1. Introduction
It is well known that Ryogo Hirota’s bilinear method [1, 2] is a powerful tool in the study of integrable systems. With regard to finding solutions, nonlinear equations can be transformed into bilinear forms and then soliton solutions can be derived. According to the bilinear forms, bilinear equations were classified as the Korteweg–de Vries (KdV) type, the modified KdV (mKdV) type, the sine-Gordon (sG) type and the nonlinear Schrödinger (NLS) type by Jarmo Hietarinta [3–6]. It is Hirota who first realized that the KdV-type bilinear equations always have one-soliton and two-soliton solutions [7] but having a three-soliton solution indicates a kind of integrability, which is now known as the integrability of bilinear equations in Hirota’s sense [2, 3, 7]. In a series of celebrated papers in 1987 and 1988, Hietarinta examined the KdV-type, mKdV-type and sG-type bilinear equations that have three-soliton solutions [3–5] and the NLS-type bilinear equations that have two-soliton solutions [6]. Some new bilinear equations integrable in Hirota’s sense were found and so far bilinear forms of some of these equations are still not known.
It is also well known that bilinear equations have remarkable mathematical structures. Sato found that the bilinear Kadomtsev–Petviashvili (KP) hierarchy are the Plücker relations on the infinitely dimensional Grassmannians [8], which has led to a set of beautiful and profound theory for integrable systems developed by the Kyoto group. On the other hand, Lambert, Gilson and their collaborators found connections between Hirota’s bilinear derivatives and the Bell polynomials [9, 10], which led to a mechanism to bilinearize nonlinear equations. This mechanism was later extended to deriving bilinear Bäcklund transformations and Lax pairs, etc [11–13]. It was also extended to the study of supersymmetric systems [14] and can be implemented using symbolic computations [15]. Note that the Bell polynomials approach is in principle applicable for the KdV-type equations (see [16–18] as examples), while the application to the mKdV-type, sG-type and NLS-type equations was less developed.
In this paper, we will describe a procedure to convert bilinear equations into their nonlinear forms. This will enable us to obtain nonlinear forms of the new integrable bilinear equations found by Hietarinta in [3, 4]. Our procedure is also based on the Bell polynomials. In fact, in [9, 10], the connection between Hirota’s bilinear derivatives and the Bell polynomials has been revealed. However, in [9, 10], all attention was paid to bilinearize nonlinear equations rather than the reverse direction. In this paper, we explain how the Bell polynomial approach works in nonlinearization bilinear equations.
The paper is organized as follows. First, in section 2 , we recall the Bell polynomials, binary Bell polynomials and their connections with Hirota’s bilinear derivatives. In section 3 , we show how such a nonlinearization technique works in the KdV-type and mKdV-type equations. Illustrative examples include integrable equations and also non-integrable ones. As a result, we obtain nonlinear forms of some integrable bilinear equations found by Hietarinta in [3, 4]. Finally, concluding remarks are given in section 4 .
2. Bell polynomials theory
2.1. Bell polynomials
The Bell polynomials of our interest are defined as the following.
[19] Let h = h(x) be a C∞ function of $x\in {\mathbb{R}}$ and denote ${h}_{r}={\partial }_{x}^{r}h$ for r = 1, 2, ⋯ . Then,
$\begin{eqnarray}{Y}_{[nx]}(h)\doteq {Y}_{n}({h}_{1},{h}_{2},\cdots \,{,}{h}_{n})={{\rm{e}}}^{-h}{\partial }_{x}^{n}{{\rm{e}}}^{h},\,\,\,\,n=1,2,\cdots \end{eqnarray}$
are known as the Bell polynomials.In 1934, Bell [19] studied these polynomials for several special choices of h(x). These polynomials can be generated recursively by
$\begin{eqnarray}\begin{array}{l}{Y}_{0}=1,\\ {Y}_{n+1}({h}_{1},{h}_{2},\cdots \,{,}{h}_{n+1})=({\partial }_{x}+{h}_{1}){Y}_{n}({h}_{1},{h}_{2},\cdots \,{,}{h}_{n}),\end{array}\end{eqnarray}$
which agrees with the (potential) Burgers hierarchy (see [9, 10]). They can also be represented with a single formula by means of the Arbogast formula [20] or the Fa$\mathop{\,\rm{a}\,}\limits^{\unicode{x00300}}$ di Bruno formula [21, 22]3(3 For this formula one may also refer to [23].) $\begin{eqnarray}\begin{array}{rcl}{Y}_{[nx]}(h) & = & \sum \frac{n!}{{c}_{1}!\cdots {c}_{n}!{(1!)}^{{c}_{1}}\cdots {(n!)}^{{c}_{n}}}{h}_{1}^{{c}_{1}}\cdots {h}_{n}^{{c}_{n}},\\ n & = & 1,2,\cdots \,{,}\end{array}\end{eqnarray}$
where the sum is to be taken over all partitions of n = c1 + 2c2 + ⋯ + ncn. The first few Bell polynomials are $\begin{eqnarray*}\begin{array}{rcl}{Y}_{[x]} & = & {h}_{1},\\ {Y}_{[2x]} & = & {h}_{2}+{h}_{1}^{2},\\ {Y}_{[3x]} & = & {h}_{3}+3{h}_{1}{h}_{2}+{h}_{1}^{3}.\end{array}\end{eqnarray*}$
In the following, for convenience, we call the polynomials defined in (2.1 ) Y-polynomials. Using the relation
$\begin{eqnarray}{(FG)}^{-1}{\partial }_{x}^{n}(FG)=\displaystyle \sum _{p=0}^{n}\left(\genfrac{}{}{0.0pt}{}{n}{p}\right)({F}^{-1}{\partial }_{x}^{n-p}F)({G}^{-1}{\partial }_{x}^{p}G),\end{eqnarray}$
and let F = eh, $G={{\rm{e}}}^{{h}^{{\prime} }}$, one can obtain an addition formula for Y-polynomials: $\begin{eqnarray}\begin{array}{rcl}{Y}_{[nx]}(h+{h}^{{\prime} }) & = & {{\rm{e}}}^{-(h+{h}^{{\prime} })}{\partial }_{x}^{n}{{\rm{e}}}^{(h+{h}^{{\prime} })}=({{\rm{e}}}^{-h}{{\rm{e}}}^{-{h}^{{\prime} }}){\partial }_{x}^{n}({{\rm{e}}}^{h}{{\rm{e}}}^{{h}^{{\prime} }})\\ & = & \displaystyle \sum _{p=0}^{n}\left(\genfrac{}{}{0.0pt}{}{n}{p}\right)({{\rm{e}}}^{-h}{\partial }_{x}^{n-p}{{\rm{e}}}^{h})({{\rm{e}}}^{-{h}^{{\prime} }}{\partial }_{x}^{p}{{\rm{e}}}^{{h}^{{\prime} }})\\ & = & \displaystyle \sum _{p=0}^{n}\left(\genfrac{}{}{0.0pt}{}{n}{p}\right){Y}_{[(n-p)x]}(h){Y}_{[px]}({h}^{{\prime} }).\end{array}\end{eqnarray}$
In addition, replacing h(x) by h( − x) in (2.1 ), one can immediately find that
$\begin{eqnarray}{Y}_{n}(-{h}_{1},{h}_{2},\cdots \,{,}{(-1)}^{n}{h}_{n})={(-1)}^{n}{Y}_{[nx]}(h).\end{eqnarray}$
This also indicates that $\begin{eqnarray}{Y}_{[(2n+1)x]}(h){| }_{{h}_{1}={h}_{3}=\cdots ={h}_{2n+1}=0}=0.\end{eqnarray}$
2.2. Bilinear derivatives and Bell polynomials
For two C∞ functions F(x) and G(x), their Nth order bilinear derivative was introduced by Hirota in 1974 [24]:
$\begin{eqnarray}{D}_{x}^{n}F\cdot G={({\partial }_{x}-{\partial }_{{x}^{{\prime} }})}^{n}F(x)G({x}^{{\prime} }){| }_{{x}^{{\prime} }=x},\end{eqnarray}$
where D is called Hirota’s bilinear operator. Since $\begin{eqnarray}{D}_{x}^{n}F\cdot G=\displaystyle \sum _{p=0}^{n}{(-1)}^{p}\left(\genfrac{}{}{0.0pt}{}{n}{p}\right)({\partial }_{x}^{n-p}F)({\partial }_{x}^{p}G),\end{eqnarray}$
if we introduceF = ef(x), G = eg(x),
it is easy to see that 2.5 ) and (2.6 ), the above relation can be written as
$\begin{eqnarray}\begin{array}{rcl}{(FG)}^{-1}{D}_{x}^{n}F\cdot G & = & \displaystyle \sum _{p=0}^{n}{(-1)}^{p}\left(\genfrac{}{}{0.0pt}{}{n}{p}\right)({F}^{-1}{\partial }_{x}^{n-p}F)({G}^{-1}{\partial }_{x}^{p}G)\\ & = & \displaystyle \sum _{p=0}^{n}{(-1)}^{p}\left(\genfrac{}{}{0.0pt}{}{n}{p}\right){Y}_{[(n-p)x]}(f){Y}_{[px]}(g),\end{array}\end{eqnarray}$
which is expressed in terms of Y-polynomials of f and g. Next, using the formulae ( $\begin{eqnarray}{(FG)}^{-1}{D}_{x}^{n}F\cdot G={Y}_{n}({h}_{1},\cdots \,{,}{h}_{n}){| }_{{h}_{m}={f}_{m}+{(-1)}^{m}{g}_{m}}.\end{eqnarray}$
Equations (2.10 ) and (2.11 ) reveal the relations between Hirota’s bilinear derivatives and the Bell polynomials.
In practice, it is more convenient to use (2.11 ) in its version described in terms of binary Bell polynomials:
$\begin{eqnarray}{{ \mathcal Y }}_{[nx]}(v,u)={Y}_{n}({h}_{1},\cdots \,{,}{h}_{n}){| }_{{h}_{2j-1}={v}_{2j-1},\,{h}_{2j}={u}_{2j}}.\end{eqnarray}$
Note again that for function f(x), by fk we denote ${\partial }_{x}^{k}f(x)$. For convenience, we call ${{ \mathcal Y }}_{[nx]}(v,u)$ ${ \mathcal Y }$-polynomials. The first few of them read
$\begin{eqnarray}{{ \mathcal Y }}_{[x]}(v,u)={v}_{1},\end{eqnarray}$
$\begin{eqnarray}{{ \mathcal Y }}_{[2x]}(v,u)={u}_{2}+{v}_{1}^{2},\end{eqnarray}$
$\begin{eqnarray}{{ \mathcal Y }}_{[3x]}(v,u)={v}_{3}+3{u}_{2}{v}_{1}+{v}_{1}^{3},\end{eqnarray}$
$\begin{eqnarray}{{ \mathcal Y }}_{[4x]}(v,u)={u}_{4}+4{v}_{3}{v}_{1}+3{u}_{2}^{2}+6{u}_{2}{v}_{1}^{2}+{v}_{1}^{4}.\end{eqnarray}$
Thus, the relation (2.11 ) is written as 2.12 ) together with property (2.7 ), we have
$\begin{eqnarray}{(FG)}^{-1}{D}_{x}^{n}F\cdot G={{ \mathcal Y }}_{[nx]}(v={\mathrm{ln}}\,(F/G),u={\mathrm{ln}}\,(FG)),\end{eqnarray}$
where we have taken u = f + g and v = f − g. In particular, the bilinear derivatives ${D}_{x}^{n}F\cdot F$ can be expressed using ${{ \mathcal Y }}_{[nx]}(0,u)$. When n is odd, from the definition ( $\begin{eqnarray}{{ \mathcal Y }}_{[(2j+1)x]}(v=0,u)=0,\end{eqnarray}$
which agrees with the property ${D}_{x}^{2j+1}F\cdot F=0$. If n is even, we denote (P-polynomials for short) $\begin{eqnarray}{P}_{[2jx]}(u)\equiv {{ \mathcal Y }}_{[2jx]}(0,u),\end{eqnarray}$
the first few of which are $\begin{eqnarray}{P}_{[0]}(u)=1,\end{eqnarray}$
$\begin{eqnarray}{P}_{[2x]}(u)={u}_{2},\end{eqnarray}$
$\begin{eqnarray}{P}_{[4x]}(u)={u}_{4}+3{u}_{2}^{2},\end{eqnarray}$
$\begin{eqnarray}{P}_{[6x]}(u)={u}_{6}+15{u}_{2}{u}_{4}+15{u}_{2}^{3}.\end{eqnarray}$
Thus, for n = 2j being even, we have
$\begin{eqnarray}{F}^{-2}{D}_{x}^{2j}F\cdot F={P}_{[2jx]}(u=2\,{\mathrm{ln}}\,F).\end{eqnarray}$
Note that the binary Bell polynomials ${{ \mathcal Y }}_{[nx]}(v,u)$ can be expressed using Y-polynomials and P-polynomials. In fact, 2.5 ) has been used. Then, using (2.12 ), (2.15 ) and (2.16 ), we have
$\begin{eqnarray*}\begin{array}{l}{{ \mathcal Y }}_{[nx]}(v,u)\,=\,{{ \mathcal Y }}_{[nx]}(v,v+u-v)\,\\ \,={Y}_{n}({v}_{1}+0,{v}_{2}+{(u-v)}_{2},\cdots \,)\\ \,={Y}_{[nx]}(v+h){| }_{{h}_{2j-1}=0,{h}_{2j}={(u-v)}_{2j}}\\ \,=\displaystyle \sum _{r=0}^{n}\left(\genfrac{}{}{0.0pt}{}{n}{r}\right){Y}_{[(n-r)x]}(v){Y}_{[rx]}(h){| }_{{h}_{2j-1}=0,{h}_{2j}={(u-v)}_{2j}},\end{array}\end{eqnarray*}$
where the addition formula ( $\begin{eqnarray}\begin{array}{l}{{ \mathcal Y }}_{[nx]}(v,u)=\displaystyle \sum _{j=0}^{[n/2]}\left(\genfrac{}{}{0.0pt}{}{n}{2j}\right){Y}_{[(n-2j)x]}(v){P}_{[2jx]}(u-v),\end{array}\end{eqnarray}$
where [n/2] denotes the floor function of n/2.2.3. Multidimensional generalization
Assume h = h(x1, x2, ⋯ , xℓ) is an ℓ-dimension C∞ function of (x1, x2, ⋯ , xℓ). Denote
$\begin{eqnarray*}{h}_{{r}_{1},\cdots \,{,}{r}_{\ell }}={\partial }_{{x}_{1}}^{{r}_{1}}\cdots {\partial }_{{x}_{\ell }}^{{r}_{\ell }}h({x}_{1},\cdots \,{,}{x}_{\ell }).\end{eqnarray*}$
The ℓ-dimension Bell polynomials of h are defined by $\begin{eqnarray}{Y}_{[{n}_{1}{x}_{1},\cdots \,{,}{n}_{\ell }{x}_{\ell }]}(h)\doteq {Y}_{{n}_{1},\cdots \,{,}{n}_{\ell }}(\{{h}_{{r}_{1},\cdots \,{,}{r}_{\ell }}\})={{\rm{e}}}^{-h}{\partial }_{{x}_{1}}^{{n}_{1}}\cdots {\partial }_{{x}_{\ell }}^{{n}_{\ell }}{{\rm{e}}}^{h},\end{eqnarray}$
which are alternatively expressed through $\begin{eqnarray}{Y}_{[{n}_{1}{x}_{1},\cdots \,{,}{n}_{\ell }{x}_{\ell }]}(h)=\sum \frac{{n}_{1}!{n}_{2}!\cdots {n}_{\ell }!}{{c}_{1}!{c}_{2}!\cdots {c}_{k}!}\displaystyle \prod _{j=1}^{k}{\left(\frac{{h}_{{r}_{1j},\cdots \,{,}{r}_{\ell {j}}}}{{r}_{1j}!\cdots {r}_{\ell {j}}!}\right)}^{{c}_{j}}.\end{eqnarray}$
Here we just give two examples. When h = h(x1, x2), we have $\begin{eqnarray*}\begin{array}{rcl}{Y}_{[{x}_{1},{x}_{2}]} & = & {h}_{1,0}{h}_{0,1}+{h}_{1,1},\\ {Y}_{[2{x}_{1},{x}_{2}]} & = & 2{h}_{1,0}{h}_{1,1}+{h}_{1,0}^{2}{h}_{0,1}+{h}_{2,0}{h}_{0,1}+{h}_{2,1},\\ {Y}_{[3{x}_{1},{x}_{2}]} & = & 3{h}_{1,0}^{2}{h}_{1,1}+3{h}_{1,1}{h}_{2,0}+3{h}_{1,0}{h}_{2,1}\\ & & +\,{h}_{0,1}({h}_{1,0}^{3}+3{h}_{1,0}{h}_{2,0}+{h}_{3,0})+{h}_{3,1},\end{array}\end{eqnarray*}$
and when h = h(x1, x2, x3), we have $\begin{eqnarray*}\begin{array}{rcl}{Y}_{[{x}_{1},{x}_{2},{x}_{3}]} & = & {h}_{1,0,0}{h}_{0,1,1}+{h}_{1,0,1}{h}_{0,1,0}+{h}_{1,1,0}{h}_{0,0,1}\\ & & +\,{h}_{1,0,0}{h}_{0,1,0}{h}_{0,0,1}+{h}_{1,1,1}.\end{array}\end{eqnarray*}$
These polynomials allow addition formula
$\begin{eqnarray}{Y}_{[{n}_{1}{x}_{1},\cdots \,{,}{n}_{\ell }{x}_{\ell }]}(f+g)=\displaystyle \sum _{{p}_{1}=0}^{{n}_{1}}\cdots \displaystyle \sum _{{p}_{\ell }=0}^{{n}_{\ell }}\displaystyle \prod _{i=1}^{\ell }\left(\genfrac{}{}{0.0pt}{}{{n}_{i}}{{p}_{i}}\right){Y}_{[({n}_{1}-{p}_{1}){x}_{1},\cdots \,{,}({n}_{\ell }-{p}_{\ell }){x}_{\ell }]}(f){Y}_{[{p}_{1}{x}_{1},\cdots \,{,}{p}_{\ell }{x}_{\ell }]}(g)\end{eqnarray}$
and an identity $\begin{eqnarray}\begin{array}{l}{Y}_{{n}_{1},\cdots \,{,}{n}_{\ell }}(\{{(-1)}^{{r}_{1}+\cdots +{r}_{\ell }}{h}_{{r}_{1},\cdots \,{,}{r}_{\ell }}\})\\ =\,{(-1)}^{{n}_{1}+\cdots +{n}_{\ell }}{Y}_{{n}_{1},\cdots \,{,}{n}_{\ell }}(\{{h}_{{r}_{1},\cdots \,{,}{r}_{\ell }}\}).\end{array}\end{eqnarray}$
For F and G being C∞ functions of x = (x1, x2, ⋯ , xℓ), their bilinear derivative is defined as
$\begin{eqnarray*}\begin{array}{l}{D}_{{x}_{1}}^{{n}_{1}}\cdots {D}_{{x}_{\ell }}^{{n}_{\ell }}F({\boldsymbol{x}})\cdot G({\boldsymbol{x}})\\ \,=\,{({\partial }_{{x}_{1}}-{\partial }_{{x}_{1}^{{\prime} }})}^{{n}_{1}}\cdots {({\partial }_{{x}_{\ell }}-{\partial }_{{x}_{\ell }^{{\prime} }})}^{{n}_{\ell }}F({\boldsymbol{x}})G({{\boldsymbol{x}}}^{{\prime} }){| }_{{{\boldsymbol{x}}}^{{\prime} }={\boldsymbol{x}}}.\end{array}\end{eqnarray*}$
Introducing F = ef(x) and G = eg(x), we can have
$\begin{eqnarray}\begin{array}{l}{(FG)}^{-1}{D}_{{x}_{1}}^{{n}_{1}}\cdots {D}_{{x}_{\ell }}^{{n}_{\ell }}F\cdot G\\ \,={Y}_{{n}_{1},\cdots \,{,}{n}_{\ell }}(\{{h}_{{r}_{1},\cdots \,{,}{r}_{\ell }}={f}_{{r}_{1},\cdots \,{,}{r}_{\ell }}+{(-1)}^{{r}_{1}+\cdots +{r}_{\ell }}{g}_{{r}_{1},\cdots \,{,}{r}_{\ell }}\}).\end{array}\end{eqnarray}$
We now introduce multidimensional binary Bell polynomials [9, 10]:
$\begin{eqnarray}{{ \mathcal Y }}_{[{n}_{1}{x}_{1},\cdots \,{,}{n}_{\ell }{x}_{\ell }]}(v,u)\doteq {Y}_{{n}_{1},\cdots \,{,}{n}_{\ell }}(\{{h}_{{r}_{1},\ldots ,{r}_{\ell }}\}),\end{eqnarray}$
where $\begin{eqnarray}{h}_{{r}_{1},\cdots \,{,}{r}_{\ell }}=\left\{\begin{array}{ll}{u}_{{r}_{1},\cdots \,{,}{r}_{\ell }}, & {\rm{if}}\,{r}_{1}+\cdots +{r}_{\ell }\,{\rm{is}}\,{\rm{even}},\\ {v}_{{r}_{1},\cdots \,{,}{r}_{\ell }}, & {\rm{if}}\,{r}_{1}+\cdots +{r}_{\ell }\,{\rm{is}}\,{\rm{odd}}.\end{array}\right.\end{eqnarray}$
Similar to the one-dimension case, we have
$\begin{eqnarray*}\begin{array}{l}{{ \mathcal Y }}_{[{n}_{1}{x}_{1},\cdots \,{,}{n}_{\ell }{x}_{\ell }]}(v=0,u)=0,\,\,\,\\ {\rm{if}}\,{n}_{1}+\cdot \cdot \cdot +{n}_{\ell }\,{\rm{is}}\,{\rm{odd}}.\end{array}\end{eqnarray*}$
When n1 + ⋯ + nℓ is even, we denote
$\begin{eqnarray}{P}_{[{n}_{1}{x}_{1},\cdots \,{,}{n}_{\ell }{x}_{\ell }]}(u)={{ \mathcal Y }}_{[{n}_{1}{x}_{1},\cdots \,{,}{n}_{\ell }{x}_{\ell }]}(v=0,u).\end{eqnarray}$
In two-dimension case, i.e., ℓ = 2 and x1 = x, x2 = y, the first few ${ \mathcal Y }$-polynomials are
$\begin{eqnarray}{{ \mathcal Y }}_{[x]}(v,u)={v}_{x},\end{eqnarray}$
$\begin{eqnarray}{{ \mathcal Y }}_{[2x]}(v,u)={u}_{2x}+{v}_{x}^{2},\end{eqnarray}$
$\begin{eqnarray}{{ \mathcal Y }}_{[x,y]}(v,u)={u}_{x,y}+{v}_{x}{v}_{y},\end{eqnarray}$
$\begin{eqnarray}{{ \mathcal Y }}_{[3x]}(v,u)={v}_{3x}+3{v}_{x}{u}_{2x}+{v}_{x}^{3},\end{eqnarray}$
$\begin{eqnarray}{{ \mathcal Y }}_{[2x,y]}(v,u)={v}_{2x,y}+2{v}_{x}{u}_{x,y}+{v}_{x}^{2}{v}_{y}+{u}_{2x}{v}_{y},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{{ \mathcal Y }}_{[3x,y]}(v,u)=3{v}_{x}^{2}{u}_{x,y}+3{u}_{x,y}{u}_{2x}+3{v}_{x}{v}_{2x,y}\\ +{v}_{y}({v}_{x}^{3}+3{v}_{x}{u}_{2x}+{v}_{3x})+{u}_{3x,y}.\end{array}\end{eqnarray}$
The simplest example of three-dimension case ((x1, x2, x3) = (x, y, z)) is
$\begin{eqnarray}{{ \mathcal Y }}_{[x,y,z]}(v,u)={v}_{x,y,z}+{v}_{x}{u}_{y,z}+{v}_{y}{u}_{x,z}+{v}_{z}{u}_{x,y}+{v}_{x}{v}_{y}{v}_{z}.\end{eqnarray}$
The first few P-polynomials are
$\begin{eqnarray}{P}_{[2x]}(u)={u}_{2x},\end{eqnarray}$
$\begin{eqnarray}{P}_{[x,y]}(u)={u}_{x,y},\end{eqnarray}$
$\begin{eqnarray}{P}_{[2x,2y]}(u)=2{u}_{x,y}^{2}+{u}_{2x}{u}_{2y}+{u}_{2x,2y},\end{eqnarray}$
$\begin{eqnarray}{P}_{[3x,y]}(u)={u}_{3x,y}+3{u}_{2x}{u}_{x,y}.\end{eqnarray}$
Here, we use ukx,sy to denote ${\partial }_{x}^{k}{\partial }_{y}^{s}u(x,y)$ without making any confusion.
With regard to the connection between the multidimensional binary Bell polynomials and bilinear derivatives of F and G, by taking F = ef, G = eg, u = f + g and v = f − g, we can have (cf.(2.24 ))
$\begin{eqnarray}\begin{array}{l}{(FG)}^{-1}{D}_{{x}_{1}}^{{n}_{1}}\cdots {D}_{{x}_{\ell }}^{{n}_{\ell }}F\cdot G\\ ={{ \mathcal Y }}_{[{n}_{1}{x}_{1},\ldots ,{n}_{\ell }{x}_{\ell }]}(v={\mathrm{ln}}\,(F/G),u={\mathrm{ln}}\,(FG)),\end{array}\end{eqnarray}$
i.e., $\begin{eqnarray}\begin{array}{l}{(FG)}^{-1}{D}_{{x}_{1}}^{{n}_{1}}\cdots {D}_{{x}_{\ell }}^{{n}_{\ell }}F\cdot G\\ \,=\displaystyle \sum _{{p}_{1}=0}^{{n}_{1}}\cdots \displaystyle \sum _{{p}_{\ell }=0}^{{n}_{\ell }}\displaystyle \prod _{i=1}^{\ell }\left(\genfrac{}{}{0.0pt}{}{{n}_{i}}{{p}_{i}}\right){Y}_{[({n}_{1}-{p}_{1}){x}_{1},\cdots \,{,}({n}_{\ell }-{p}_{\ell }){x}_{\ell }]}(v){P}_{[{p}_{1}{x}_{1},\cdots \,{,}{p}_{\ell }{x}_{\ell }]}(2g),\end{array}\end{eqnarray}$
and in particular, when F = G, we have $\begin{eqnarray}{F}^{-2}{D}_{{x}_{1}}^{{n}_{1}}\cdots {D}_{{x}_{\ell }}^{{n}_{\ell }}F\cdot F={P}_{[{n}_{1}{x}_{1},\ldots ,{n}_{\ell }{x}_{\ell }]}(u=2\,{\mathrm{ln}}\,F),\end{eqnarray}$
where n1 + ⋯ + nℓ is even. These formulae provide relations that will be used to convert Hirota’s bilinear equations into nonlinear forms.Finally, we also point out that, from the definition (2.20 ), any Bell polynomial define in (2.20 ) can be linearized through a logarithmic transformation $h({\boldsymbol{x}})={\mathrm{ln}}\,\psi ({\boldsymbol{x}})$:
$\begin{eqnarray}\psi \,{Y}_{[{n}_{1}{x}_{1},\cdots \,{,}{n}_{\ell }{x}_{\ell }]}(h){| }_{h={\mathrm{ln}}\,\psi }={\psi }_{{n}_{1}{x}_{1},\cdots \,{,}{n}_{\ell }{x}_{\ell }}.\end{eqnarray}$
This is useful in deriving linear problems (Lax pairs) from bilinear Bäcklund transformations, e.g.[12].
3. Convert Hirota bilinear equations to nonlinear forms
In this section, we provide examples to show how nonlinear equations are recovered from their bilinear forms.
3.1. Examples of the KdV-type
The KdV-type bilinear equations are [7]3.1 ). In 1980, Hirota showed that a KdV-type bilinear equation always has one-soliton and two-soliton solutions and he proposed three-soliton condition as an integrable criteria for bilinear equations of the KdV-type [7]. Later, Hietarinta searched the KdV-type bilinear equations that have three-soliton solutions [3]. Some examples on the Hietarinta’s list are (see table 4 in [3])3.2a ) is the well known bilinear KP equation. (3.2b ) and (3.2c ) belongs to the bilinear BKP hierarchy (see page 768 and 769 in [25]) and allows various reductions (see [26] and the references therein). (3.2d ) is new, now called Hietarinta’s equation (cf.[27]), and its nonlinear form has not yet been known. The first three equations allow dimension reductions of traveling wave type, e.g. Dy = a1Dx + a2Dy and the resulting two-dimensional equations still admit three-soliton solutions [3]. Apart from the above equations, all two-dimensional equations in the ‘Accepted final result’ columns in table 1, 2 and 3 in [3] provide one-dimensional and two-dimensional bilinear equations that have three-soliton solutions.
$\begin{eqnarray}Q({D}_{x},{D}_{y},{D}_{t},\cdots \,)F\cdot F=0,\end{eqnarray}$
where Q is a polynomial with constant coefficients and F = 0 must be a solution of ( $\begin{eqnarray}(1)\,\,\,({D}_{x}^{4}-4{D}_{x}{D}_{t}+3{D}_{y}^{2})F\cdot F=0,\end{eqnarray}$
$\begin{eqnarray}(2)\,\,\,({D}_{x}^{3}{D}_{t}+a{D}_{x}^{2}+{D}_{t}{D}_{y})F\cdot F=0,\end{eqnarray}$
$\begin{eqnarray}(3)\,\,\,({D}_{x}^{6}+5{D}_{x}^{3}{D}_{t}-5{D}_{t}^{2}+{D}_{x}{D}_{y})F\cdot F=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}(4)\,\,\,[{D}_{x}{D}_{t}({D}_{x}^{2}+\sqrt{3}{D}_{x}{D}_{t}+{D}_{t}^{2})\\ +a{D}_{x}^{2}+b{D}_{x}{D}_{t}+c{D}_{t}^{2}]F\cdot F=0,\end{array}\end{eqnarray}$
where a, b, c are constants. Among them, (In what follows, we show how bilinear equations are converted into nonlinear forms by using the formulae we got in section 2 . First, for the bilinear KP equation (3.2a ), taking 2.32 ), we have
$\begin{eqnarray}q=2\,{\mathrm{ln}}\,F\end{eqnarray}$
and using formula ( $\begin{eqnarray}\begin{array}{l}{F}^{-2}({D}_{x}^{4}-4{D}_{x}{D}_{t}+3{D}_{y}^{2})F\cdot F\\ ={P}_{[4x]}(q)-4{P}_{[x,t]}(q)+3{P}_{[2y]}(q)\\ ={q}_{4x}+3{q}_{2x}^{2}-4{q}_{x,t}+3{q}_{2y}.\end{array}\end{eqnarray}$
Letting u = q2x, we get the well known KP equation
$\begin{eqnarray}{({u}_{xxx}+6u{u}_{x}-4{u}_{t})}_{x}+3{u}_{yy}=0.\end{eqnarray}$
For the second equation in (3.2), using (3.3 ) and (2.32 ), the equation (3.2b ) gives to 3.2b )) in [25] as a member in the BKP hierarchy.4(4 This equation is given in item (4) in page 699 of [25], corresponding a plane wave factor ${{\rm{e}}}^{kx+{k}^{3}y+{k}^{-1}t}$ up to some scaling. It can be considered as the first member in the negative BKP hierarchy. So, strictly speaking, it should be called the negative BKP equation or BKP( − 1) equation for short.) It is also a (2+1)-dimensional extension of the model equation for shallow water waves (y = x) (cf equation (2) and (7) [28]). In addition, introducing w and v by 3.7 ) is written as 2.16 ) in [29]), which is connected with the bilinear form (3.2b ) by $w=2{({\mathrm{ln}}\,F)}_{xt}$ and $v=2{({\mathrm{ln}}\,F)}_{xx}$. Of course, it is actually the BKP(−1) equation (3.7 ) via (3.8 ). Note that the term avx in this equation can be removed by a simple transformation $w=\bar{w}-\frac{a}{3}$. Thus, on the nonlinear level, the parameter a in the bilinear equation (3.2b ) maybe not important.
$\begin{eqnarray}\begin{array}{l}{P}_{[3x,t]}(q)+a{P}_{[2x]}(q)+{P}_{[t,y]}(q)\\ =\,{q}_{3x,t}+3{q}_{2x}{q}_{x,t}+a{q}_{2x}+{q}_{t,y}=0.\end{array}\end{eqnarray}$
Setting u = qx we get $\begin{eqnarray}{u}_{xxxt}+a{u}_{xx}+3{({u}_{x}{u}_{t})}_{x}+{u}_{yt}=0,\end{eqnarray}$
which was first derived (together with the bilinear form ( $\begin{eqnarray}w={u}_{t},\,\,v={u}_{x},\end{eqnarray}$
equation ( $\begin{eqnarray}{w}_{y}+{w}_{xxx}+a{v}_{x}+3{(wv)}_{x}=0,\,\,\,{w}_{x}={v}_{t}.\end{eqnarray}$
This is known as the Boiti–Leon–Manna–Pempinelli (BLMP) equation (see equation (For the third equation (3.2c ), by using (3.3 ) and (2.32 ), it yields
$\begin{eqnarray}\begin{array}{l}{q}_{6x}+15{q}_{2x}{q}_{4x}+15{q}_{2x}^{3}\\ \,+\,5({q}_{3x,t}+3{q}_{2x}{q}_{x,t})-5{q}_{2t}+{q}_{x,y}=0,\end{array}\end{eqnarray}$
which, by setting u = qx, leads to $\begin{eqnarray}\begin{array}{l}({u}_{y}+15{u}_{x}{u}_{xxx}+15{u}_{x}^{3}+15{u}_{x}{u}_{t}+{u}_{xxxxx}{)}_{x}\\ \,+\,5{u}_{xxxt}-5{u}_{tt}=0,\end{array}\end{eqnarray}$
which is the BKP equation (see item (2) on page 768 in [25], corresponding to the plane wave factor ${{\rm{e}}}^{kx+{k}^{3}t+{k}^{5}y}$ up to some scaling).For the Hietarinta equations (3.2d ), from (3.3 ) and (2.32 ) we get
$\begin{eqnarray}\begin{array}{l}{q}_{3x,t}+3{q}_{2x}{q}_{x,t}+\sqrt{3}({q}_{2x,2t}+{q}_{2x}{q}_{2t}+2{q}_{x,t}^{2})\\ +{q}_{x,3t}+3{q}_{2t}{q}_{x,t}+a{q}_{2x}+b{q}_{x,t}+c{q}_{2t}=0.\end{array}\end{eqnarray}$
We can either use it as the nonlinear form of (3.2d ), or after introducing u = qx, get 3.2d ) is (by introducing u = qx, v = qt)
$\begin{eqnarray}\begin{array}{l}{u}_{xxt}+{u}_{ttt}+3{u}_{t}({u}_{x}+{\partial }_{x}^{-1}{u}_{tt})\\ +\sqrt{3}({u}_{xtt}+{u}_{x}{\partial }_{x}^{-1}{u}_{tt}+2{u}_{t}^{2})\\ +a{u}_{x}+b{u}_{t}+c{\partial }_{x}^{-1}{u}_{tt}=0,\end{array}\end{eqnarray}$
where ${\partial }_{x}^{-1}\cdot ={\int }_{-\infty }^{x}\cdot \,{\rm{d}}x$. A third nonlinear form of ( $\begin{eqnarray}\begin{array}{l}{v}_{xxx}+{v}_{xtt}+3{v}_{x}({u}_{x}+{v}_{t})\\ +\sqrt{3}({v}_{xxt}+{u}_{x}{v}_{t}+2{v}_{x}^{2})+a{u}_{x}+b{v}_{x}+c{v}_{t}=0,\,\,\,{v}_{x}={u}_{t}.\end{array}\end{eqnarray}$
All the equations in Hietarinta’s list (3.2) allow traveling-wave extension, in other words, e.g. replacing Dy by Dy + Dz in (3.2a ), the obtained bilinear equation still has three-soliton solution. For the bilinear KP equation (3.2a ), such an extension yields
$\begin{eqnarray}({D}_{x}^{4}-4{D}_{x}{D}_{t}+3{({D}_{y}+{D}_{z})}^{2})F\cdot F=0.\end{eqnarray}$
The resulting nonlinear equation (via $q=2\,{\mathrm{ln}}\,F$ and u = q2x) reads
$\begin{eqnarray}{({u}_{xxx}+6u{u}_{x}-4{u}_{t})}_{x}+3({u}_{yy}+2{u}_{yz}+{u}_{zz})=0.\end{eqnarray}$
Such type of extended KP equations have been frequently studied, e.g. [30, 31]. However, the traveling-wave type extension is trivial since solution of the extended equation can be easily obtained by making corresponding replacement of independent variables in the elementary plane wave factor, e.g., for (3.16 ), just replacing y → y + z in the plane wave factor of the KP solutions.
One more integrable example is a four-dimensional bilinear equation 3.2b ) by taking z = x. It is interesting that the bilinear equation is also related to toroidal algebra ${{\rm{sl}}}_{2}^{{\rm{tor}}}$ (see (3.6) in [34] and degree 3 case in table 1 in [35], up to some scaling). Note that the bilinear equation (3.17 ) coupled with the bilinear KdV equation $({D}_{x}^{4}-4{D}_{x}{D}_{t})F\cdot F=0$ provide a bilinear form for Bogoyavlensky’s breaking soliton KdV equation [35]. Using $q=2\,{\mathrm{ln}}\,F$ and formula (2.32 ), from the above bilinear equation we have 3.17 ): (via $q=2\,{\mathrm{ln}}\,F$ and u = qx) 3.17 ) via $w=2{({\mathrm{ln}}\,F)}_{xy}$ and $v=2{({\mathrm{ln}}\,F)}_{xx}$.
$\begin{eqnarray}({D}_{x}^{3}{D}_{y}+2{D}_{y}{D}_{t}-3{D}_{x}{D}_{z})F\cdot F=0,\end{eqnarray}$
which appears in the bilinear KP hierarchy (see page 995 of [32] and equation (4.36b) in [33]), but not the BKP hierarchy. However, it will recover the bilinear BKP(−1) equation ( $\begin{eqnarray*}\begin{array}{l}{P}_{[3x,y]}(q)+2{P}_{[y,t]}(q)-3{P}_{[x,z]}(q)\\ =\,{q}_{3x,y}+3{q}_{2x}{q}_{x,y}+2{q}_{y,t}-3{q}_{x,z}=0.\end{array}\end{eqnarray*}$
Thus, we obtain a nonlinear form of ( $\begin{eqnarray}{u}_{xxxy}+3{({u}_{x}{u}_{y})}_{x}+2{u}_{yt}-3{u}_{xz}=0.\end{eqnarray}$
If we further introduce w = uy and v = ux, we get $\begin{eqnarray}{w}_{xxx}+3{(wv)}_{x}+2{w}_{t}-3{v}_{z}=0,\,\,\,{v}_{y}={w}_{x},\end{eqnarray}$
which is now connected with the bilinear equation (Compared with the BLMP equation (3.9 ) (switching t ↔ y), we may consider (3.19 ) as a four-dimensional extension of (3.9 ), or, we say (3.18 ) is a four-dimensional extension of the BKP(− 1) equation (3.7 ).
3.2. Non-integrable examples of the KdV-type
As we mentioned, Hirota showed that a KdV-type bilinear equation always has one-soliton and two-soliton solutions [7] but may not have a three-soliton solution. These means some nonlinear PDEs might not be integrable although they have two-soliton solutions. In the following we provide three such non-integrable examples of the KdV-type.
The first example is 3.20 ).) By setting $q=2\,{\mathrm{ln}}\,F$ and using formula (2.32 ), we obtain
$\begin{eqnarray}({D}_{x}^{3}{D}_{y}+{D}_{x}{D}_{t})F\cdot F=0.\end{eqnarray}$
Note that this equation is not integrable in Hirota’s sense because it does not have a three-soliton solution.5(5 In fact, this equation has the following solutions $\begin{eqnarray*}\begin{array}{l}F=1+{{\rm{e}}}^{{\xi }_{1}},\,\,F=1+{{\rm{e}}}^{{\xi }_{1}}+{{\rm{e}}}^{{\xi }_{2}}+{A}_{12}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}},\\ {\xi }_{i}={k}_{i}x+{s}_{i}y-{s}_{i}{k}_{i}^{2}t+{\xi }_{i}^{(0)},\,\,\,{A}_{ij}=\frac{({k}_{i}-{k}_{j})[{s}_{j}{k}_{i}({k}_{i}-2{k}_{j})+{s}_{i}(2{k}_{i}-{k}_{j}){k}_{j}]}{({k}_{i}+{k}_{j})[{s}_{j}{k}_{i}({k}_{i}+2{k}_{j})+{s}_{i}(2{k}_{i}+{k}_{j}){k}_{j}]},\end{array}\end{eqnarray*}$
where ki, si and ${\xi }_{i}^{(0)}$ are constants. However, it can be checked that $\begin{eqnarray*}\begin{array}{l}F=1+{{\rm{e}}}^{{\xi }_{1}}+{{\rm{e}}}^{{\xi }_{2}}+{{\rm{e}}}^{{\xi }_{3}}+{A}_{12}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}}+{A}_{13}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{3}}+{A}_{23}{{\rm{e}}}^{{\xi }_{2}+{\xi }_{3}}\\ \,+\,{A}_{12}{A}_{13}{A}_{23}{{\rm{e}}}^{{\xi }_{1}+{\xi }_{2}+{\xi }_{3}}\end{array}\end{eqnarray*}$
is not a solution of ( $\begin{eqnarray}{P}_{3x,y}(q)+{P}_{x,t}(q)={q}_{xxxy}+3{q}_{xx}{q}_{xy}+{q}_{xt}=0.\end{eqnarray}$
If we just take w = qx, we have a simple equation
$\begin{eqnarray}{w}_{t}+{w}_{xxy}+3{w}_{x}{w}_{y}=0.\end{eqnarray}$
The second example is 2.32 ), and then taking u = qx, one can get its nonlinear form
$\begin{eqnarray}({D}_{t}{D}_{y}-{D}_{x}^{3}{D}_{y}-3{D}_{x}^{2}+3{D}_{z}^{2})F\cdot F=0,\end{eqnarray}$
which was first considered as an example allows resonance of solitons in [39]. Setting $q=2\,{\mathrm{ln}}\,F$ and using formula ( $\begin{eqnarray}{u}_{ty}-{u}_{xxxy}-3{({u}_{x}{u}_{y})}_{x}-3{u}_{xx}+3{u}_{zz}=0.\end{eqnarray}$
Since the KdV-type bilinear equations always admit one-soliton and two-soliton solutions, even they are not integrable, one can freely choose a KdV-type bilinear equation, then the corresponding nonlinear form will always have one-soliton and two-soliton solutions. Here we just add one more non-integrable example, which is related to the bilinear form
$\begin{eqnarray}({D}_{x}{D}_{t}+{D}_{x}^{2}{D}_{y}^{2})F\cdot F=0,\end{eqnarray}$
and the nonlinear form reads (via $w(x,y,t)=2{({\mathrm{ln}}\,F)}_{x}$) $\begin{eqnarray}{w}_{t}+2{w}_{y}^{2}+{w}_{xyy}+{w}_{x}{\int }_{-\infty }^{x}{w}_{yy}\,{\rm{d}}x=0.\end{eqnarray}$
3.3. Examples of the mKdV-type
In addition to the KdV-type bilinear equations, the mKdV-type and the sG-type bilinear equations automatically admit one-soliton and two-soliton solutions as well [4, 5]. Note that the NLS-type bilinear equations considered by Hietarinta always have one-soliton solutions [6]. Apart from the KdV-type bilinear equations, Hietarinta also did the searching of the mKdV-type and sG-type bilinear equations that have three-soliton solutions [4, 5].
In the following, we provide some examples of the mKdV-type bilinear equations, of which the general form (in two-dimension as an example) considered in [4] is a coupled system2.30 ) or (2.31 ), and the transformations
$\begin{eqnarray}{Q}_{1}({D}_{x},{D}_{t})F\cdot G=0,\end{eqnarray}$
$\begin{eqnarray}{Q}_{2}({D}_{x},{D}_{t})F\cdot G=0,\end{eqnarray}$
where Q1 and Q2 are some two-component polynomials, Q1 is odd and Q2 is quadratic and even. To convert them into nonlinear equations, we need to use formula ( $\begin{eqnarray}u={\mathrm{ln}}\,(FG),\,\,\,v={\mathrm{ln}}\,(F/G).\end{eqnarray}$
The first example of the mKdV-type is
$\begin{eqnarray}({D}_{t}+{D}_{x}^{3})F\cdot G=0,\end{eqnarray}$
$\begin{eqnarray}{D}_{x}^{2}F\cdot G=0.\end{eqnarray}$
Based on formula (2.30 ), we can derive 3.30b ), we have 3.31 ) to eliminate uxx, we arrive at the potential (defocusing) mKdV equation
$\begin{eqnarray}\begin{array}{rcl} & & ({D}_{t}+{D}_{x}^{3})F\cdot G\\ & = & {{ \mathcal Y }}_{[t]}(v,u)+{{ \mathcal Y }}_{[3x]}(v,u)\\ & = & {v}_{t}+{v}_{xxx}+3{v}_{x}{u}_{xx}+{v}_{x}^{3}=0.\end{array}\end{eqnarray}$
Meanwhile, from equation ( $\begin{eqnarray}{D}_{x}^{2}F\cdot G={{ \mathcal Y }}_{[2x]}(v,u)={u}_{xx}+{v}_{x}^{2}=0.\end{eqnarray}$
Substituting the above equation into ( $\begin{eqnarray}{v}_{t}+{v}_{xxx}-2{v}_{x}^{3}=0,\end{eqnarray}$
which yields the mKdV equation ($w={v}_{x}={\partial }_{x}{\mathrm{ln}}\,(f/G)$) $\begin{eqnarray}{w}_{t}+{w}_{xxx}-6{w}^{2}{w}_{x}=0.\end{eqnarray}$
Another example of the mKdV-type is (see table I (Generalizations with Y) of [4])
$\begin{eqnarray}({D}_{x}^{2}{D}_{t}+{D}_{y})F\cdot G=0,\end{eqnarray}$
$\begin{eqnarray}{D}_{x}^{2}F\cdot G=0.\end{eqnarray}$
These equations were also derived in [42] as bilinear forms related to the Bogoyavlensky-mKdV hierarchy and toroidal Lie algebra ${{\rm{sl}}}_{2}^{{\rm{tor}}}$, see degree 2 case in table 1 in [42]. Using formula (2.30 ) we have 3.32 ). Eliminating u we obtain 3.35a ), equation (3.30a ) is also involved.
$\begin{eqnarray}\begin{array}{rcl}({D}_{x}^{2}{D}_{t}+{D}_{y})F\cdot G & = & {{ \mathcal Y }}_{[2x,t]}(v,u)+{{ \mathcal Y }}_{[y]}(v,u)\\ & = & {v}_{xxt}+2{v}_{x}{u}_{xt}+{v}_{x}^{2}{v}_{t}+{u}_{xx}{v}_{t}+{v}_{y}=0\end{array}\end{eqnarray}$
and ( $\begin{eqnarray}{v}_{xxt}-4{v}_{x}{\int }_{-\infty }^{x}{v}_{x}{v}_{xt}\,{\rm{d}}x+{v}_{y}=0,\end{eqnarray}$
which leads us to (with $w={v}_{x}={\partial }_{x}{\mathrm{ln}}\,(F/G)$): $\begin{eqnarray}{w}_{y}+{w}_{xxt}-4{w}^{2}{w}_{t}-2{w}_{x}{\int }_{-\infty }^{x}{({w}^{2})}_{t}\,{\rm{d}}x=0.\end{eqnarray}$
This is an integrable equation known as the breaking soliton mKdV equation (see equation (1.4) on page 47 of [43]). Lax pair of this equation has been constructed from a generic sense in [42]. In addition, its alternative bilinear form has been given in [44], where apart from (The next example is (also see table 1 (Generalizations with Y) of [4])
$\begin{eqnarray}({D}_{x}^{5}+{D}_{x}^{2}{D}_{t}+{D}_{y})F\cdot G=0,\end{eqnarray}$
$\begin{eqnarray}{D}_{x}^{2}F\cdot G=0.\end{eqnarray}$
The related Bell polynomial expressions are 3.32 ), which are given in terms of (v, u) as 3.38 ), and it is connected with the bilinear form (3.39a ) via $w={v}_{x}={\partial }_{x}{\mathrm{ln}}\,(F/G)$.
$\begin{eqnarray*}{{ \mathcal Y }}_{[5x]}(v,u)+{{ \mathcal Y }}_{[2x,t]}(v,u)+{{ \mathcal Y }}_{[y]}(v,u)=0\end{eqnarray*}$
and ( $\begin{eqnarray*}\begin{array}{l}{v}_{5x}+{v}_{x}^{5}+10{v}_{x}^{2}{v}_{3x}+5{v}_{x}{u}_{4x}+10{v}_{3x}{u}_{2x}+15{v}_{x}{u}_{2x}^{2}\\ +10{v}_{x}^{3}{u}_{2x}+{v}_{2x,t}+2{v}_{x}{u}_{x,t}+{v}_{x}^{2}{v}_{t}+{u}_{2x}{v}_{t}+{v}_{y}=0\end{array}\end{eqnarray*}$
and ${u}_{2x}+{v}_{x}^{2}=0$. Again, eliminating u and introducing w = vx, we arrive at $\begin{eqnarray}\begin{array}{l}{w}_{xxxxx}+30{w}^{4}{w}_{x}+{w}_{xxt}+{w}_{y}\\ -{\left(10{w}^{2}{w}_{xx}+10w{w}_{x}^{2}+2w{\displaystyle \int }_{-\infty }^{x}{({w}^{2})}_{t}\,{\rm{d}}x\right)}_{x}=0.\end{array}\end{eqnarray}$
It can be considered as a fifth-order breaking soliton mKdV equation, cf.(The last example is (see equation (34) in [4])
$\begin{eqnarray}({D}_{x}{D}_{y}{D}_{t}+a{D}_{x}+b{D}_{t})F\cdot G=0,\end{eqnarray}$
$\begin{eqnarray}{D}_{x}{D}_{t}F\cdot G=0,\end{eqnarray}$
where a, b are parameters. It corresponds to $\begin{eqnarray*}\begin{array}{l}{{ \mathcal Y }}_{[x,y,t]}(v,u)+a{{ \mathcal Y }}_{[x]}(v,u)+b{{ \mathcal Y }}_{[t]}(v,u)=0,\\ {{ \mathcal Y }}_{[x,t]}(v,u)=0,\end{array}\end{eqnarray*}$
i.e. $\begin{eqnarray*}\begin{array}{l}{v}_{xyt}+{v}_{x}{u}_{yt}+{v}_{y}{u}_{xt}+{v}_{t}{u}_{xy}+{v}_{x}{v}_{y}{v}_{t}+a{v}_{x}+b{v}_{t}=0,\\ {u}_{xt}+{v}_{x}{v}_{t}=0.\end{array}\end{eqnarray*}$
By introducing w = uy, we have the following nonlinear form: 3.41a ) via $v={\mathrm{ln}}\,(F/G)$ and $w={\partial }_{y}{\mathrm{ln}}\,(FG)$. This equation is integrable in Hirota’s sense, while all its integrability characteristics remain open to find.
$\begin{eqnarray}{v}_{xyt}+{v}_{x}{w}_{t}+{v}_{t}{w}_{x}+a{v}_{x}+b{v}_{t}=0,\,\,\,{w}_{xt}={({v}_{x}{v}_{t})}_{y},\end{eqnarray}$
which is connected with the bilinear equation (4. Concluding remarks
In this paper, we have shown how bilinear equations are converted into their nonlinear forms by using the Bell polynomials. The KdV-type and mKdV-type bilinear equations served as illustrative examples, where the key roles are played by the formulae (2.30 ) and (2.32 ).
Among the examples of the KdV-type, a nonlinear form of the Hietartinta equation (3.2d ) has been found, which is given in (3.13 ) (or (3.14 )) and is not known before. We also presented a nonlinear form of an integrable four-dimensional equation in the KP family, see (3.18 ) or (3.19 ). Note that the related bilinear equation (3.17 ) is interesting in the sense that it allows elliptic τ functions but no need to introduce elliptic curve moduli parameters in the bilinear equation (see (4.36b) in [33]). It can also be considered as a four-dimensional extension of the bilinear BKP(−1) equation (3.2b ). In addition to the integrable equations, we also gave three non-integrable examples in section 3.2 , although two of them used to be incorrectly studied as integrable equations.
Apart from the KdV-type equations, some integrable mKdV-type bilinear equations have been nonlinearized in section 3.3 . It is notable that the bilinear system (3.35a ) is shown to connected with the breaking soliton mKdV equation (3.38 ), the study of which is much less compared with the breaking soliton KdV equation. In addition, to our knowledge, (3.40 ) and (3.42 ) are new integrable systems. The former maybe recognized as a fifth-order breaking soliton mKdV equation, while for equation (3.42 ), it seems its integrability context remains unknown except having three-soliton solutions.
It is well known that integrable bilinear equations can be categorized according to different affine Lie algebras (e.g. [32]) and toroidal Lie algebras (e.g. [34, 35, 42, 45, 46]), but nonlinear forms of many bilinear equations are not yet known. Since the Bell polynomials also can be used to construct bilinear Bäcklund transformations and then Lax pairs for the KdV-type equation [12, 13], one may keep using the Bell polynomials to study equations (3.13 ), (3.19 ), (3.40 ) and (3.42 ) so as to achieve more insight of their integrability. This will be considered in the future. In addition, in this paper, we only investigated some bilinear KdV-type and mKdV-type equations. It would be interesting to study nonlinearization of the integrable sG-type and NLS-type bilinear equations presented in [5, 6], which involve complex conjugate operations but the corresponding Bell polynomials approach is much less developed compared with the wide application in the KdV-type equations. Finally, note that the Bell polynomials are related to the Burgers hierarchy [9, 10] and the discrete Burgers equation has been well understood in [47, 48] recently. A related topic that might be interesting is whether there is a discrete version of the Bell polynomial approach to study the discrete KdV-type nonlinear equations and also bilinear ones such as those having three-soliton solutions found in [49].