1. Introduction
The pursuit of exact solutions for nonlinear partial differential equations (PDEs) has spurred the development of a multitude of methodologies, each offering unique insights and approaches. From the inverse scattering method [1–3] to Hirota's method [4–8], and the group theoretical method [9] to direct methods [10–20], researchers have explored diverse avenues to unravel the complexities of nonlinear PDEs. Central to these methodologies is the Bäcklund transformation (BT), a powerful analytical tool that establishes crucial connections between solutions of specific nonlinear PDEs, thereby advancing the field of integrability [21, 22]. The rich tapestry of research surrounding BTs underscores their enduring appeal to mathematicians and physicists alike. With active developments spanning differential geometry, algebra, and nonlinear science, BTs have emerged as a cornerstone in addressing integrability challenges across various disciplines. The wealth of literature dedicated to BTs reflects the profound interest and significance attributed to these transformative tools [23–26].
Within this landscape of methodologies, the decomposition method based on the formal variable separation approach (FVSA) emerges as a notable approach for constructing BTs [27–29]. In this paper, we introduce a novel decomposition technique in section 2 , that not only enables the construction of BTs, but also demonstrates remarkable efficacy in tackling two prominent nonlinear PDEs: the B-type and C-type Kadomtsev–Petviashvili equations (BKP and CKP). Our assertion of the straightforward and natural approach to constructing BTs heralds novel discoveries with potential applications across a spectrum of equations. Sections 3 and 4 of this paper delve into a detailed exploration of the utilization of the decomposition method in analyzing the potential BKP equation and potential CKP equation, respectively. Through a systematic investigation, we unveil BTs and Lax pairs for both equations in a highly intuitive manner, underscoring the coherence and significance of the results. The paper culminates in a comprehensive summary of the findings in section 5 , with intricate calculations thoughtfully relocated to the appendix to enhance readability and focus on the key insights derived from the application of the decomposition method.
2. Modified formal variable separation approach
The modified FVSA outlined in this section focuses on deriving the general decomposition solution1 ) and (2 ). Here, w is a function of x, y, t, with F and G depending on these variables and derivatives of w with respect to x.
$\begin{eqnarray}{w}_{y}=F(x,y,t,w,{w}_{x},{w}_{{xx}},\ldots ,{w}_{{xm}}),\end{eqnarray}$
$\begin{eqnarray}{w}_{t}=G(x,y,t,w,{w}_{x},{w}_{{xx}},\ldots ,{w}_{{xn}})\end{eqnarray}$
for a (2+1)-dimensional PDE. The key objective is to maintain the integrability condition $\begin{eqnarray}{w}_{{yt}}-{w}_{{ty}}=[F,G]=0,\end{eqnarray}$
while seeking a solution in the form of equations (The method involves a systematic procedure:
| 1. | (1) Determination of admissible values: by analyzing the terms in the (2+1)-dimensional PDE and comparing their magnitudes, suitable values of integers m and n in equations ( |
| 2. | (2) Substitution analysis: substituting the decomposed forms into the PDE and the integrability condition ( |
| 3. | (3) Solutions for F and G: by solving the system of equations from step (2), the functions F and G are determined, resulting in the desired decomposition solutions, the corresponding BT and Lax pair. |
The systematic methodology not only guarantees the maintenance of integrability, but also establishes a well-defined framework for constructing BTs and Lax pairs. The upcoming application of this technique is primed to shed light on the dynamics of (2+1)-dimensional PDEs, demonstrating the effectiveness and potential of the modified FVSA.
3. The decomposition solutions, Bäcklund transformation, Lax pair and linear superposition solutions of the potential BKP equation
An illustrative starting point often resides in examples. Let us consider the BKP equation4 ). Consequently, the equation governing w can be expressed as follows:
$\begin{eqnarray}\begin{array}{l}{u}_{{xt}}+\left({u}_{x4}+15{{uu}}_{x2}+15{u}^{3}-15{uv}\right.\\ \,{\left.-\,5{u}_{{xy}}\right)}_{{xx}}-5{u}_{{yy}}=0,{v}_{x}={u}_{y},\end{array}\end{eqnarray}$
where ${u}_{x}={\partial }_{x}u,{u}_{x2}={\partial }_{x}^{2}u,{u}_{x3}={\partial }_{x}^{3}u,\ldots ,$ a (2+1)-dimensional extension of the Sawada–Kotera (SK) equation, renowned for its manifestation of elegant integrable properties within the realm of academic discourse [30, 31]. Analytical treatment of a potential function w is advantageous, achieved by defining u = wx within the context of equation ( $\begin{eqnarray}\begin{array}{l}{w}_{{xt}}=5{w}_{{yy}}-\left({w}_{x5}+15{w}_{x}{w}_{x3}\right.\\ \,{\left.+\,15{w}_{x}^{3}-15{w}_{x}{w}_{y}-5{w}_{{xxy}}\right)}_{x}.\end{array}\end{eqnarray}$
The analysis of the potential BKP equation (5 ) reveals that for the chosen leading orders m = 3 and n = 5, the expressions of wy and wt can be determined in terms of6 ) and (7 ) while ensuring the satisfaction of equation (5 ). The derivation of the functions F and G is straightforward but involves a detailed procedure, which is provided in the appendix for reference. The outcome of the calculations leads directly to the formulation of the following theorems.
$\begin{eqnarray}{w}_{y}=F(x,y,t,w,{w}_{x},{w}_{{xx}},{w}_{{xxx}}),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{w}_{t}=G(x,y,t,w,{w}_{x},{w}_{{xx}},{w}_{{xxx}},{w}_{{xxxx}},{w}_{{xxxxx}}).\end{array}\end{eqnarray}$
Consequently, it becomes necessary to impose the integrability condition $\begin{eqnarray}\begin{array}{l}{F}_{t}={G}_{y}\end{array}\end{eqnarray}$
on equations (If w satisfies the decomposition system of the consistent variable coefficient potential Korteweg–de Vries (KdV) equation
$\begin{eqnarray}{v}_{y}={v}_{{xxx}}-\displaystyle \frac{a}{2}{v}_{x}^{2}+\displaystyle \frac{{m}_{0}}{a}x+{n}_{y},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{v}_{t}=9{v}_{{xxxxx}}-15{{av}}_{x}{v}_{{xxx}}-\displaystyle \frac{15a}{2}{v}_{{xx}}^{2}\\ +\displaystyle \frac{5{a}^{2}}{2}{v}_{x}^{3}+\left[15{m}_{2}+15{m}_{1}y-\left(5{m}_{0}^{2}+\displaystyle \frac{1}{2}{m}_{0t}\right){y}^{2}\right.\\ \left.-5{m}_{0}x\Space{0ex}{3.25ex}{0ex}\right]{v}_{x}-5{m}_{0}v+[(10{m}_{0}^{2}+{m}_{0t})y\\ -15{m}_{1}]\displaystyle \frac{x}{a}+\displaystyle \frac{{y}^{3}}{6a}{m}_{0}(10{m}_{0}^{2}+{m}_{0t})\\ -\displaystyle \frac{15{y}^{2}}{2a}{m}_{0}{m}_{1}-\displaystyle \frac{15y}{a}{m}_{0}{m}_{2}\\ +5{m}_{0}n+{n}_{t}+{m}_{3},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{w}_{y}={w}_{{xxx}}+3{w}_{x}^{2}+{{av}}_{x}{w}_{x}\\ +\displaystyle \frac{{a}^{2}}{6}{v}_{x}^{2}-\displaystyle \frac{1}{3}{m}_{0}(x+{m}_{0}{y}^{2})\\ -\displaystyle \frac{{m}_{0t}{y}^{2}}{30}+{m}_{1}y+{m}_{2},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{w}_{t}=9{w}_{{xxxxx}}+15{\left({av}+6w\right)}_{x}{w}_{{xxx}}\\ +5a{\left({av}+3w\right)}_{x}{v}_{{xxx}}+15{\left({av}+3w\right)}_{{xx}}{w}_{{xx}}\\ +\displaystyle \frac{15}{2}{w}_{x}[{\left({av}+3w\right)}_{x}^{2}+3{w}_{x}^{2}]\\ +\displaystyle \frac{5{a}^{2}}{2}{v}_{{xx}}^{2}+{w}_{x}(15{m}_{2}+15{m}_{1}y\\ -5{m}_{0}^{2}{y}^{2}-5{m}_{0}x-\displaystyle \frac{1}{2}{m}_{0t}{y}^{2})-5{m}_{0}w\\ -\displaystyle \frac{{xy}}{3}({m}_{0t}+10{m}_{0}^{2})+5{m}_{1}x\\ -\displaystyle \frac{{y}^{3}}{90}({m}_{0{tt}}+30{m}_{0}{m}_{0t}+100{m}_{0}^{3})\\ +\displaystyle \frac{{y}^{2}}{2}({m}_{1t}+10{m}_{0}{m}_{1})\\ +({m}_{2t}+10{m}_{0}{m}_{2})y+{m}_{4},\end{array}\end{eqnarray}$
where ${m}_{i},i=0,1,...,4$ are arbitrary functions of t, n is an arbitrary function of y and t, and a is an arbitrary constant, then w also satisfies the potential BKP equation (The arbitrary parameter a plays a crucial role in determining whether the function v, defined by equations (
Considering the decomposition presented in theorem
$\begin{eqnarray}\begin{array}{l}a=-3,n={m}_{2}y+\displaystyle \frac{1}{2}{m}_{1}{y}^{2}\\ -\displaystyle \frac{1}{90}(10{m}_{0}^{2}+{m}_{0t}){y}^{3}+{n}_{0}(t),\end{array}\end{eqnarray}$
it can be established that both v and w, as defined by equations (We hereby present an additional corollary stemming from theorem
By considering the functions v and w determined through equations (
$\begin{eqnarray}\begin{array}{l}a=-6,n={m}_{2}y+{n}_{0}(t),\,\,{m}_{0}={m}_{1}=0,\end{array}\end{eqnarray}$
it can be concluded that these functions satisfy the potential BKP equation (The implications derived from Corollaries
(Bäcklund Transformation) The decomposition relation
$\begin{eqnarray}\begin{array}{l}{p}_{y}={p}_{{xxx}}+\displaystyle \frac{3}{2}{\left(w+v\right)}_{{xx}}p\\ +\displaystyle \frac{3}{2}({w}_{x}^{2}-{v}_{x}^{2})+\displaystyle \frac{1}{4}{\left({p}^{3}\right)}_{x},p\equiv w-v,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{p}_{t}=9{p}_{{xxxxx}}+\displaystyle \frac{9}{16}{\left({p}^{5}\right)}_{x}\\ +\displaystyle \frac{45}{8}{\left(w+v\right)}_{{xx}}{p}^{3}\\ -\displaystyle \frac{45}{8}(3{v}_{x}^{2}-3{w}_{x}^{2}-2{p}_{{xxx}}){p}^{2}\\ +\displaystyle \frac{15}{4}(9{v}_{x}^{2}+9{w}_{x}^{2}-6{v}_{x}{w}_{x}\\ +{\left.4{v}_{{xxx}}+4{w}_{{xxx}}+2{v}_{y}+2{w}_{y}\right)}_{x}p\\ -15{w}_{{xxx}}{\left(v-4w\right)}_{x}-15{v}_{{xxx}}{\left(4v-w\right)}_{x}\\ +45{w}_{{xx}}^{2}-45{v}_{{xx}}^{2}\\ +\displaystyle \frac{45}{2}{p}_{x}({v}_{x}^{2}+{w}_{x}^{2})+\displaystyle \frac{15}{2}{\left(w+v\right)}_{y}{p}_{x},\end{array}\end{eqnarray}$
serves as a BT connecting two solutions, denoted as w and v, of the potential BKP equation (The BT exhibits a distinctive characteristic in the form of a conservation law
$\begin{eqnarray*}\begin{array}{l}{p}_{y}={\left[{p}_{{xx}}+\displaystyle \frac{3}{2}p({w}_{x}+{v}_{x})+\displaystyle \frac{{p}^{3}}{4}\right]}_{x},\\ {p}_{t}=\left[9{p}_{{xxxx}}+\displaystyle \frac{9{p}^{5}}{16}\right.\\ +\displaystyle \frac{45}{8}{\left(w+v\right)}_{x}{p}^{3}+\displaystyle \frac{45}{4}{p}^{2}{p}_{{xx}}\\ +\left(15{\left(w+v\right)}_{{xxx}}+\displaystyle \frac{45}{2}({w}_{x}^{2}+{v}_{x}^{2})\right.\\ \left.+\displaystyle \frac{15}{2}{\left(w+v\right)}_{y}\right)p\\ {\left.+\displaystyle \frac{45}{2}{\left({p}_{x}{\left(w+v\right)}_{x}\right)}_{x}\right]}_{x}.\end{array}\end{eqnarray*}$
The identification of the BT for a PDE is equivalent to the discovery of its corresponding Lax pair, as indicated in the references [33, 34]. Consequently, it is reasonable to explore the BT further to facilitate the construction of the associated Lax pair. Building upon the insights provided by theorem $\begin{eqnarray}{\psi }_{y}={\psi }_{{xxx}}+3{w}_{x}{\psi }_{x},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\psi }_{t}=9{\psi }_{{xxxxx}}+45{w}_{x}{\psi }_{{xxx}}\\ +45{w}_{{xx}}{\psi }_{{xx}}+15(2{w}_{{xxx}}-3{w}_{x}^{2}+{w}_{y}){\psi }_{x}\end{array}\end{eqnarray}$
of the potential BKP equation ( $\begin{eqnarray}\begin{array}{l}{S}_{{xxx}}+{C}_{x}+4{S}_{x}^{2}-5{K}_{x}^{2}\\ +5({S}_{x}K-{S}_{y}-{K}_{y})=0\end{array}\end{eqnarray}$
with Schwarzian derivatives $S=\tfrac{{\psi }_{{xxx}}}{{\psi }_{x}}-\tfrac{3{\psi }_{{xx}}^{2}}{2{\psi }_{x}^{2}},C=\tfrac{{\psi }_{t}}{{\psi }_{x}}$ and $K=\tfrac{{\psi }_{y}}{{\psi }_{x}}$. It is notable that each of the two corollaries and theorem
Let the function w be a solution to the consistent variable coefficient Svinolupov–Sokolov (SS) system
$\begin{eqnarray}\begin{array}{l}{w}_{y}={w}_{{xxx}}+\displaystyle \frac{3{w}_{{xx}}^{2}}{2W}+\displaystyle \frac{3}{2}{w}_{x}^{2}-\displaystyle \frac{27}{4}{M}^{2}\\ +\displaystyle \frac{3}{2}{m}_{1}x+\displaystyle \frac{3y}{20}{\left({m}_{1}y+2{m}_{2}\right)}_{t}+{m}_{3},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{w}_{t}=9{w}_{{xxxxx}}+45{w}_{x}{w}_{{xxx}}\\ +\displaystyle \frac{45}{2}{w}_{x}^{3}+\displaystyle \frac{3}{4}[30{m}_{1}x-135{M}^{2}\\ +3y{\left({m}_{1}y+2{m}_{2}\right)}_{t}+20{m}_{3}]{w}_{x}\\ +\displaystyle \frac{45}{2}{m}_{1}w-\displaystyle \frac{3}{2}(45{m}_{1}M-{m}_{1t}y\\ -{m}_{2t})x+\displaystyle \frac{{y}^{2}}{20}{\left({m}_{1}y+3{m}_{2}\right)}_{{tt}}\\ -\displaystyle \frac{27}{4}y{\left({m}_{1}y+2{m}_{2}\right)}_{t}M\\ +y({m}_{3t}-45{m}_{1}{m}_{3})+{m}_{4}\\ +\displaystyle \frac{45}{2W}[2{w}_{{xx}}{w}_{{xxxx}}+{w}_{{xxx}}^{2}\\ +(3M+{w}_{x}){w}_{{xx}}^{2}]+\displaystyle \frac{180}{{W}^{2}}{w}_{{xx}}^{2}{w}_{{xxx}}\\ +\displaystyle \frac{315}{2{W}^{3}}{w}_{{xx}}^{4}+\displaystyle \frac{405}{4}{M}^{3},\end{array}\end{eqnarray}$
where $M\equiv {m}_{1}y+{m}_{2},W\equiv 3M-2{w}_{x}$ and ${m}_{i},i=1,2,3,4$ are arbitrary functions of t, then w is a solution of the potential BKP equation (The potential BKP equation (
$\begin{eqnarray}\begin{array}{l}{w}_{y}=-\displaystyle \frac{1}{2}{w}_{{xxx}}-\displaystyle \frac{3}{2}{w}_{x}^{2}+6{{Mw}}_{x}\\ +\displaystyle \frac{y}{10}{\left({m}_{1}y+2{m}_{2}\right)}_{t}+{m}_{1}x-6{M}^{2}\\ +{m}_{3},M\equiv {m}_{1}y+{m}_{2},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{w}_{t}=-\displaystyle \frac{9}{4}{w}_{{xxxxx}}-\displaystyle \frac{45}{4}(2{w}_{x}{w}_{{xxx}}\\ +{w}_{{xx}}^{2}+2{w}_{x}^{3})+15{m}_{1}w+\displaystyle \frac{3}{2}y\left({m}_{1}y\right.\\ {\left.+2{m}_{2}\right)}_{t}{w}_{x}\\ +15(6{M}^{2}+{m}_{1}x+{m}_{3}){w}_{x}\\ +\displaystyle \frac{{y}^{2}}{30}{\left({m}_{1}y+3{m}_{2}\right)}_{{tt}}-3({y}^{2}{m}_{1t}\\ +10{m}_{1}x+2{{ym}}_{2t})M+{M}_{t}x+({m}_{3t}\\ -30{m}_{1}{m}_{3})y-60{M}^{3}+{m}_{4},\end{array}\end{eqnarray}$
where ${m}_{i},i=1,2,3,4$ are arbitrary functions of t.The variable coefficient potential SK decomposition solution of the potential BKP equation (
$\begin{eqnarray}\begin{array}{l}{w}_{y}={{Mw}}_{x}+\displaystyle \frac{y}{30}{\left(M+{m}_{2}\right)}_{t}+\displaystyle \frac{1}{3}{m}_{1}x\\ -\displaystyle \frac{1}{2}{M}^{2}+{m}_{3},M\equiv {m}_{1}y+{m}_{2},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{w}_{t}=-{w}_{{xxxxx}}+5(M-3{w}_{x}){w}_{{xxx}}\\ +15(M-{w}_{x}){w}_{x}^{2}\\ +\left(\displaystyle \frac{1}{2}{y}^{2}{m}_{1t}+{{ym}}_{2t}+5{m}_{1}x\right.\\ -\left.\displaystyle \frac{5}{2}{M}^{2}+15{m}_{3}\right){w}_{x}\\ +5{m}_{1}w+\displaystyle \frac{{y}^{2}}{90}{\left({m}_{1}y+3{m}_{2}\right)}_{{tt}}\\ -\displaystyle \frac{y}{3}({My}-x){m}_{1t}+\displaystyle \frac{1}{3}(x-2{M}_{y}){m}_{2t}\\ -\displaystyle \frac{10}{3}{m}_{1}({Mx}+3{m}_{3}y)\\ +{m}_{3t}y+{m}_{4}\end{array}\end{eqnarray}$
with ${m}_{i},i=1,2,...,4$ being arbitrary functions of t. Suppose that arbitrary functions ${m}_{i}=0,i=1,2,...,4$, then by theorems
If a BT represents one of the different aspects of the property of integrability for a PDE, the existence of a linear combination principle further allows explicitly building some classes of solutions depending on the decomposition system (
If v1 and v2 are solutions of the variable coefficient potential KdV decompositions
$\begin{eqnarray}\begin{array}{l}{v}_{1y}={v}_{1{xxx}}-\displaystyle \frac{1}{2}{a}_{1}{v}_{1x}^{2}+\displaystyle \frac{1}{{a}_{1}}{m}_{0}x\\ +{n}_{y},M\equiv {m}_{1}y+{m}_{2},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{v}_{1t}=9{v}_{1{xxxxx}}+\displaystyle \frac{5}{2}{a}_{1}^{2}{v}_{1x}^{3}\\ -\displaystyle \frac{15}{2}{a}_{1}(2{v}_{1x}{v}_{1{xxx}}+{v}_{1{xx}}^{2})\\ +\left[15M-5{m}_{0}x-\displaystyle \frac{1}{2}\left(10{m}_{0}^{2}+{m}_{0t}\right){y}^{2}\right]{v}_{1x}\\ -5{m}_{0}{v}_{1}+5{m}_{0}n+{n}_{t}+{m}_{3}\\ +\displaystyle \frac{1}{{a}_{1}}\{[(10{m}_{0}^{2}+{m}_{0t})y-15{m}_{1}]x\\ +\displaystyle \frac{5}{3}{y}^{3}{m}_{0}^{3}+\displaystyle \frac{1}{6}{m}_{0}{m}_{0t}{y}^{3}\\ -\displaystyle \frac{15}{2}{m}_{0}({m}_{1}y+2{m}_{2})y\},\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}{v}_{2y}={v}_{2{xxx}}-\displaystyle \frac{1}{2}{a}_{2}{v}_{2x}^{2}+\displaystyle \frac{1}{{a}_{2}}\\ \times \left(\displaystyle \frac{1}{5}{m}_{0t}{y}^{2}-{a}_{1}{n}_{y}+{m}_{0}x+2{y}^{2}{m}_{0}^{2}-6M\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{v}_{2t}=9{v}_{2{xxxxx}}+\displaystyle \frac{5}{2}{a}_{2}^{2}{v}_{2x}^{3}\\ -\displaystyle \frac{15}{2}{a}_{2}(2{v}_{2x}{v}_{2{xxx}}+{v}_{2{xx}}^{2})\\ -\displaystyle \frac{1}{2}{y}^{2}{v}_{2x}{m}_{0t}+(15M-5{m}_{0}^{2}{y}^{2}\\ -5{m}_{0}x){v}_{2x}-5{m}_{0}{v}_{2}+{m}_{4}\\ +\displaystyle \frac{1}{{a}_{2}}\left[\displaystyle \frac{1}{15}{m}_{0{tt}}{y}^{3}+\displaystyle \frac{1}{6}(6{xy}+11{m}_{0}{y}^{3}){m}_{0t}\right.\\ -3{m}_{1t}{y}^{2}-6{m}_{2t}y-{a}_{1}{n}_{t}\\ +5(2{m}_{0}^{2}y-3{m}_{1})x\\ +\left.5{y}^{3}{m}_{0}^{3}-\displaystyle \frac{45}{2}{m}_{0}{m}_{1}{y}^{2}-45{m}_{0}{m}_{2}y-5{m}_{0}{a}_{1}n\right],\end{array}\end{eqnarray}$
respectively, then their linear combination $\begin{eqnarray}\begin{array}{l}w=-\displaystyle \frac{1}{6}({a}_{1}{v}_{1}+{a}_{2}{v}_{2})\end{array}\end{eqnarray}$
is a solution of the potential BKP equation (In general, the functions v1 and v2 presented in theorem
4. The decomposition solutions, Bäcklund transformation and Lax pair of the potential CKP equation
As a second illustration, we replicate the comprehensive procedure outlined in the preceding section to ascertain the decomposition solutions, Bäcklund transformation, and Lax pair for the potential CKP equation31 ) corresponds to the widely known Kaup–Kupershmidt (KK) equation [36, 37]. Likewise, a direct calculation establishes the following theorems.
$\begin{eqnarray}\begin{array}{l}9{u}_{{xt}}+\left({u}_{x4}+15{{uu}}_{x2}+15{u}^{3}\right.\\ {\left.+\displaystyle \frac{45}{4}{u}_{x}^{2}-15{uv}-5{u}_{{xy}}\right)}_{{xx}}\\ -5{u}_{{yy}}=0,{v}_{x}={u}_{y},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}9{w}_{{xt}}=5{w}_{{yy}}-\left({w}_{x5}+15{w}_{x}{w}_{x3}\right.\\ +15{w}_{x}^{3}+\displaystyle \frac{45{w}_{{xx}}^{2}}{4}\\ {\left.-15{w}_{x}{w}_{y}-5{w}_{{xxy}}\right)}_{x},\end{array}\end{eqnarray}$
in which u denotes the conservative field and w represents the potential one. In the limit where uy approaches zero, the reduced form of the CKP equation ((Bäcklund Transformation) Let v be an arbitrary solution of the potential CKP equation (
$\begin{eqnarray}\begin{array}{l}{p}_{y}={\left[{p}_{{xx}}-\displaystyle \frac{3{p}_{x}^{2}}{4p}+\displaystyle \frac{3}{2}p({w}_{x}+{v}_{x})+\displaystyle \frac{{p}^{3}}{4}\right]}_{x},\\ p=w-v,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{p}_{t}=\left[{p}_{{xxxx}}-\displaystyle \frac{5}{2p}{p}_{x}{p}_{{xxx}}+\displaystyle \frac{5p}{3}{\left(w+v\right)}_{{xxx}}\right.\\ -\displaystyle \frac{5}{4p}{p}_{{xx}}^{2}+\left(\displaystyle \frac{5{p}^{2}}{4}\right.\\ \left.+\displaystyle \frac{5{p}_{x}^{2}}{{p}^{2}}+\displaystyle \frac{5}{2}{\left(w+v\right)}_{x}\right){p}_{{xx}}\\ +\displaystyle \frac{5}{4}{p}_{x}{\left(w+v\right)}_{{xx}}\\ -\displaystyle \frac{35{p}_{x}^{4}}{16{p}^{3}}-\displaystyle \frac{15{\left(w+v\right)}_{x}}{8p}{p}_{x}^{2}\\ +\displaystyle \frac{15}{8}p({w}_{x}^{2}+{v}_{x}^{2})+\displaystyle \frac{{p}^{5}}{16}+\displaystyle \frac{5p}{6}{\left(w+v\right)}_{y}\\ {\left.+\displaystyle \frac{5{p}^{3}}{8}{\left(w+v\right)}_{x}+\displaystyle \frac{5}{4}{{pw}}_{x}{v}_{x}\right]}_{x}.\end{array}\end{eqnarray}$
The BT given by equations ( $\begin{eqnarray}\begin{array}{l}{\psi }_{y}={\psi }_{{xxx}}+3{w}_{x}{\psi }_{x}+\displaystyle \frac{3}{2}{w}_{{xx}}\psi ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\psi }_{t}={\psi }_{{xxxxx}}+5{w}_{x}{\psi }_{{xxx}}\\ +\displaystyle \frac{15}{2}{w}_{{xx}}{\psi }_{{xx}}\\ +5\left(\displaystyle \frac{7}{6}{w}_{{xxx}}+{w}_{x}^{2}+\displaystyle \frac{{w}_{y}}{3}\right){\psi }_{x}\\ +5\left(\displaystyle \frac{{w}_{{xxxx}}}{3}+{w}_{x}{w}_{{xx}}+\displaystyle \frac{{w}_{{xy}}}{6}\right)\psi \end{array}\end{eqnarray}$
followed by a relation $v=w+\mathrm{ln}{\left({\partial }_{x}^{-1}{\psi }^{2}\right)}_{x}$. When v = 0, the BT is reduced to the modified STO decomposition solution $\begin{eqnarray}\begin{array}{l}{w}_{y}={\left[{w}_{{xx}}-\displaystyle \frac{3{w}_{x}^{2}}{4w}+\displaystyle \frac{3}{2}{{ww}}_{x}+\displaystyle \frac{{w}^{3}}{4}\right]}_{x},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{w}_{t}=\left[{w}_{{xxxx}}-\displaystyle \frac{5}{2w}{w}_{x}{w}_{{xxx}}+\displaystyle \frac{5}{3}{{ww}}_{{xxx}}\right.\\ -\displaystyle \frac{5}{4w}{w}_{{xx}}^{2}+5\left(\displaystyle \frac{{w}^{2}}{4}+\displaystyle \frac{{w}_{x}^{2}}{{w}^{2}}+\displaystyle \frac{3}{4}{w}_{x}\right){w}_{{xx}}\\ -\displaystyle \frac{35{w}_{x}^{4}}{16{w}^{3}}-\displaystyle \frac{15{w}_{x}^{3}}{8w}+\displaystyle \frac{15{{ww}}_{x}^{2}}{8}\\ {\left.+\displaystyle \frac{{w}^{5}}{16}+\displaystyle \frac{5{{ww}}_{y}}{6}+\displaystyle \frac{5{w}^{3}{w}_{x}}{8}\right]}_{x}\end{array}\end{eqnarray}$
of the potential CKP equation ( $\begin{eqnarray}\begin{array}{l}{f}_{y}={\left[{f}_{{xx}}+\displaystyle \frac{1}{4}{f}^{3}+\displaystyle \frac{3}{2}({f}_{x}f)\right]}_{x},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{f}_{t}=\left[{f}_{{xxxx}}+5{f}_{x}{f}_{{xx}}+5\left(\displaystyle \frac{{{ff}}_{{xxx}}}{2}\right.\right.\\ +\displaystyle \frac{{f}^{2}{f}_{{xx}}}{2}+\displaystyle \frac{{f}^{3}{f}_{x}}{4}\\ {\left.\left.+\displaystyle \frac{3{{ff}}_{x}^{2}}{4}\right)+\displaystyle \frac{{f}^{5}}{16}\right]}_{x}\end{array}\end{eqnarray}$
through the relationship ${w}_{x}={fw}-{w}^{2}$, which precisely mirrors the STO decomposition observed in the potential BKP equation (The function w is a solution of the potential CKP equation (
$\begin{eqnarray}\begin{array}{l}{w}_{y}={w}_{{xxx}}+\displaystyle \frac{3{w}_{{xx}}^{2}}{2W}+3{w}_{x}^{2}-\displaystyle \frac{9{{Mw}}_{x}}{2}\\ -\displaystyle \frac{27}{8}{M}^{2}+\displaystyle \frac{3}{2}{m}_{1}x\\ +\displaystyle \frac{27y}{20}{\left({m}_{1}y+2{m}_{2}\right)}_{t}+\displaystyle \frac{27{m}_{2}^{2}}{8}+{m}_{3},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{w}_{t}={w}_{{xxxxx}}+\displaystyle \frac{5{w}_{{xx}}{w}_{{xxxx}}}{W}+\displaystyle \frac{5{w}_{{xxx}}^{2}}{2W}\\ +\left(\displaystyle \frac{20{w}_{{xx}}^{2}}{{W}^{2}}+10{w}_{x}-\displaystyle \frac{15M}{2}\right){w}_{{xxx}}\\ +\displaystyle \frac{35{w}_{{xx}}^{4}}{2{W}^{3}}+\displaystyle \frac{5(3M+4{w}_{x}){w}_{{xx}}^{2}}{4W}\\ +10{w}_{x}^{3}-\displaystyle \frac{45M}{2}{w}_{x}^{2}\\ +\left(\displaystyle \frac{45}{8}{M}^{2}+\displaystyle \frac{45{m}_{2}^{2}}{8}+\displaystyle \frac{5{m}_{1}x}{2}+\displaystyle \frac{5{m}_{3}}{3}\right){w}_{x}\\ +\displaystyle \frac{(9{w}_{x}{y}^{2}+6{xy}-27{y}^{2}M){m}_{1t}}{4}\\ +\displaystyle \frac{(18{w}_{x}y+6x-27y(2{m}_{1}y+{m}_{2})){m}_{2t}}{4}\\ +\displaystyle \frac{9{y}^{2}{\left({m}_{1}y+3{m}_{2}\right)}_{{tt}}}{20}-\displaystyle \frac{15{m}_{1}{Mx}}{2}\\ +\displaystyle \frac{45{M}^{3}}{4}-\displaystyle \frac{135{m}_{1}{m}_{2}^{2}y}{8}\\ +\displaystyle \frac{5{m}_{1}w}{2}-5{m}_{1}{m}_{3}y+{m}_{3t}y+{m}_{4},\end{array}\end{eqnarray}$
where $M\equiv {m}_{1}y+{m}_{2},W\equiv 3M-2{w}_{x}$ and ${m}_{i},i=1,2,3,4$ are arbitrary functions of t.The potential CKP equation (
$\begin{eqnarray}\begin{array}{l}{w}_{y}=\displaystyle \frac{1}{4}{w}_{{xxx}}+\displaystyle \frac{3}{2}{w}_{x}^{2}+\displaystyle \frac{3{m}_{0}x}{5}\\ +\displaystyle \frac{27{y}^{2}({m}_{0t}-2{m}_{0}^{2})}{50}+M,M\\ \equiv {m}_{1}y+{m}_{2},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{w}_{t}=\displaystyle \frac{1}{16}{w}_{{xxxxx}}+\displaystyle \frac{5}{8}(2{w}_{x}{w}_{{xxx}}+{w}_{{xx}}^{2}+4{w}_{x}^{3})\\ +\left({m}_{0}x-\displaystyle \frac{9{m}_{0}^{2}{y}^{2}}{5}+\displaystyle \frac{5}{3}M+\displaystyle \frac{9{m}_{0t}{y}^{2}}{10}\right){w}_{x}\\ +{m}_{0}w+\left(\left(\displaystyle \frac{3{m}_{0t}}{5}-\displaystyle \frac{6{m}_{0}^{2}}{5}\right)y+\displaystyle \frac{5{m}_{1}}{9}\right)x\\ +\left(\displaystyle \frac{18{m}_{0}^{3}}{25}-\displaystyle \frac{27{m}_{0}{m}_{0t}}{25}+\displaystyle \frac{9{m}_{0{tt}}}{50}\right){y}^{3}\\ +\left(\displaystyle \frac{{m}_{1t}}{2}-{m}_{0}{m}_{1}\right){y}^{2}\\ +({m}_{2t}-2{m}_{0}{m}_{2})y+{m}_{3},\end{array}\end{eqnarray}$
where ${m}_{i},i=1,2,3,4$ are arbitrary functions of t.The variable coefficient potential KK decomposition solution of the potential CKP equation (
$\begin{eqnarray}\begin{array}{l}{w}_{y}={{Mw}}_{x}+\displaystyle \frac{3y}{10}{\left(M+{m}_{2}\right)}_{t}+\displaystyle \frac{1}{3}{m}_{1}x\\ -\displaystyle \frac{1}{2}{M}^{2}+\displaystyle \frac{{m}_{2}^{2}}{2}+{m}_{3},M\equiv {m}_{1}y+{m}_{2},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{w}_{t}=-\displaystyle \frac{{w}_{{xxxxx}}}{9}+\displaystyle \frac{5}{9}(M-3{w}_{x}){w}_{{xxx}}\\ -\displaystyle \frac{5{w}_{{xx}}^{2}}{4}-\displaystyle \frac{5{w}_{x}^{3}}{3}+\displaystyle \frac{5M}{3}{w}_{x}^{2}\\ +\left(\displaystyle \frac{5{m}_{1}x}{9}-\displaystyle \frac{5{M}^{2}}{18}+\displaystyle \frac{5{m}_{2}^{2}}{6}+\displaystyle \frac{5{m}_{3}}{3}\right.\\ \left.+\displaystyle \frac{{m}_{1t}{y}^{2}}{2}+{m}_{2t}y\right){w}_{x}+\displaystyle \frac{5{m}_{1}w}{9}\\ +\displaystyle \frac{{y}^{2}}{10}{\left({m}_{1}y+3{m}_{2}\right)}_{{tt}}-\displaystyle \frac{{m}_{1t}{y}^{2}M}{3}\\ +\displaystyle \frac{{{xM}}_{t}}{3}+\displaystyle \frac{{m}_{2t}y({m}_{2}-2{m}_{1}y)}{3}\\ +{m}_{3t}y-\displaystyle \frac{10{m}_{1}{xM}}{27}-\displaystyle \frac{5{m}_{1}y({m}_{2}^{2}+2{m}_{3})}{9}+{m}_{4}\end{array}\end{eqnarray}$
with ${m}_{i},i=1,2,...,4$ being arbitrary functions of t. The proof of theorems
Overall, the decomposition technology offers advantages in:
| 1. | (1) Simplifying analysis: by decomposing the higher-dimensional equation into a lower-dimensional system, the analysis becomes more manageable and can be tackled using techniques specifically designed for lower-dimensional systems. This simplification aids in understanding the behavior and properties of the system under consideration. |
| 2. | (2) Enhancing solvability: variable coefficient potential decompositions can lead to equations with more structured forms, making them potentially easier to solve. This can facilitate the study of the system's solutions, stability, and other important characteristics. |
| 3. | (3) Providing insights into system dynamics: by breaking down the original equation into two separate equations, each focusing on different aspects of the system, this separation can help in identifying and understanding the individual contributions of different factors affecting the system's behavior. |
| 4. | (4) Allowing for flexible parameter analysis: variable coefficient decompositions allow for a more flexible analysis of the system's behavior under varying conditions. By studying how the coefficients affect the solutions, researchers can gain a deeper understanding of the system's response to different parameters and external influences. |
| 5. | (5) Finding practical applications in the study of physical systems: this decomposition technique finds applications in various physical systems where the higher-dimensional equation arises, such as in the study of wave propagation, fluid dynamics, and quantum mechanics. The insights gained from the variable coefficient decompositions can be applied to understand and predict the behavior of these systems in real-world scenarios. |
5. Conclusions
In this research endeavor, we have extended the findings elucidated in a preceding scholarly publication [32] by introducing a modified formal variable separation approach to derive decomposition solutions for the potential BKP and CKP equations. This heuristic method provides a unified framework for obtaining decomposition solutions of the (2+1)-dimensional PDEs. The result makes clear that we establish the connection between the potential BKP and CKP equations with several classical integrable systems. Consequently, one might therefore hope to deduce the solutions of the potential BKP and CKP equations from solutions of the classical integrable systems.
Notably, we have effectively constructed the BTs through the decomposition process. It is widely accepted that the existence of a BT serves as a sign to the integrability of a PDE. While the BTs for the (1+1)-dimensional SK and KK equations have been previously known [33], the BTs for the (2+1)-dimensional SK (4 ) and KK (31 ) equations have not been reported before. In this study, we present the explicit forms of the BTs for the potential BKP and CKP equations. Although other standard analytic techniques exist for obtaining BTs of nonlinear PDEs, the construction of BTs through decomposition proves to be an exceptionally straightforward approach.
For the potential BKP and CKP equations, the system defining the identified BTs exhibits two pivotal characteristics. Firstly, it is linearizable, culminating in the establishment of a Lax pair. Secondly, it showcases a conservation form. Furthermore, it is noteworthy that the solution to the potential BKP equation in theorem 6 possesses a distinctive trait: it adheres to the renowned linear superposition principle under specific parameter values. It is imperative to underscore that despite the apparent resemblance between the potential BKP and CKP equations, they are fundamentally disparate. No scaling transformation exists that can convert one equation into the other. Nevertheless, upon scrutiny of the system (39 )-(40 ), the STO decomposition elucidates a connection between the potential BKP and CKP equations.
Decomposition, as a fundamental tool for formulating solutions of PDEs, proves to be as accessible to apply as we contend. We look forward to delving into additional theoretical facets associated with this approach in forthcoming research pursuits.
Appendix
We elucidate the decomposition methodology by considering the case of the potential BKP equation (5 ).
By substituting (6 ) and (7 ), where $m\gt 3$, into the potential BKP equation (5 ), along with the decomposition compatibility condition (8 ), it is observed that the decomposition does not generally hold for $m\gt 3$. Consequently, we consider the specific case of m = 3 and subsequently set n = 5 in the decomposition relations (6 ) and (7 ). Then substituting (6 ) and (7 ) into (5 ) yields:A2 ), (A1 ) undergoes a transformation, yielding the following expression:
$\begin{eqnarray}\begin{array}{l}{w}_{x6}(1-5{F}_{{x}_{3}}^{2}-5{F}_{{x}_{3}}+{G}_{{x}_{5}})+W=0,\end{array}\end{eqnarray}$
where $F=F(x,y,t,w,{w}_{x},{w}_{x2},{w}_{x3})\equiv F(x,y,t,{x}_{0},{x}_{1},{x}_{2},{x}_{3}),$ $G=G(x,y,t,w,{w}_{x},{w}_{x2},{w}_{x3},{w}_{x4},{w}_{x5})\equiv G(x,y,t,{x}_{0},{x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5}),$ and $W=W(x,y,t,{x}_{0},{x}_{1},\ldots ,{x}_{5})$ is a complicated expression of $x,y,t,{x}_{0},{x}_{1},\ldots ,{x}_{5}$. With the coefficient of wx6 vanishing, we can deduce the following result: $\begin{eqnarray}\begin{array}{l}G=(5{F}_{{x}_{3}}+5{F}_{{x}_{3}}^{2}-1){x}_{5}+{G}_{1},\end{array}\end{eqnarray}$
where ${G}_{1}={G}_{1}(x,y,t,{x}_{0},{x}_{1},\ldots ,{x}_{4})$ is a function of $\{x,y,t,{x}_{0},{x}_{1},\ldots ,{x}_{4}\}$. By employing the relation ( $\begin{eqnarray}\begin{array}{l}{w}_{x5}[{G}_{1{x}_{4}}-5({F}_{{x}_{3}}+2)({x}_{1}{F}_{{x}_{0}{x}_{3}}\\ +{x}_{2}{F}_{{x}_{1}{x}_{3}}+{x}_{3}{F}_{{x}_{2}{x}_{3}}+{x}_{4}{F}_{{x}_{3}{x}_{3}})\\ -5{F}_{{x}_{2}}(1+2{F}_{{x}_{3}})-5{F}_{{{xx}}_{3}}(2+{F}_{{x}_{3}})]\\ +{W}_{1}=0,\end{array}\end{eqnarray}$
where ${W}_{1}={W}_{1}(x,y,t,{x}_{0},{x}_{1},\ldots ,{x}_{4})$ is wx5 independent. Eliminating the coefficient of wx5 yields $\begin{eqnarray}\begin{array}{l}{G}_{1}=5({F}_{{x}_{3}}+2)\left(\displaystyle \frac{1}{2}{x}_{4}{F}_{{x}_{3}{x}_{3}}+{x}_{3}{F}_{{x}_{2}{x}_{3}}\right.\\ \left.+{x}_{2}{F}_{{x}_{1}{x}_{3}}+{x}_{1}{F}_{{x}_{0}{x}_{3}}\right){x}_{4}\\ +5{x}_{4}{F}_{{x}_{2}}(1+2{F}_{{x}_{3}})\\ +5{x}_{4}{F}_{{{xx}}_{3}}(2+{F}_{{x}_{3}})+{G}_{2},\end{array}\end{eqnarray}$
with ${G}_{2}\equiv {G}_{2}(x,y,t,{x}_{0},{x}_{1},{x}_{2},{x}_{3})$ being a function of $\{x,y,t,{x}_{0},{x}_{1},{x}_{2},{x}_{3}\}$.Similarly, by substituting the decomposition (6 ) and (7 ) with m = 3 and n = 5, along with the results (A2 ) and (A4 ), into the compatibility condition (8 ), we obtain the following expression:A5 ) to zero, we obtain the following result:A6 ) into (A5 ) and requiring the coefficient of wx7 being zero result ${F}_{1}(x,y,t,{x}_{0},{x}_{1},{x}_{2})={H}_{1}(y,t)$. Thus, we haveA7 ), (A3 ) becomesA8 ) leads to
$\begin{eqnarray}\begin{array}{l}5{w}_{x7}{\left(1-{F}_{{x}_{3}}\right)}^{2}({x}_{1}{F}_{{x}_{0}{x}_{3}}\\ +{x}_{2}{F}_{{x}_{1}{x}_{3}}+{x}_{3}{F}_{{x}_{2}{x}_{3}}+{x}_{4}{F}_{{x}_{3}{x}_{3}}+{F}_{{{xx}}_{3}})\\ +{\rm{\Gamma }}=0,\end{array}\end{eqnarray}$
where ${\rm{\Gamma }}={\rm{\Gamma }}(x,y,t,{x}_{0},\ldots ,{x}_{6})$ is a wx7 independent function of the lower-order differentiations of w with respect to x. By setting the coefficient of ${w}_{x7}{w}_{x4}$ in ( $\begin{eqnarray}\begin{array}{l}F={F}_{1}(x,y,t,{x}_{0},{x}_{1},{x}_{2}){x}_{3}+H(x,y,t,{x}_{0},{x}_{1},{x}_{2}).\end{array}\end{eqnarray}$
Substituting ( $\begin{eqnarray}\begin{array}{l}F={H}_{1}(y,t){x}_{3}+H(x,y,t,{x}_{0},{x}_{1},{x}_{2}).\end{array}\end{eqnarray}$
Taking account of ( $\begin{eqnarray}\begin{array}{l}{w}_{x4}\left[{G}_{2{x}_{3}}-5({H}_{1}+2){x}_{3}{H}_{{x}_{2}{x}_{2}}\right.\\ -5({H}_{1}+2){x}_{2}{H}_{{x}_{1}{x}_{2}}\\ +5{x}_{1}(3-{H}_{1}{H}_{{x}_{0}{x}_{2}}-3{H}_{1}-2{H}_{{x}_{0}{x}_{2}})\\ -10{H}_{1}{H}_{{x}_{1}}-5{H}_{{x}_{2}}^{2}-5{H}_{{x}_{1}}\\ \left.-5{H}_{1}{H}_{{{xx}}_{2}}-10{H}_{{{xx}}_{2}}\right]+{W}_{2}=0\end{array}\end{eqnarray}$
with ${W}_{2}={W}_{2}(x,y,t,{x}_{0},{x}_{1},{x}_{2},{x}_{3})$. Vanishing the coefficient of wx4 in ( $\begin{eqnarray}\begin{array}{l}{G}_{2}=5\left[\displaystyle \frac{{H}_{1}+2}{2}{x}_{3}{H}_{{x}_{2}{x}_{2}}+({H}_{1}+2){x}_{2}{H}_{{x}_{1}{x}_{2}}\right.\\ -{x}_{1}(3-{H}_{1}{H}_{{x}_{0}{x}_{2}}-3{H}_{1}-2{H}_{{x}_{0}{x}_{2}})\\ +2{H}_{1}{H}_{{x}_{1}}+{H}_{{x}_{2}}^{2}+{H}_{{x}_{1}}\\ \left.+{H}_{1}{H}_{{{xx}}_{2}}+2{H}_{{{xx}}_{2}}\right]{x}_{3}+J,\end{array}\end{eqnarray}$
where $J=J(x,y,t,{x}_{0},{x}_{1},{x}_{2})$. Up to now, the decomposition relation is simplified to $\begin{eqnarray}{w}_{y}={H}_{1}{w}_{x3}+H,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{w}_{t}=(5{H}_{1}+5{H}_{1}^{2}-1){w}_{{x}_{5}}+5{H}_{{x}_{2}}(1+2{H}_{1}){w}_{{x}_{4}}\\ +5\left[\displaystyle \frac{{H}_{1}+2}{2}{w}_{{x}_{3}}{H}_{{x}_{2}{x}_{2}}+({H}_{1}+2){w}_{{x}_{2}}{H}_{{x}_{1}{x}_{2}}\right.\\ -{w}_{{x}_{1}}(3-{H}_{1}{H}_{{x}_{0}{x}_{2}}-3{H}_{1}-2{H}_{{x}_{0}{x}_{2}})\\ +2{H}_{1}{H}_{{x}_{1}}+{H}_{{x}_{2}}^{2}+{H}_{{x}_{1}}+{H}_{1}{H}_{{{xx}}_{2}}\\ \left.+2{H}_{{{xx}}_{2}}\right]{w}_{{x}_{3}}+J\end{array}\end{eqnarray}$
with three undetermined functions ${H}_{1}={H}_{1}(y,t),H\,=H(x,y,t,{x}_{0},{x}_{1},{x}_{2})$ and $J=J(x,y,t,{x}_{0},{x}_{1},{x}_{2})$.Inserting (A10 ) and (A11 ) into the potential BKP equation (5 ) and the compatibility condition (8 ), subsequently, vanishing the coefficients of wxk for $k\geqslant 3$ leave the set of determining equations on $\{H,{H}_{1},J\}$. Solving this set of equations, we can identify several cases that correspond to nontrivial solutions for the unknowns.
Case 1. When ${H}_{1}=0$, further calculation then leads to the expression
$\begin{eqnarray}\begin{array}{l}\left\{\begin{array}{l}H={{Mw}}_{x}+\displaystyle \frac{y}{30}{\left(M+{m}_{2}\right)}_{t}+\displaystyle \frac{1}{3}{m}_{1}x-\displaystyle \frac{1}{2}{M}^{2}+{m}_{3},M\equiv {m}_{1}y+{m}_{2},\\ J=15(M-{w}_{x}){w}_{x}^{2}+(\displaystyle \frac{1}{2}{y}^{2}{m}_{1t}+{{ym}}_{2t}+5{m}_{1}x-\displaystyle \frac{5}{2}{M}^{2}+15{m}_{3}){w}_{x}+5{m}_{1}w+\displaystyle \frac{{y}^{2}}{90}{\left({m}_{1}y+3{m}_{2}\right)}_{{tt}}\\ -\displaystyle \frac{y}{3}({My}-x){m}_{1t}+\displaystyle \frac{1}{3}(x-2{M}_{y}){m}_{2t}-\displaystyle \frac{10}{3}{m}_{1}({Mx}+3{m}_{3}y)+{m}_{3t}y+{m}_{4}\end{array}\right.\end{array}\end{eqnarray}$
with ${m}_{i},i=1,2,...,4$ being arbitrary functions of t.Case 2. Taking ${H}_{1}=1$ in the result of this case, we can identify three distinct solutions.
The first one is as follows:
$\begin{eqnarray}\begin{array}{l}\left\{\begin{array}{l}H=\displaystyle \frac{3{w}_{{xx}}^{2}}{2W}+\displaystyle \frac{3}{2}{w}_{x}^{2}-\displaystyle \frac{27}{4}{M}^{2}+\displaystyle \frac{3}{2}{m}_{1}x+\displaystyle \frac{3y}{20}{\left({m}_{1}y+2{m}_{2}\right)}_{t}+{m}_{3},\\ J=\displaystyle \frac{45}{2}{w}_{x}^{3}+\displaystyle \frac{3}{4}[30{m}_{1}x-135{M}^{2}+3y{\left({m}_{1}y+2{m}_{2}\right)}_{t}+20{m}_{3}]{w}_{x}+\displaystyle \frac{45}{2}{m}_{1}w-\displaystyle \frac{3}{2}(45{m}_{1}M-{m}_{1t}y\\ -{m}_{2t})x+\displaystyle \frac{{y}^{2}}{20}{\left({m}_{1}y+3{m}_{2}\right)}_{{tt}}-\displaystyle \frac{27}{4}y{\left({m}_{1}y+2{m}_{2}\right)}_{t}M+y({m}_{3t}-45{m}_{1}{m}_{3})+{m}_{4}\\ +\displaystyle \frac{45}{2W}[(3M+{w}_{x}){w}_{{xx}}^{2}]+\displaystyle \frac{315}{2{W}^{3}}{w}_{{xx}}^{4}+\displaystyle \frac{405}{4}{M}^{3},\end{array}\right.\end{array}\end{eqnarray}$
where $M\equiv {m}_{1}y+{m}_{2},W\equiv 3M-2{w}_{x}$ and ${m}_{i},i=1,2,3,4$ are arbitrary functions of t.The second solution is given by:5 ).
$\begin{eqnarray}\left\{\begin{array}{l}H=3{w}_{x}^{2}+{{av}}_{x}{w}_{x}+\displaystyle \frac{{a}^{2}}{6}{v}_{x}^{2}-\displaystyle \frac{1}{3}{m}_{0}(x+{m}_{0}{y}^{2})-\displaystyle \frac{{m}_{0t}{y}^{2}}{30}+{m}_{1}y+{m}_{2},\\ J=5a{\left({av}+3w\right)}_{x}{v}_{{xxx}}+15{\left({av}+3w\right)}_{{xx}}{w}_{{xx}}+\displaystyle \frac{15}{2}{w}_{x}[{\left({av}+3w\right)}_{x}^{2}+3{w}_{x}^{2}]+\displaystyle \frac{5{a}^{2}}{2}{v}_{{xx}}^{2}+{w}_{x}(15{m}_{2}\\ +15{m}_{1}y-5{m}_{0}^{2}{y}^{2}-5{m}_{0}x-\displaystyle \frac{1}{2}{m}_{0t}{y}^{2})-5{m}_{0}w-\displaystyle \frac{{xy}}{3}({m}_{0t}+10{m}_{0}^{2})+5{m}_{1}x-\displaystyle \frac{{y}^{3}}{90}({m}_{0{tt}}\\ +30{m}_{0}{m}_{0t}+100{m}_{0}^{3})+\displaystyle \frac{{y}^{2}}{2}({m}_{1t}+10{m}_{0}{m}_{1})+{\left({m}_{2t}+10{m}_{0}{m}_{2}\right)}_{y}+{m}_{4}\end{array}\right.\end{eqnarray}$
with v satisfying $\begin{eqnarray}\left\{\begin{array}{l}{v}_{y}={v}_{{xxx}}-\displaystyle \frac{a}{2}{v}_{x}^{2}+\displaystyle \frac{{m}_{0}}{a}x+{n}_{y},\\ {v}_{t}=9{v}_{{xxxxx}}-15{{av}}_{x}{v}_{{xxx}}-\displaystyle \frac{15a}{2}{v}_{{xx}}^{2}+\displaystyle \frac{5{a}^{2}}{2}{v}_{x}^{3}+[15{m}_{2}+15{m}_{1}y-(5{m}_{0}^{2}+\displaystyle \frac{1}{2}{m}_{0t}){y}^{2}-5{m}_{0}x]{v}_{x}\\ -5{m}_{0}v+[(10{m}_{0}^{2}+{m}_{0t})y-15{m}_{1}]\displaystyle \frac{x}{a}+\displaystyle \frac{{y}^{3}}{6a}{m}_{0}(10{m}_{0}^{2}+{m}_{0t})-\displaystyle \frac{15{y}^{2}}{2a}{m}_{0}{m}_{1}-\displaystyle \frac{15y}{a}{m}_{0}{m}_{2}\\ +5{m}_{0}n+{n}_{t}+{m}_{3}.\end{array}\right.\end{eqnarray}$
The third solution can be expressed as: $\begin{eqnarray}\left\{\begin{array}{l}H=-{v}_{{xxx}}+{v}_{y}+\displaystyle \frac{3}{2}{\left(w+v\right)}_{{xx}}(w-v)+\displaystyle \frac{3}{2}({w}_{x}^{2}-{v}_{x}^{2})+\displaystyle \frac{1}{4}{\left[{\left(w-v\right)}^{3}\right]}_{x},p\equiv w-v,\\ J={v}_{t}-9{v}_{{xxxxx}}+\displaystyle \frac{15}{2}{{pv}}_{{xxxx}}+(\displaystyle \frac{15}{2}{w}_{x}-\displaystyle \frac{105}{2}{v}_{x}){v}_{{xxx}}-45{v}_{{xx}}^{2}+\displaystyle \frac{45}{2}{{pv}}_{x}{v}_{{xx}}+15{{pv}}_{{xy}}+45{w}_{{xx}}^{2}\\ +(\displaystyle \frac{225}{2}{{pw}}_{x}-45{{pv}}_{x}+\displaystyle \frac{45}{4}{p}^{3}){w}_{{xx}}+\displaystyle \frac{45}{4}{v}_{x}^{2}{p}_{x}+\displaystyle \frac{135}{4}{p}^{2}{w}_{x}{p}_{x}+15{p}_{x}{v}_{y}+\displaystyle \frac{135}{4}{w}_{x}^{2}{p}_{x}+\displaystyle \frac{45}{16}{p}^{4}{p}_{x},\end{array}\right.\end{eqnarray}$
in which case, the decomposition solution depends on another solution v of the potential BKP equation (Case 3. ${H}_{1}=-\tfrac{1}{2}$. In this particular case, the functions H and J are determined as follows:A12 )–(A17 ) into the decomposition relations (A10 ) and (A11 ), the validity of theorems 1 ‒ 5 is established.
$\begin{eqnarray}\left\{\begin{array}{l}H=-\displaystyle \frac{3}{2}{w}_{x}^{2}+6{{Mw}}_{x}+\displaystyle \frac{y}{10}{\left({m}_{1}y+2{m}_{2}\right)}_{t}+{m}_{1}x-6{M}^{2}+{m}_{3},M\equiv {m}_{1}y+{m}_{2},\\ J=-\displaystyle \frac{45}{4}({w}_{{xx}}^{2}+2{w}_{x}^{3})+15{m}_{1}w+\displaystyle \frac{3}{2}y{\left({m}_{1}y+2{m}_{2}\right)}_{t}{w}_{x}+15(6{M}^{2}+{m}_{1}x+{m}_{3}){w}_{x}+\displaystyle \frac{{y}^{2}}{30}\left({m}_{1}y\right.\\ {\left.+3{m}_{2}\right)}_{{tt}}-3({y}^{2}{m}_{1t}+10{m}_{1}x+2{{ym}}_{2t})M+{M}_{t}x+({m}_{3t}-30{m}_{1}{m}_{3})y-60{M}^{3}+{m}_{4},\end{array}\right.\end{eqnarray}$
where ${m}_{i},i=1,2,3,4$ are arbitrary functions of t. By substituting the obtained solutions (