-Petviashvili (KP) equation and Davey-Stewartson (DS) equation. The latter two equations are used to describe the surface wave and internal wave in channel or straits with different depths and widths, respectively[25-28].

The (4+1)-dimensional Fokas Eq.(1.1) has been studied by different scholars. He et al.[29, 30] obtained a few new exact solutions of Eq.(1.1) by applying extended F-expansion method. Zhang et al.[31] constructed multiple-soliton of Eq.(1.1). And Two different methods are used to look for exact traveling wave solutions of a modified simple equation about Eq.(1.1) in Ref [32]. More recently, Cheng et al.[33] have derived lump-type solutions of a reduced Fokas equation utilizing the positive quadratic function method. Wazwaz[34] made use of the simplified Hirota's method[35], whereafter demonstrated a variety of multiple soliton solutions. And the M-lump solutions of the Fokas equation are ascertained by taking limit method on multi-soliton solutions by Zhang and Xia [36]. As far as we know, the high order rogue wave solution, breathers solution and high order breathers solution of the Fokas equation by the simplified Hirota's method have not been given.

The paper is arranged as follows. Section 2, the $ (4+1) $-dimensional Fokas equation is transformed into a (1 + 1) dimensional equation by variable transformation. Then based on the Hirota bilinear form of the $ (1+1) $-dimensional equation, high order rogue wave solutions are constructed. Further, their typical dynamics behaviors are analyzed and explained. In Section 3, by means of the simplified Hirota's method for the $ (4+1) $-dimensional equation, breathers solution and high order breathers solution are obtained, then the corresponding dynamics behaviors are illustrated. Some conclusions will be given in the final section.

If one sets $ X = \alpha x + \beta y + \theta t, Z = \gamma z + \varepsilon w $ in Eq.(1.1), Eq.(1.1) can be transformed into the following $ (1+1) $-dimensional equation

$ \begin{equation} \theta {u_{XX}} + \frac{1}{4}\beta ({\beta ^2} - {\alpha ^2}){u_{XXXX}} + 3\beta {(u{u_X})_X} - \frac{{3\gamma \varepsilon }}{{2\alpha }}{u_{ZZ}} = 0, \end{equation} $

(2.1)

where $ \alpha , \beta , \theta , \gamma $ and $ \varepsilon $ are five real parameters.

Under the logarithmic transformation

$ \begin{equation} u = ({\beta ^2} - {\alpha ^2}){(\ln f)_{XX}}+u_0, \end{equation} $

(2.2)

where $ \left| \alpha \right| \ne \left| \beta \right| $ is a constant, the bilinear form to Eq.(2.1) is generated as

$ \begin{equation} ((\theta + 3\beta {u_0})D_X^2 + \frac{1}{4}\beta ({\beta ^2} - {\alpha ^2})D_X^4 - \frac{{3\gamma \varepsilon }}{{2\alpha }}D_Z^2)f \cdot f = 0, \end{equation} $

(2.3)

where the bilinear operator[37] $ D $ is defined by

$ \begin{equation} \begin{array}{l} D_X^mD_Z^nf(X, Z)g(X', Z') = {(\frac{\partial }{{\partial X}} - \frac{\partial }{{\partial X'}})^m}{(\frac{\partial }{{\partial Z}} - \frac{\partial }{{\partial Z'}})^n}f(X, Z)g(X', Z')\left| {_{X' = X, Z' = Z}} \right., \end{array} \end{equation} $

(2.4)

$ m $ and $ n $ are nonnegative integers. Eq.(2.1) is reduced to the bilinear equation

$ \begin{equation} \begin{array}{l} 2(\theta + 3\beta {u_0})({f_{XX}}f - f_X^2) - \frac{{3\gamma \varepsilon }}{\alpha }({f_{ZZ}}f - f_Z^2) + \frac{1}{2}\beta ({\beta ^2} - {\alpha ^2})({f_{XXXX}}f - 4{f_{XXX}}{f_X} + f_{XX}^2) = 0. \end{array} \end{equation} $

(2.5)

An improved ansatz was proposed about multiple rogue solutions in 2018, as shown below[13]

$ \begin{equation} f = {F_{n + 1}}(X, Z) + 2{\alpha _1}Z{P_n}(X, Z) + 2{\beta _1}X{Q_n}(X, Z) + (\alpha _1^2 + \beta _1^2){F_{n - 1}}(X, Z), \end{equation} $

(2.6)

with

$ \begin{equation} \begin{array}{l} {F_n}(X, Z){\text{ = }}\sum\limits_{k = 0}^{n(n + 1)/2} {\sum\limits_{i = 0}^k {{a_{n(n + 1) - 2k, 2i}}{X^{n(n + 1) - 2k}}{Z^{2i}}} } , \\ {P_n}(X, Z) = \sum\limits_{k = 0}^{n(n + 1)/2} {\sum\limits_{i = 0}^k {{b_{n(n + 1) - 2k, 2i}}{X^{n(n + 1) - 2k}}{Z^{2i}}} } , \\ {Q_n}(X, Z) = \sum\limits_{k = 0}^{n(n + 1)/2} {\sum\limits_{i = 0}^k {{c_{n(n + 1) - 2k, 2i}}{X^{n(n + 1) - 2k}}{Z^{2i}}} } , \end{array} \end{equation} $

(2.7)

$ {F_0} = 1, {F_{ - 1}} = {P_0} = {Q_0} = 0, $ where $ {a_{m, l}}, {b_{m, l}}, {c_{m, l}}(m, l \in \{ 0, 2, 4, ..., n(n + 1)/2\} ) $ and $ {\alpha _1}, {\beta _1} \in \mathbb{R} $. The coefficients $ {a_{m, l}}, {b_{m, l}}, {c_{m, l}} $ can be determined and the wave center can be controlled by arbitrary constants $ {\alpha _1}, {\beta _1} $. Then by substituting these values into (2.2), some rational solutions of the Eq. (1.1) are obtained. Rogue wave solutions come from these solutions. And this kind of rogue wave is localized in $ X $ and $ Z $.

Through research and Maple direct symbolic computations, we derive the central controllable higher order rogue wave solutions of the Eq. (2.1).

We will receive the first order rogue wave solution to the Eq.(2.1) if one sets $ n = 0 $ in (2.6)

$ \begin{equation} f = {F_1}(X, Z) = {X^2} + {a_{0, 2}}{Z^2} + {a_{0, 0}}. \end{equation} $

(2.8)

Then substituting (2.8) into (2.5) and we conduct a direct symbolic computation, a polynomial equation system is obtained. By solving the polynomial equations, we get

$ \begin{equation} \{ {{a_{0, 0}} = \frac{{3\beta ({\alpha ^2} - {\beta ^2})}}{{4(3\beta {u_0} + \theta )}}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {a_{0, 2}} = - \frac{{2(3\beta {u_0} + \theta )\alpha }}{{3\gamma \varepsilon }}} \}. \end{equation} $

(2.9)

Further

$ \begin{equation} f = {F'_1}(X, Z) = {(X - {\alpha _1})^2} - \frac{{2(3\beta {u_0} + \theta )\alpha }}{{3\gamma \varepsilon }}{(Z - {\beta _1})^2} + \frac{{3\beta ({\alpha ^2} - {\beta ^2})}}{{4(3\beta {u_0} + \theta )}}, \end{equation} $

(2.10)

is a solution to Eq. (2.5), where $ {\alpha _1}, {\beta _1} \in \mathbb{R} $.

When $ {a_{0, 0}} = \frac{{3\beta ({\alpha ^2} - {\beta ^2})}}{{4(3\beta {u_0} + \theta )}} > 0 $ and $ {a_{0, 2}} = \frac{{2(3\beta {u_0} + \theta )\alpha }}{{3\gamma \varepsilon }} < 0, $ Eq.(2.5) will have a positive polynomial solution. Substituting (2.10) into (2.2), the rogue wave solution of Eq. (2.1) is obtained

$ \begin{equation} u = {u_0} + ({\beta ^2} - {\alpha ^2})(\frac{2}{{{F'_1}(X, Z)}} - \frac{{4{{(X - {\alpha _1})}^2}}}{{{F'_1}{{(X, Z)}^2}}}). \end{equation} $

(2.11)

Figure. 1 shows the first order rogue wave. From the figure, we can clearly see that the rogue wave has three peaks. Two of the peaks are higher and the other is lower than the water level. Two parameters $ ({\alpha _1}, {\beta _1}) $ are used to control the center of the rogue wave. By computation we can find that when $ {a_{0, 0}} = \frac{{3\beta ({\alpha ^2} - {\beta ^2})}}{{4(3\beta {u_0} + \theta )}} > 0 $ and $ {a_{0, 2}} = \frac{{2(3\beta {u_0} + \theta )\alpha }}{{3\gamma \varepsilon }} < 0 $. The rogue wave reaches the peaks at the point $ ({\alpha _1}, {\beta _1}) $ and $ (\frac{{ \pm 3\sqrt {\beta (3\beta {u_0} + \theta )({\alpha ^2} - {\beta ^2})} + (6\beta {u_0} + 2\theta ){\alpha _1}}}{{6\beta {u_0} + 2\theta }}, {\beta _1}) $ in the $ (X, Z) $-plane, respectively. The first order rogue wave have extreme values $ \frac{{6\beta {u_0} + \theta }}{{3\beta }} $ and $ \frac{{ - 21\beta {u_0} - 8\theta }}{{3\beta }} $. In figures. 1 (a) and 1 (c), it is shown that the rogue waves are concentrated near $ (0, 0) $ and $ (10, 10) $, respectively.

By (2.6) we can derive

$ \begin{equation} f = {F'_2}(X, Z) = {F_2}(X, Z) + 2{\alpha _1}Z{P_1}(X, Z) + 2{\beta _1}X{Q_1}(X, Z) + (\alpha _{^1}^2 + \beta _1^2){F_0}(X, Z), \end{equation} $

(2.12)

where

$ \begin{equation} \begin{array}{l} {F_2}(X, Z) = {X^6} + {a_{4, 0}}{X^4} + {a_{4, 2}}{Z^2}{X^4} + ({a_{2, 0}} + {a_{2, 2}}{Z^2} + {a_{2, 4}}{Z^4}){X^2} + {a_{0, 0}} + {a_{0, 4}}{Z^4} + {a_{0, 6}}{Z^6}, \\ {P_1}(X, Z) = {b_{0, 0}} + {b_{0, 2}}{X^2} + {b_{2, 0}}{Z^2}, {Q_1}(X, Z) = {c_{0, 0}} + {c_{0, 2}}{Z^2} + {c_{2, 0}}{X^2}, \\ {F_0}(X, Z) = 1. \end{array} \end{equation} $

(2.13)

Substituting (2.12) into (2.5) and setting all the coefficients of the different powers of $ {Z^p}{X^q} $ to zero. The set of solutions read as

$ \begin{equation} \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} {a_{0, 0}} = \frac{1}{{192\alpha {{(3\beta {u_0} + \theta )}^3}}}(5625{\alpha ^3}{\beta ^3}({\alpha ^4} - 3{\alpha ^2}{\beta ^2} + 3{\beta ^4}) - 5625{\beta ^9}\alpha - 192\alpha (\beta _1^2 + \alpha _1^2\\ - \beta _1^2c_{2, 0}^2){(3\beta {u_0} + \theta )^3} - 288\varepsilon \gamma \alpha _1^2b_{2, 0}^2{(\beta {u_0} + \frac{\theta }{3})^2}), {a_{0, 2}} = - \frac{{475\alpha {{({\alpha ^2} - {\beta ^2})}^2}{\beta ^2}}}{{24\varepsilon \gamma (3\beta {u_0} + \theta )}}, \\ {a_{0, 4}} = \frac{{17(3\beta {u_0} + \theta ){\alpha ^2}({\alpha ^2} - {\beta ^2})\beta }}{{9{\varepsilon ^2}{\gamma ^2}}}, {a_{2, 0}} = \frac{{ - 125{{({\alpha ^2} - {\beta ^2})}^2}{\beta ^2}}}{{16{{(3\beta {u_0} + \theta )}^2}}}, {a_{0, 6}} = \frac{{8{\alpha ^3}{{(3\beta {u_0} + \theta )}^3}}}{{ - 27{\gamma ^3}{\varepsilon ^3}}}, \end{array}\\ {{a_{2, 2}} = \frac{{15\beta \alpha ({\beta ^2} - {\alpha ^2})}}{{\varepsilon \gamma }}, {a_{2, 4}} = \frac{{4{\alpha ^2}{{(3\beta {u_0} + \theta )}^2}}}{{3{\varepsilon ^2}{\gamma ^2}}}, {a_{4, 0}} = \frac{{25\beta ({\alpha ^2} - {\beta ^2})}}{{4(3\beta {u_0} + \theta )}}, {b_{2, 0}} = {b_{2, 0}}}, \\ {{b_{0, 0}} = \frac{{5\beta ({\alpha ^2} - {\beta ^2}){b_{2, 0}}}}{{12(3\beta {u_0} + \theta )}}, {b_{0, 2}} = \frac{{2{b_{2, 0}}(3\beta {u_0} + \theta )\alpha }}{{9\gamma \varepsilon }}, {c_{0, 0}} = \frac{{\beta ({\beta ^2} - {\alpha ^2}){c_{2, 0}}}}{{4(3\beta {u_0} + \theta )}}, {c_{2, 0}} = {c_{2, 0}}, }\\ {{c_{0, 2}} = \frac{{2\alpha (3\beta {u_0} + \theta ){c_{2, 0}}}}{{\gamma \varepsilon }}, {a_{4, 2}} = \frac{{ - 2\alpha (3\beta {u_0} + \theta )}}{{\gamma \varepsilon }}}, \end{array}} \right\} \end{equation} $

(2.14)

where $ {b_{2, 0}}, {c_{2, 0}} \in C $. The second order rogue wave solution of Eq. (2.1) is obtained by Substituting (2.14) and (2.12) into (2.2).

$ \begin{equation} u = {u_0} + ({\beta ^2} - {\alpha ^2}){(\ln {F'_2}(X, Z))_{XX}} \end{equation} $

(2.15)

in which $ X = \alpha x + \beta y + \theta z, Z = \gamma z + \varepsilon w, \alpha , \beta , \theta , \gamma , \varepsilon , {u_0} \in C $ that guarantee the analyticity of $ u $. When $ {\alpha _1} = {\beta _1} = 0 $, the generated rogue wave is shown in figure 2. The second order rogue wave solution wave is concentrated around $ (0, 0) $. When the parameter values $ {\alpha _1} $ and $ {\beta _1} $ reach certain level, the second order waves begin to separate, forming three first order rogue waves. The second-order rogue waves are drawn in figure 3. From (b) and (c) of figure 3, it has a clear display of three first-order rogue waves with three centers forming a triangle. So the second order rogue wave is also called triplet lump wave.

To construct the third order rogue wave solutions to the equation (2.1), we set the function $ f $ as

$ \begin{equation} f = {F'_3}(X, Z) = {F_3}(X, Z) + 2{\alpha _1}Z{P_2}(X, Z) + 2{\beta _1}X{Q_2}(X, Z) + (\alpha _1^2 + \beta _1^2){F_1}(X, Z), \end{equation} $

(2.16)

with

$ \begin{eqnarray*} \label{18} {F_3}(X, Z) & = & {X^{12}} + ({a_{10, 0}} + {a_{10, 2}}{Z^2}){X^{10}} + ({a_{8, 0}} + {a_{8, 2}}{Z^2} + {a_{8, 4}}{Z^4}){X^8} + ({a_{6, 0}}\nonumber\\ && + {a_{6, 2}}{Z^2} + {a_{6, 4}}{Z^4} + {a_{6, 6}}{Z^6}){X^6} + ({a_{4, 0}} + {a_{4, 2}}{Z^2} + {a_{4, 4}}{Z^4} + {a_{4, 6}}{Z^6} + {a_{4, 8}}{Z^8}){X^4} \nonumber\\ &&{ + ({a_{2, 0}} + {a_{2, 2}}{Z^2} + {a_{2, 4}}{Z^4} + {a_{2, 6}}{Z^6} + {a_{2, 8}}{Z^8} + {a_{2, 10}}{Z^{10}}){X^2} + {a_{0, 0}} + {a_{0, 2}}{Z^2}}\nonumber\\ &&{ + {a_{0, 4}}{Z^4} + {a_{0, 6}}{Z^6} + {a_{0, 8}}{Z^8} + {a_{0, 10}}{Z^{10}} + {a_{0, 12}}{Z^{12}}, } \nonumber\\ {P_2}(X, Z) & = & {Z^6} + ({b_{4, 0}} + {b_{4, 2}}{X^2}){Z^4} + ({b_{2, 0}} + {b_{2, 2}}{X^2} + {b_{2, 4}}{X^4}){Z^2}\nonumber\\ && + ({b_{0, 0}} + {b_{0, 2}}{X^2} + {b_{0, 4}}{X^4} + {b_{0, 6}}{X^6}), \end{eqnarray*} $

$ \begin{eqnarray} {Q_2}(X, Z) & = & {X^6} + ({c_{4, 0}} + {c_{4, 2}}{Z^2}){X^4} + ({c_{2, 0}} + {c_{2, 2}}{Z^2} + {c_{2, 4}}{Z^4}){X^2}\\ && + ({c_{0, 0}} + {c_{0, 2}}{Z^2} + {c_{0, 4}}{Z^4} + {c_{0, 6}}{Z^6}), \\ {F_1}(X, Z) & = & {X^2} - \frac{{2(3\beta {u_0} + \theta )\alpha }}{{3\gamma \varepsilon }}{Z^2} + \frac{{3\beta ({\alpha ^2} - {\beta ^2})}}{{4(3\beta {u_0} + \theta )}}. \end{eqnarray} $

(2.17)

Similarly, substituting (2.16) into (2.5), we can obtain the following set of constraining equations for the parameters

$ \begin{equation} \left\{ \begin{array}{l} {a_{0, 0}} = \frac{{6561({\alpha ^2} - {\beta ^2})}}{{2{\alpha ^{43}}{\beta ^{19}}}}\left[ {\frac{{{\text{70212289292481875}}A}}{{{\text{120932352}}}}} \right. - \frac{{878826025{\beta ^{15}}}}{{13436928}} + (\beta _1^2 + \frac{{\alpha _1^2}}{2}){({u_0}\beta + \frac{\theta }{3})^5}{\alpha ^7}\\ \left. { - \frac{{{\gamma ^7}\alpha _1^2{\varepsilon ^7}}}{{256}}} \right], {a_{0, 2}} = \frac{{B - \frac{{145}}{{77}}A}}{{2304\gamma \varepsilon {{(3{u_0}\beta + \theta )}^6}{\alpha ^6}}}, {a_{0, 4}} = \frac{{16391725{\alpha ^{11}}{\beta ^9}{D^4}}}{{1728{\gamma ^2}{\varepsilon ^2}C}}, {a_{0, 6}} = \frac{{199745{D^3}{\alpha ^3}{\beta ^3}}}{{162{\gamma ^3}{\varepsilon ^3}}}, \\ {a_{0, 8}} = \frac{{1445{D^2}C{\alpha ^5}{\beta ^3}}}{{27{\gamma ^4}{\varepsilon ^4}}}, {a_{0, 10}} = \frac{{464{C^2}{D^2}}}{{243{\gamma ^5}{\varepsilon ^5}{\alpha ^{13}}{\beta ^9}}}, {a_{0, 12}} = \frac{{{\text{46656}}{\alpha ^{33}}{\beta ^{15}}}}{{{C^3}{\gamma ^6}{\varepsilon ^6}}}, {a_{2, 8}} = \frac{{760DC}}{{ - 9{\alpha ^5}{\beta ^4}{\gamma ^4}{\varepsilon ^4}}}, \\ {a_{2, 0}} = \frac{{A - B}}{{1536{\alpha ^7}{{(3{u_0}\beta + \theta )}^7}}}, {a_{2, 2}} = \frac{{94325{D^4}\alpha {\beta ^4}}}{{ - 64\gamma \varepsilon {{(3{u_0}\beta + \theta )}^3}}}, {a_{2, 4}} = \frac{{1225{D^3}{\alpha ^6}{\beta ^5}}}{{ - 12{\gamma ^2}{\varepsilon ^2}}}\sqrt {\frac{{\alpha \beta }}{C}} , \\ {a_{2, 6}} = \frac{{17710{D^2}}}{{ - 27\alpha {\gamma ^3}{\varepsilon ^3}}}\sqrt {\frac{C}{{\alpha \beta }}} , {a_{2, 10}} = \frac{{192{{({u_0}\beta + \frac{\theta }{3})}^5}{\alpha ^5}}}{{ - {\gamma ^5}{\varepsilon ^5}}}, {a_{4, 0}} = \frac{{5187875{D^4}{\beta ^{12}}{\alpha ^{16}}}}{{ - 768{C^2}}}.\\ {a_{4, 2}} = \frac{{18375{D^3}{\alpha ^{10}}{\beta ^8}}}{{ - 8\gamma \varepsilon C}}, {a_{4, 4}} = \frac{{18725{D^2}{\alpha ^2}{\beta ^2}}}{{18{\gamma ^2}{\varepsilon ^2}}}, {a_{4, 6}} = \frac{{2920DC}}{{ - 27{\gamma ^3}{\varepsilon ^3}{\alpha ^6}{\beta ^4}}}, {a_{4, 8}} = \frac{{80{C^2}}}{{27{\alpha ^{14}}{\beta ^{10}}{\gamma ^4}{\varepsilon ^4}}}, \\ {a_{6, 0}} = \frac{{18865{D^3}{\beta ^3}}}{{48{{(3{u_0}\beta + \theta )}^3}}}, {a_{6, 2}} = \frac{{4655{D^2}\alpha {\beta ^2}}}{{ - 6(3{u_0}\beta + \theta )\gamma \varepsilon }}, {a_{6, 4}} = \frac{{1540D}}{{9{\alpha ^2}\beta {\gamma ^2}{\varepsilon ^2}}}\sqrt {\frac{C}{{\alpha \beta }}} , {a_{8, 0}} = \frac{{735{D^2}{\beta ^7}{\alpha ^9}}}{{16C}}, \\ {a_{6, 6}} = \frac{{160{\alpha ^3}{{({u_0}\beta + \frac{\theta }{3})}^3}}}{{ - {\gamma ^3}{\varepsilon ^3}}}, {a_{8, 2}} = \frac{{115D\alpha \beta }}{{ - \gamma \varepsilon }}, {a_{8, 4}} = \frac{{20C}}{{3\alpha {\gamma ^2}{\varepsilon ^2}{\beta ^5}}}, {a_{10, 0}} = \frac{{49\beta D{\alpha ^4}{\beta ^2}}}{{2(3{u_0}\beta + \theta )}}\sqrt {\frac{{\alpha \beta }}{C}} , \\ {a_{10, 2}} = \frac{{ - 4(3{u_0}\beta + \theta )\alpha }}{{\gamma \varepsilon }}, {b_{0, 0}} = \frac{{169785{D^3}{\alpha ^{24}}{\beta ^{15}}{\gamma ^3}{\varepsilon ^3}}}{{512{C^3}}}, {b_{0, 2}} = \frac{{17955{D^2}{\beta ^2}{\gamma ^3}{\varepsilon ^3}}}{{128{{(3{u_0}\beta + \theta )}^5}{\alpha ^3}}}, \\ {b_{0, 4}} = \frac{{2835D{\alpha ^{15}}{\beta ^{10}}{\gamma ^3}{\varepsilon ^3}}}{{32C}}, {b_{0, 6}} = \frac{{135{\gamma ^3}{\varepsilon ^3}}}{{8{{(3{u_0}\beta + \theta )}^3}{\alpha ^3}}}, {b_{2, 0}} = \frac{{2205{D^2}{\beta ^8}{\alpha ^{16}}{\gamma ^2}{\varepsilon ^2}}}{{64{C^2}}}, \\ {b_{2, 2}} = \frac{{855D\beta {\gamma ^2}{\varepsilon ^2}}}{{8{{(3{u_0}\beta + \theta )}^3}{\alpha ^2}}}, {b_{2, 4}} = \frac{{ - 45{\gamma ^2}{\varepsilon ^2}}}{{4{\beta ^5}{\alpha ^{11}}}}, {b_{4, 0}} = \frac{{21D\gamma \varepsilon {\alpha ^8}{\beta ^6}}}{{8C}}, {c_{2, 0}} = \frac{{245{D^2}{\beta ^7}{\alpha ^9}}}{{ - 16C}}, \\ {b_{4, 2}} = \frac{{27{\alpha ^8}{\beta ^5}\gamma \varepsilon }}{{2C}}, {c_{0, 0}} = \frac{{12005{D^3}{\beta ^3}}}{{192{{(3{u_0}\beta + \theta )}^3}}}, {c_{0, 2}} = \frac{{535{D^2}\alpha {\beta ^2}}}{{ - 24(3{u_0}\beta + \theta )\gamma \varepsilon }}, {c_{0, 4}} = \frac{{5D}}{{{\alpha ^2}\beta {\gamma ^2}{\varepsilon ^2}}}\sqrt {\frac{C}{{\alpha \beta }}} , \\ {c_{0, 6}} = \frac{{40{{({u_0}\beta + \frac{\theta }{3})}^3}{\alpha ^3}}}{{ - {\gamma ^3}{\varepsilon ^3}}}, {c_{2, 2}} = \frac{{115D\alpha \beta }}{{3\gamma \varepsilon }}, {c_{2, 4}} = \frac{{20C}}{{9{\alpha ^5}{\beta ^7}{\gamma ^2}{\varepsilon ^2}}}, {c_{4, 0}} = \frac{{13D\beta }}{{4(3{u_0}\beta + \theta )}}, \\ {c_{4, 2}} = \frac{6}{{{\alpha ^3}{\beta ^2}\gamma \varepsilon }}\sqrt {\frac{C}{{\alpha \beta }}} , A = 79893275C({\alpha ^8} - 5{\alpha ^6}{\beta ^2} + 10{\alpha ^4}{\beta ^4} - 10{\alpha ^2}{\beta ^6} + 5{\beta ^8}), \\ B = {\text{373248}}\frac{C}{{{\beta ^5}{\alpha ^9}}}(\frac{{79893275{\beta ^{15}}}}{{373248}} + \frac{{C\alpha _1^2}}{{9{\beta ^5}{\alpha ^9}}}){\alpha ^7} + 26244{\gamma ^7}\alpha _1^2{\varepsilon ^7}, \\ C = {\alpha ^9}{\beta ^5}{(3{u_0} + \theta )^2}, D = {\beta ^2} - {\alpha ^2}, \end{array} \right\} \end{equation} $

(2.18)

where $ \alpha, \beta, \theta, \gamma, \varepsilon, {u_0}, {\alpha _1}, {\beta _1} $ are arbitrary constants. Similarly, substituting (2.16) and (2.18) into (2.2), we have the rogue wave solution of Eq.(2.1) as

$ \begin{equation} u = {u_0} + 2\ln {({F'_3}(X, Z))_{XX}}. \end{equation} $

(2.19)

When $ \alpha = - 5, \beta = 2, \theta = 1, \varepsilon = 2, \gamma = 2, {u_0} = 2, {\alpha _1} = 0 $ and $ {\beta _1} = 0 $, from the figure 4 we find that the third order rogue wave has four peaks of wave and three troughs of wave. When we take $ {\alpha _1} = {\beta _1} = 1200000 $ in figure 5, the rogue wave (figure 4) will gradually split into six first order rogue waves. In this case, from the figure, it is found that the structure has six peaks, which form a pentagram.

In this section, We apply the simplified Hirota's bilinear method to investigate the breathers solution of the Eq.(1.1). Substituting

$ \begin{equation} u = {e^{{\theta _i}}}, {\theta _i} = {{k_i}x + {r_i}y + {s_i}z+{q_i}w-{c_i}t}, i = 1, 2, ..., N, \end{equation} $

(3.1)

into the linear terms of (1.1) and solve it. We find that there exists the soliton solution when the following relation is satisfied

$ \begin{equation} {c_i} = - \frac{{k_i^3{r_i} - {k_i}r_i^3 + 6{s_i}{q_i}}}{{4{k_i}}}, i = 1, 2, ..., N. \end{equation} $

(3.2)

Under the transformations

$ \begin{equation} u(x, y, z, w, t) = R{(\ln f(x, y, z, w, t))_{xx}}, \end{equation} $

(3.3)

where

$ \begin{equation} f = 1 + {e^{{k_1}x + {r_1}y + {s_1}z + {q_1}w + \frac{{k_1^3{r_1} - {k_1}r_1^3 + 6{s_1}{q_1}}}{{4{k_1}}}t}}. \end{equation} $

(3.4)

Substituting (3.4) into Eq.(1.1), we find that

$ \begin{equation} m{k_1}^2 = {r_1}^2 , m = R+1, m\neq-1, \end{equation} $

(3.5)

further

$ \begin{equation} {\theta _i} = {k_i}x + \sqrt m {k_i}y + {s_i}z +{q_i}w+ \frac{{\sqrt m (1 - m)k_i^4 + 6{s_i}{q_i}}}{{4{k_i}}}t, \end{equation} $

(3.6)

under the constraint (3.3). Through the above calculation, we obtain the following two-soliton solution

$ \begin{equation} f = 1 + {e^{{\theta _1}}} + {e^{{\theta _2}}} + {a_{12}}{e^{{\theta _1} + {\theta _2}}}. \end{equation} $

(3.7)

Therefore, the solution (3.3) can also be written as

$ \begin{equation} u(x, y, z, w, t) = (m - 1){(\ln f(x, y, z, w, t))_{xx}}. \end{equation} $

(3.8)

Based on (3.7), (3.8), the $ {a_{12}} $ is expressed as

$ \begin{equation} {a_{12}} = \frac{{ - k_1^2k_2^2{{({k_1} - {k_2})}^2}{m^{\frac{3}{2}}} + k_1^2k_2^2{{({k_1} - {k_2})}^2}{m^{\frac{1}{2}}} - 2({k_1}{s_2} - {k_2}{s_1})({k_1}{q_2} - {k_2}{q_1})}}{{ - k_1^2k_2^2{{({k_1} + {k_2})}^2}{m^{\frac{3}{2}}} + k_1^2k_2^2{{({k_1} + {k_2})}^2}{m^{\frac{1}{2}}} - 2({k_1}{s_2} - {k_2}{s_1})({k_1}{q_2} - {k_2}{q_1})}}, \end{equation} $

(3.9)

and the phase shift can be generalized in the following ways

$ \begin{equation} {a_{ij}} = \frac{{ - k_i^2k_j^2{{({k_i} - {k_j})}^2}{m^{\frac{3}{2}}} + k_i^2k_j^2{{({k_i} - {k_j})}^2}{m^{\frac{1}{2}}} - 2({k_i}{s_j} - {k_j}{s_i})({k_i}{q_j} - {k_j}{q_i})}}{{ - k_i^2k_j^2{{({k_i} + {k_j})}^2}{m^{\frac{3}{2}}} + k_i^2k_j^2{{({k_i} + {k_j})}^2}{m^{\frac{1}{2}}} - 2({k_i}{s_j} - {k_j}{s_i})({k_i}{q_j} - {k_j}{q_i})}}. \end{equation} $

(3.10)

The breathers solution to the Fokas equation can be presented by choosing

$ \begin{equation} {k_1} = k_2^ * = {a_1} + {b_1}I, {s_1} = s_2^ * = {a_2} + {b_2}I, {q_1} = q_2^ * = {a_3} + {b_3}I, \end{equation} $

(3.11)

where $ {\ast} $ denotes the conjugate operator and $ {I^2} = - 1 $.

Setting $ {k_1} = I, {s_1} = 2 + I, {q_1} = 2, m = 1 $, the $ f $ can be expressed as

$ \begin{eqnarray} f & = & 1 + 2(\cosh (2z + 2w + 3t) + \sinh (2z + 2w + 3t))\cos (\frac{{t + 4y}}{4}\sqrt 2 - 6t + x + z)\\ && + (1 + \frac{{\sqrt 2 }}{8})(\cosh (4z + 4w + 6t) + \sinh (4z + 4w + 6t). \end{eqnarray} $

(3.12)

By analyzing function $ f $ (Eq. (3.12)), we can see that the spatial variables $ x $ and $ y $ are similar, but the spatial variable $ z $ is essentially different from the spatial variables $ x $ and $ y $ when $ w = 0 $. Eq. (3.12) constitutes the general solution of the Fokas Eq.(1.1). Figure 6(a) shows that the behavior of the general breathers is periodic in both space and time. The solution shown in Figure 6 is Akhmediev breather solution, which is periodic in $ x $ and localized in $ t $.

To acquire the high order breathers solution of the Eq.(1.1), we analyse the following four-soliton solution

$ \begin{eqnarray} f & = & 1 + {e^{{\theta _1}}} + {e^{{\theta _2}}} + {e^{{\theta _3}}} + {e^{{\theta _4}}} + {a_{12}}{e^{{\theta _1}{\text{ + }}{\theta _2}}}{\text{ + }}{a_{1{\text{3}}}}{e^{{\theta _1}{\text{ + }}{\theta _3}}}{\text{ + }}{a_{14}}{e^{{\theta _1}{\text{ + }}{\theta _4}}} + {a_{23}}{e^{{\theta _2}{\text{ + }}{\theta _3}}}\\ &&{\text{ + }}{a_{24}}{e^{{\theta _2}{\text{ + }}{\theta _4}}} + {a_{34}}{e^{{\theta _3}{\text{ + }}{\theta _4}}} + {a_{12{\text{3}}}}{e^{{\theta _1}{\text{ + }}{\theta _2} + {\theta _3}}} + {a_{124}}{e^{{\theta _1}{\text{ + }}{\theta _2}{\text{ + }}{\theta _4}}} + {a_{134}}{e^{{\theta _1}{\text{ + }}{\theta _3}{\text{ + }}{\theta _4}}} \\ &&{ + {a_{234}}{e^{{\theta _2} + {\theta _3}{\text{ + }}{\theta _4}}} + {a_{1234}}{e^{{\theta _1} + {\theta _2} + {\theta _3}{\text{ + }}{\theta _4}}}, } \end{eqnarray} $

(3.13)

where

$ \begin{equation} \begin{array}{l} {a_{12{\text{3}}}} = {a_{12}}{a_{1{\text{3}}}}{a_{23}}, {a_{124}} = {a_{12}}{a_{14}}{a_{24}}, {a_{134}} = {a_{13}}{a_{14}}{a_{34}}, \\ {a_{234}} = {a_{23}}{a_{24}}{a_{34}}, {a_{1234}} = {a_{12}}{a_{1{\text{3}}}}{a_{14}}{a_{23}}{a_{24}}{a_{34}}. \end{array} \end{equation} $

(3.14)

Meanwhile $ {\theta _i}, 1 \le i \le 4, $ and $ {a_{ij}}, 1 \le i < j \le 4 $ are given in (3.1), (3.10), respectively.

In a similar way, by choosing the following parameters

$ \begin{equation} \begin{array}{l} {k_1} = k_2^* = {a_1} + {b_1}I, {k_3} = k_4^* = {a_2} + {b_2}I, {s_1} = s_2^* = {a_3} + {b_3}I, {s_3} = s_4^* = {a_4} + {b_4}I, \\ {q_1} = q_2^* = {a_5} + {b_5}I, {q_3} = q_4^* = {a_6} + {b_6}I. \end{array} \end{equation} $

(3.15)

And by the Eq. (3.13), the high order breathers solution are obtained. As shown

$ \begin{eqnarray} f & = & \frac{1}{{17904\sqrt 2 + 43832}}\left( {2(8408 + 7920\sqrt 2 )\cos (\frac{{(9t + 12y)\sqrt 2 }}{4} - \frac{{15t}}{2} + 3x + 2z)} \right.\\ &&\times {e^{3z + 4w + \frac{{9t}}{2}}} - 2(408 + 1888\sqrt 2 )\sin (\frac{{(9t + 12y)\sqrt 2 }}{4} - \frac{{15t}}{2} + 3x + 2z){e^{3z + 4w + \frac{{9t}}{2}}} \\ && + 2(18612 + 13719\sqrt 2 )\cos ((2t + 2y)\sqrt 2 - \frac{{3t}}{2} + 2x + z){e^{5z + 6w + \frac{{15t}}{2}}}\\ && - 2(4512 + 3168\sqrt 2 )\sin ((2t + 2y)\sqrt 2 - \frac{{3t}}{2} + 2x + z){e^{5z + 6w + \frac{{15t}}{2}}}\\ && + 2(110520 + 77520\sqrt 2 )\cos (\frac{{(t + 4y)\sqrt 2 }}{4} - 6t + x + z){e^{4z + 6w + 6t}} \\ && - 2(11712 + 5184\sqrt 2 )\sin (\frac{{(t + 4y)\sqrt 2 }}{4} - 6t + x + z){e^{4z + 6w + 6t}}\\ && + 2(59544 + 39408\sqrt 2 )\cos (\frac{{(7t + 4y)\sqrt 2 }}{4} + x + \frac{9}{2}t){e^{3z + 4w + \frac{9}{2}t}} \\ && - 2(2592 + 4128\sqrt 2 )\sin (\frac{{(7t + 4y)\sqrt 2 }}{4} + x + \frac{9}{2}t){e^{3z + 4w + \frac{9}{2}t}}\\ && + 2(43832 + 17904\sqrt 2 )\cos (\frac{{(t + 4y)\sqrt 2 }}{4} + x + z - 6t){e^{2z + 2w + 3t}} \\ && + (193232\sqrt 2 + 187064){e^{2z + 4w + 3t}} + 17904\sqrt 2 + 43832 + (23383\sqrt 2 + 48308){e^{6z + 8w + 9t}}), \end{eqnarray} $

(3.16)

by taking

$ \begin{equation} {k_1} = I, {k_3} = 2I, {s_1} = 2 + I, {s_3} = 1 + I, {q_1} = 2, {q_3} = 2. \end{equation} $

(3.17)

The corresponding dynamics characteristics of solution $ u $ are shown in figure 7.

The high order general breathers solution of the Fokas Eq. (1.1) are shown in figure 7(a) that the behavior of the general breathers is periodic in both space and time. The solution shown in figure 7(c) is Akhmediev breather solution, which is periodic in $ x $ and localized in $ t $.

Remark In this paper, based on the bilinear form of the $ (4+1) $-dimensional Fokas equation, the rogue wave solution and breathers solution of Eq. (2.1) were constructed. And high order rogue wave solutions and high order breathers solution were presented also. The correctness of the results in this work has been verified. The results of this paper enrich the types of solutions of the $ (4+1) $-dimensional Fokas equation, which is helpful to promote our understanding of various kinds wave phenomena in fluid dynamics, natural phenomenon and science problems.

The mechanism of more new kinds of hybrid solutions for different nonlinear partial differential equations will be studied in the future.