The study of nonlinear evolution equations which are regarded as the models to describe nonlinear phenomena became more attractive topic in physical science [1]. Many methods were proposed over years to address the nonlinear evolution equations, such as inverse scattering transformation [2, 3], Darboux transformation [4], Hirota method [5], Bäcklund transformation [6], Bell polynomials [7]. Through these methods, many kinds of solutions are presented. As a kind of solutions, the rational solution to some nonlinear partial differential equations are studied. The aim of this study is to use the Hirota bilinear equations to generate the generalized (3+1) dimensional variable coefficient B-type Kadomtsev-Petviashvili equation and then study the solutions to its two dimensionally reduced cases in (2+1)-dimensions.

The Kadomtsev-Petviashvili equation [8],

$ \begin{equation} P_{KP}(u) = u_{t}+6uu_{t}+u_{xxx}+\alpha u_{yy} = 0 \end{equation} $

(1.1)

is a nonlinear partial differential equation. The rational solutions were exhibited by symbolic computation [9]. Furthermore, the bilinear formulation plays a key role in the study of rational solutions, which is defined as [10]:

$ \begin{align*} &D_{p, x}^{m}D_{p, t}^{n}f\cdot f\\ = &(\frac{\partial}{\partial_{x}}+\alpha_{p}\frac{\partial}{\partial_{x^{'}}})^{m}(\frac{\partial}{\partial_{t}}+\alpha_{p}\frac{\partial}{\partial_{t^{'}}})^{n}f(x, t)f(x^{'}, t^{'})|_{x^{'} = x, t^{'} = t}\\ = & \sum^{m}_{i = 0} \sum^{n}_{j = 0}(_{i}^{m})(_{j}^{n})\alpha_{p}^{i}\alpha_{p}^{j}\frac{\partial^{m-i}}{\partial x^{m-i}}\frac{\partial^{i}}{\partial x^{'(i)}}\frac{\partial^{n-j}}{\partial t^{n-j}}\frac{\partial^{j}}{\partial t^{'(j)}}f(x, t)f(x^{'}, t^{'})|_{x^{'} = x, t^{'} = t}\\ = & \sum^{m}_{i = 0} \sum^{n}_{j = 0}(_{i}^{m})(_{j}^{n})\alpha_{p}^{i}\alpha_{p}^{j}\frac{\partial^{m+n-i-j}f(x, t)}{\partial x^{m-i}t^{n-j}}\frac{\partial^{i+j}f(x, t)}{\partial x^{i}t^{j}}, \end{align*} $

where $ m $, $ n\geq 0 $, $ \alpha _{p}^{s} = (-1)^{r_{p}(s)} $, $ s = r_{p}(s)\; {\rm mod}\; p. $

Extended from (1.1), a generalized (3+1) dimensional variable-coefficient B-type Kadomtsev-Petviashvili equation [11],

$ \begin{equation} P_{BKP}(u) = a(t)u_{xxxy}+\rho a(t)(u_{x}u_{y})_{x}+(u_{x}+u_{y}+u_{z})_{t}+b(t)(u_{xx}+u_{zz}) = 0, \end{equation} $

(1.2)

where $ a(t) $ and $ b(t) $ are the real function of $ t $ and $ \rho $ is a real non-zero constant.

In this study, we would like to present the bilinear form for (1.2) generated from a generalized bilinear differential equation. The solutions will be generated from polynomial solutions. In Section 2, the solutions to the first dimensionally reduced case, the second dimensionally reduced case, for $ z = x $ and $ z = y $, are constructed. Finally, some conclusions are given in Section 3.

Take $ u = \frac{6}{\rho}(\ln f)_{x} $, which is suggested by bell polynomial theories [12]. A Hirota equation is proposed as follows

$ \begin{align} B_{BKP}(f) = &[a(t)D_{x}^{3}D_{y}+D_{x}D_{t}+D_{y}D_{t}+D_{z}D_{t}+b(t)(D_{x}^{2}+D_{z}^{2})]f\cdot f\\ = &2a(t)[ff_{xxxy}-f_{xxx}f_{y}+3f_{xx}f_{xy}-3f_{xxy}f_{x}]\\&+2[f_{xt}f-f_{x}f_{t}]+2[f_{yt}f-f_{y}f_{t}]+2[f_{zt}f-f_{z}f_{t}]\\&+2b(t)[f_{xx}f-f_{x}^{2}+f_{xx}f-f_{x}^{2}+f_{zz}f-f_{z}^{2}] = 0, \end{align} $

(2.1)

where $ f $ as a function of $ x $, $ y $, $ z $, and $ t $, the bilinear differential operators $ D_{x}^{3}D_{y}, \; D_{x}D_{t}, $ $ D_{y}D_{t}, \; D_{x}^{2}, \; D_{z}^{2} $ are the Hirota bilinear operators.

If $ f $ solves generalized equation (2.1), then $ u = \frac{6}{\rho}(\ln f)_{x} $ present the solutions to (1.2). In this study, we construct positive quadratic function solution to the dimensionally reduced Hirota bilinear equation (2.1) for two cases: $ z = x $ and $ z = y $, and begin with

$ \begin{eqnarray} &&f = g^{2}+h^{2}+a_{9}, \\ &&g = a_{1}x+a_{2}y+a_{3}t+a_{4}, \\ && h = a_{5}x+a_{6}y+a_{7}t+a_{8}, \end{eqnarray} $

(2.2)

where $ a_{i}\; (1 \leq i \leq 9) $ are all real parameters to be determined.

The dimensionally reduced Hirota bilinear equation (2.1) in (2+1) dimensions

$ \begin{align} &2a(t)[ff_{xxxy}-f_{xxx}f_{y}+3f_{xx}f_{xy}-3f_{xxy}f_{x}]+4[f_{xt}f-f_{x}f_{t}]\\ &+2[f_{yt}f-f_{y}f_{t}]+4b(t)[f_{xx}f-f_{x}^{2}] = 0, \end{align} $

(2.3)

which is transformed into

$ \begin{equation} a(t)[u_{xxxy}-3(u_{x}u_{y})_{x}]+2u_{xt}+u_{yt}+2b(t)u_{xx} = 0. \end{equation} $

(2.4)

Through the link between $ f $ and $ u $, substituting (2.2) into (2.3) to the following set of constraining equations for the parameters

$ \left\{ \begin{array}{l} {a_1} = {a_1}, {a_2} = {a_2}, \;\\ {a_3} = - \frac{{2(2a_1^3 + a_1^2{a_2} - {a_2}a_5^2 + 2{a_1}{a_5}({a_5} + {a_6}))b(t)}}{{{{(2{a_1} + {a_2})}^2} + {{(2{a_5} + {a_6})}^2}}}, \\ {a_4} = {a_4}, {a_5} = {a_5}, {a_6} = {a_6}, \\ {a_7} = - \frac{{2(2{a_1}{a_2}{a_5} + a_1^2(2{a_5} - {a_6}) + a_5^2(2{a_5} + {a_6}))b(t)}}{{{{(2{a_1} + {a_2})}^2} + {{(2{a_5} + {a_6})}^2}}}, \\ {a_8} = {a_8}, {a_9} = - \frac{{3(a_1^2 + a_5^2)({a_1}{a_2} + {a_5}{a_6})({{(2{a_1} + {a_2})}^2} + {{(2{a_5} + {a_6})}^2})a(t)}}{{2{{({a_2}{a_5} - {a_1}{a_6})}^2}b(t)}}, \end{array} \right. $

(2.5)

which satisfies the conditions

$ \begin{eqnarray} &&(a_{2}a_{5}-a_{1}a_{6})b(t)\neq 0, \end{eqnarray} $

(2.6)

$ \begin{eqnarray} &&(a_{1}a_{2}+a_{5}a_{6})a(t)< 0 \end{eqnarray} $

(2.7)

to guarantee the rational localization of the solutions. The parameters in set (2.5) yield a class of positive quadratic function solution to (2.2) as

$ \begin{eqnarray} f& = &(a_{1}x+a_{2}y-\frac{[2(2a^{3}_{1}+a^{2}_{1}a_{2}-a_{2}a^{2}_{5}+2a_{1}a_{5}(a_{5}+a_{6}))b(t)]t}{(2a_{1}+a_{2})^{2}+(2a_{5}+a_{6})^{2}}+a_{4})^{2}\\ &&+(a_{5}x+a_{6}y-\frac{[2(2a_{1}a_{2}a_{5}+a^{2}_{1}(2a_{5}-a_{6})+a^{2}_{5}(2a_{5}+a_{6}))b(t)]t}{(2a_{1}+a_{2})^{2}+(2a_{5}+a_{6})^{2}}+a_{8})^{2}\\ &&-\frac{3(a^{2}_{1}+a^{2}_{5})(a_{1}a_{2}+a_{5}a_{6})((2a_{1}+a_{2})^{2}+(2a_{5}+a_{6})^{2})a(t)}{2(a_{2}a_{5}-a_{1}a_{6})^{2}b(t)} \end{eqnarray} $

(2.8)

through transformation as

$ \begin{equation} u^{(1)} = \frac{12(a_{1}g+a_{5}h)}{\rho f} \end{equation} $

(2.9)

and the function $ g $, $ h $ are given as follows

$ \begin{array}{l} g = {a_1}x + {a_2}y - \frac{{[2(2a_1^3 + a_1^2{a_2} - {a_2}a_5^2 + 2{a_1}{a_5}({a_5} + {a_6}))b(t)]t}}{{{{(2{a_1} + {a_2})}^2} + {{(2{a_5} + {a_6})}^2}}} + {a_4}, \\ h = {a_5}x + {a_6}y - \frac{{[2(2{a_1}{a_2}{a_5} + a_1^2(2{a_5} - {a_6}) + a_5^2(2{a_5} + {a_6}))b(t)]t}}{{{{(2{a_1} + {a_2})}^2} + {{(2{a_5} + {a_6})}^2}}} + {a_8}. \end{array} $

The solution $ u $ involves six parameters $ a_{1} $, $ a_{2} $, $ a_{4} $, $ a_{5} $, $ a_{6} $ and $ a_{8} $. All six involved parameters are arbitrary and the rest are demanded to satisfy conditions (2.6) and (2.7). We choose the following special set of parameters

$ \begin{eqnarray*} &&a_{1} = 1, a_{2} = 1, a_{3} = -\frac{1}{3}, a_{4} = 0, a_{5} = 2, \\ && a_{6} = -1, a_{7} = \frac{7}{3}, a_{8} = 0, a_{9} = 15. \end{eqnarray*} $

The dimensionally reduced Hirota bilinear equation (2.1) turn out to be

$ \begin{eqnarray} &&2a(t)[ff_{xxxy}-f_{xxx}f_{y}+3f_{xx}f_{xy}-3f_{xxy}f_{x}]+2[f_{xt}f-f_{x}f_{t}]\\ &&+4[f_{yt}f-f_{y}f_{t}]+2b(t)[f_{xx}f-f_{x}^{2}+f_{yy}f-f_{y}^{2}] = 0, \end{eqnarray} $

(2.10)

which is transformed into

$ \begin{equation} a(t)[u_{xxxy}-3(u_{x}u_{y})_{x}]+u_{xt}+2u_{yt}+b(t)(u_{xx}+u_{yy}) = 0. \end{equation} $

(2.11)

Through the link between $ f $ and $ u $, substituting (2.2) into (2.10), the following set of constraining equations for the parameters

$ \begin{equation} \left\{\begin{array}{l} a_{1} = a_{1}, a_{2} = a_{2}, \\ a_{3} = \frac{-[a_{1}((a_{1}+a_{2})^{2}+a^{2}_{5}+4a_{5}a_{6}-a^{2}_{6})+2a_{2}(a^{2}_{2}-a^{2}_{5}+a_{5}a_{6}+a^{2}_{6})]b(t)}{(a^{2}_{1}+2a^{2}_{2})+(a^{2}_{5}+2a^{2}_{6})}, \\ a_{4} = a_{4}, a_{5} = a_{5}, a_{6} = a_{6}, \\ a_{7} = -\frac{[(a^{2}_{1}-a^{2}_{2})(a_{5}-2a_{6})+2a_{1}a_{2}(2a_{5}+a_{6})+(a_{5}+2a_{6})(a^{2}_{5}+a^{2}_{6})]b(t)}{(a^{2}_{1}+2a^{2}_{2})+(a^{2}_{5}+2a^{2}_{6})}, \\ a_{8} = a_{8}, a_{9} = -\frac{3(a^{2}_{1}+a^{2}_{5})(a_{1}a_{2}+a_{5}a_{6})[(a_{1}+2a_{2})^{2}+(a_{5}+2a_{6})^{2}]a(t)}{5(a_{2}a_{5}-a_{1}a_{6})^{2}b(t)}, \end{array}\right. \end{equation} $

(2.12)

which needs to satisfy the conditions

$ \begin{eqnarray} &&(a_{2}a_{5}-a_{1}a_{6})b(t)\neq 0, \end{eqnarray} $

(2.13)

$ \begin{eqnarray} &&(a_{1}a_{2}+a_{5}a_{6})a(t)< 0 \end{eqnarray} $

(2.14)

to guarantee the analyticity and rational localization of the solutions. The parameters in set (2.12) yield a class of positive quadratic function solution to (2.10) as

$ \begin{eqnarray} f & = &[a_{1}x+a_{2}y -\frac{[a^{3}_{1}+2a^{2}_{1}a_{2}+a_{1}(a^{2}_{2}+a^{2}_{5}+4a_{5}a_{6}-a^{2}_{6})+2a_{2}(a^{2}_{2}-a^{2}_{5}+a_{5}a_{6}+a^{2}_{6})]b(t)}{(a^{2}_{1}+2a^{2}_{2})+(a^{2}_{5}+2a^{2}_{6})}t +a_{4}]^{2}\\ &&+[a_{5}x+a_{6}y - \frac{[a^{2}_{1}(a_{5}-2a_{6})-a^{2}_{2}(a_{5}-2a_{6})+2a_{1}a_{2}(2a_{5}+a_{6})+(a_{5}+2a_{6})(a^{2}_{5}+a^{2}_{6})]b(t)}{(a^{2}_{1}+2a^{2}_{2})+(a^{2}_{5}+2a^{2}_{6})}t +a_{8}]^{2}\\&&-\frac{3(a^{2}_{1}+a^{2}_{5})(a_{1}a_{2}+a_{5}a_{6})[(a_{1}+2a_{2})^{2}+(a_{5}+2a_{6})^{2}]a(t)}{5(a_{2}a_{5}-a_{1}a_{6})^{2}b(t)} \end{eqnarray} $

(2.15)

through transformation as

$ \begin{equation} u^{(2)} = \frac{12(a_{1}g+a_{5}h)}{\rho f}, \end{equation} $

(2.16)

and the functions $ g $, $ h $ are given as follows

$ \begin{array}{l} g = {a_1}x + {a_2}y + {a_4}\\ \;\;\;\;\;\; - \frac{{[a_1^3 + 2a_1^2{a_2} + {a_1}(a_2^2 + a_5^2 + 4{a_5}{a_6} - a_6^2) + 2{a_2}(a_2^2 - a_5^2 + {a_5}{a_6} + a_6^2)]b(t)}}{{(a_1^2 + 2a_2^2) + (a_5^2 + 2a_6^2)}}t, \\ h = {a_5}x + {a_6}y + {a_8}\\ \;\;\;\;\;\; - \frac{{[a_1^2({a_5} - 2{a_6}) - a_2^2({a_5} - 2{a_6}) + 2{a_1}{a_2}(2{a_5} + {a_6}) + ({a_5} + 2{a_6})(a_5^2 + a_6^2)]b(t)}}{{(a_1^2 + 2a_2^2) + (a_5^2 + 2a_6^2)}}t. \end{array} $

The solution $ u $ involves six parameters $ a_{1} $, $ a_{2} $, $ a_{4} $, $ a_{5} $, $ a_{6} $ and $ a_{8} $. All six involved parameters are arbitrary and the rest are demanded to satisfy (2.13) and (2.14). To get the special solutions of the equation, let us choose the following special set of the parameters

$ \begin{array}{l} {a_1} = - 2, {a_2} = 1, {a_3} = \frac{1}{4}, {a_4} = 0, {a_5} = \frac{2}{3}, \\ {a_6} = 1, {a_7} = \frac{4}{3}, {a_8} = 0, {a_9} = \frac{{32}}{9}. \end{array} $

In this paper, via the Bell polynomials and Hirota method, we have derived bilinear form for (1.2). Then by searching for positive quadratic function solutions to (2.3) and (2.10), two classes of solutions to the dimensionally reduced equations are presented. These results provide some salutary information on the relevant fields in nonlinear science. Therefore, we expect that the results presented in this work will also be useful to study lump solutions in a variety of other high-dimensional nonlinear equations.