In this section, through appropriate transformation, we deduce the bilinear form and rational solutions of the potential YTSF equation (3).

First by letting

$$x-z\to x\mathrm{,}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}y\to y\mathrm{,}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}t\to t\mathrm{,}$$(4)

the potential YTSF equation (3) is transformed into

$$-4{u}_{xt}-{u}_{xxxx}-6{u}_{x}{u}_{xx}+3{u}_{yy}=0.$$(5)

Substituting the transformation

$$u\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}=2\mathrm{(}\mathrm{ln}f\mathrm{(}x\mathrm{,}\text{\hspace{0.17em}}y\mathrm{,}\text{\hspace{0.17em}}t{\mathrm{)}\mathrm{)}}_{x}$$(6)

into (5), we have

$f{f}_{xxxx}-4{f}_{x}{f}_{xxx}+3{f}_{xx}{f}_{xx}-3f{f}_{yy}+3{f}_{y}{f}_{y}+4f{f}_{xt}-4{f}_{x}{f}_{t}=0.$

Then, we obtain the bilinear form of (5)

$$\mathrm{(}{D}_{x}^{4}-3{D}_{y}^{2}+4{D}_{x}{D}_{t}\mathrm{)}f\cdot f=0.$$(7)

**Theorem 2.1** *Equation* (5) *has rational solution* (6) *with f given by N*×*N determinant f*=*τ*_{0}*, where*

$${\tau}_{n}=\underset{1\le i\mathrm{,}j\le N}{\mathrm{det}}\mathrm{(}{m}_{ij}^{\mathrm{(}n\mathrm{)}}\mathrm{}\mathrm{)}\mathrm{,}$$(8)

$${m}_{ij}^{\mathrm{(}n\mathrm{)}}={\displaystyle \sum _{k=0}^{{n}_{j}}{c}_{jk}{\mathrm{(}{p}_{j}{\partial}_{{p}_{j}}+{\widehat{\xi}}_{j}+n\mathrm{)}}^{{n}_{j}-k}}{\displaystyle \sum _{l=0}^{{n}_{i}}{d}_{il}{\mathrm{(}{q}_{i}{\partial}_{{q}_{i}}+{\widehat{\eta}}_{i}-n\mathrm{)}}^{{n}_{i}-l}\frac{1}{{p}_{j}+{q}_{i}}}\mathrm{,}$$(9)

$$\begin{array}{l}{\widehat{\xi}}_{j}={p}_{j}x+2\text{i}{p}_{j}^{2}y-3{p}_{j}^{3}t\mathrm{,}\text{\hspace{0.17em}}{\widehat{\eta}}_{i}={q}_{i}x-2\text{i}{q}_{i}^{2}y-3{q}_{i}^{3}t\mathrm{,}\text{\hspace{0.17em}}\\ {c}_{ik}={\overline{d}}_{ik}\mathrm{,}\text{\hspace{0.17em}}{q}_{j}={\overline{p}}_{j}\mathrm{,}\end{array}$$(10)

*n*_{i}, *n*_{j} *are arbitrary positive integers,* *p*_{j}, *c*_{ik} *are arbitrary complex constants.*

By scaling of *f*, we can normalise *c*_{i}_{0}=1. Thus, without the loss of generality, hereafter we set *c*_{i}_{0}=1. We call the abovementioned solution the *N*-rational solution of order (*n*_{1}, *n*_{2}, ℒ, *n*_{N}).

To prove Theorem 2.1, we introduce the following lemma. This lemma can be proved using the similar method in [20] or the ideas in [32] with determinants and Pfaffians. Thus, its proof is omitted here.

**Lemma 2.1** *Let ${m}_{ij}^{\mathrm{(}n\mathrm{)}},$ ${\phi}_{i}^{\mathrm{(}n\mathrm{)}}$*, *and ${\psi}_{j}^{\mathrm{(}n\mathrm{)}}$ be functions of x*_{1}*, x*_{2}, *and x*_{3} *satisfying the following differential and difference relations*

$\begin{array}{l}\frac{\partial {m}_{ij}^{\mathrm{(}n\mathrm{)}}}{\partial {x}_{1}}={\phi}_{i}^{\mathrm{(}n\mathrm{)}}{\psi}_{j}^{\mathrm{(}n\mathrm{)}}\mathrm{,}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\frac{\partial {m}_{ij}^{\mathrm{(}n\mathrm{)}}}{\partial {x}_{2}}={\phi}_{i}^{\mathrm{(}n-1\mathrm{)}\mathrm{}}{\psi}_{j}^{\mathrm{(}n\mathrm{)}}+{\phi}_{i}^{\mathrm{(}n\mathrm{)}}{\psi}_{j}^{\mathrm{(}n+1\mathrm{)}\mathrm{}}\mathrm{,}\\ \frac{\partial {m}_{ij}^{\mathrm{(}n\mathrm{)}}}{\partial {x}_{3}}={\phi}_{i}^{\mathrm{(}n-2\mathrm{)}\mathrm{}}{\psi}_{j}^{\mathrm{(}n\mathrm{)}}+{\phi}_{i}^{\mathrm{(}n-1\mathrm{)}\mathrm{}}{\psi}_{j}^{\mathrm{(}n+1\mathrm{)}\mathrm{}}+{\phi}_{i}^{\mathrm{(}n\mathrm{)}}{\psi}_{j}^{\mathrm{(}n+2\mathrm{)}\mathrm{}}\mathrm{,}\\ \frac{\partial {\phi}_{i}^{\mathrm{(}n\mathrm{)}}}{\partial {x}_{k}}=-{\phi}_{i}^{\mathrm{(}n-k\mathrm{)}}\mathrm{,}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\frac{\partial {\psi}_{j}^{\mathrm{(}n\mathrm{)}}}{\partial {x}_{k}}={\psi}_{j}^{\mathrm{(}n+k\mathrm{)}}\mathrm{,}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{(}k=\mathrm{1,}\text{\hspace{0.17em}}\mathrm{2,}\text{\hspace{0.17em}}3\mathrm{)}\mathrm{.}\end{array}$

*Then the determinant*

${\tau}_{n}=\underset{1\le i\mathrm{,}j\le N}{\mathrm{det}}\mathrm{(}{m}_{ij}^{\mathrm{(}n\mathrm{)}}\mathrm{)}$

*satisfies the bilinear equation*

$$\mathrm{(}{D}_{{x}_{1}}^{4}+3{D}_{{x}_{2}}^{2}-4{D}_{{x}_{1}}{D}_{{x}_{3}}\mathrm{)}{\tau}_{n}\cdot {\tau}_{n}=0.$$(11)

*Proof of Theorem*** ***2.1:* In order to prove Theorem 2.1, the functions ${m}_{ij}^{\mathrm{(}n\mathrm{)}},$ ${\phi}_{i}^{\mathrm{(}n\mathrm{)}}$, and ${\psi}_{j}^{\mathrm{(}n\mathrm{)}}$ satisfy

$$\begin{array}{rl}& {\psi}_{j}^{(n)}={A}_{j}{p}_{j}^{n}{\mathrm{e}}^{{\xi}_{j}},{\phi}_{i}^{(n)}={B}_{i}(-{q}_{i}{)}^{-n}{\mathrm{e}}^{{\eta}_{i}},\\ & {m}_{ij}^{(n)}={A}_{j}{B}_{i}\frac{1}{{p}_{j}+{q}_{i}}{\left(-\frac{{p}_{j}}{{q}_{i}}\right)}^{n}{\mathrm{e}}^{{\xi}_{j}+{\eta}_{i}},\end{array}$$(12)

where ${\xi}_{j}={p}_{j}{x}_{1}+{p}_{j}^{2}{x}_{2}+{p}_{j}^{3}{x}_{3},$ ${\eta}_{i}={q}_{i}{x}_{1}-{q}_{i}^{2}{x}_{2}+{q}_{i}^{3}{x}_{3},$ and *A*_{j} and *B*_{i} are differential operators of order *n*_{j} and *n*_{i} with respect to *p*_{j} and *q*_{i}, respectively, which are defined by

$${A}_{j}={\displaystyle \sum _{k=0}^{{n}_{j}}{c}_{jk}{\mathrm{(}{p}_{j}{\partial}_{{p}_{j}}\mathrm{)}}^{{n}_{j}-k}}\mathrm{,}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{B}_{i}={\displaystyle \sum _{l=0}^{{n}_{i}}{d}_{il}{\mathrm{(}{q}_{i}{\partial}_{{q}_{i}}\mathrm{)}}^{{n}_{i}-l}}\mathrm{.}$$(13)

Substituting *A*_{j} and *B*_{i} into (12), we have

$\begin{array}{c}({p}_{j}{\mathrm{\partial}}_{{p}_{j}}){p}_{j}^{n}{\mathrm{e}}^{{\xi}_{j}}={\mathrm{e}}^{{\xi}_{j}}{p}_{j}^{n}({p}_{j}{\mathrm{\partial}}_{{p}_{j}}+{p}_{j}{\mathrm{\partial}}_{{p}_{j}}{\xi}_{j}+n),\\ ({q}_{i}{\mathrm{\partial}}_{{q}_{i}})(-{q}_{i}{)}^{-n}{\mathrm{e}}^{{\eta}_{i}}={\mathrm{e}}^{{\eta}_{i}}(-{q}_{i}{)}^{-n}({q}_{i}{\mathrm{\partial}}_{{q}_{i}}+{q}_{i}{\mathrm{\partial}}_{{q}_{i}}{\eta}_{i}-n).\end{array}$

For simplicity, the function ${m}_{ij}^{\mathrm{(}n\mathrm{)}}$ can be rewritten as

$\begin{array}{c}{m}_{ij}^{\mathrm{(}n\mathrm{)}}={\displaystyle \sum _{k=0}^{{n}_{j}}{c}_{jk}{\mathrm{(}{p}_{j}{\partial}_{{p}_{j}}\mathrm{)}}^{{n}_{j}-k}}{\displaystyle \sum _{l=0}^{{n}_{i}}{d}_{il}{\mathrm{(}{q}_{i}{\partial}_{{q}_{i}}\mathrm{)}}^{{n}_{i}-l}\frac{1}{{p}_{j}+{q}_{i}}{\mathrm{(}-\frac{{p}_{j}}{{q}_{i}}\mathrm{)}}^{n}{\text{e}}^{{\xi}_{j}+{\eta}_{i}}}\\ ={\text{e}}^{{\xi}_{j}+{\eta}_{i}}{\mathrm{(}-\frac{{p}_{j}}{{q}_{i}}\mathrm{)}}^{n}{\displaystyle \sum _{k=0}^{{n}_{j}}{c}_{jk}{\mathrm{(}{p}_{j}{\partial}_{{p}_{j}}+{\widehat{\xi}}_{j}+n\mathrm{)}}^{{n}_{j}-k}}\\ {\displaystyle \sum _{l=0}^{{n}_{i}}{d}_{il}{\mathrm{(}{q}_{i}{\partial}_{{q}_{i}}+{\widehat{\eta}}_{i}-n\mathrm{)}}^{{n}_{i}-l}\frac{1}{{p}_{j}+{q}_{i}}}\mathrm{.}\end{array}$

Since the solution *τ*_{n} can be scaled by an arbitrary constant and the complex conjugate condition of (10), we see that *τ*_{n} with matrix elements (9) also satisfies the bilinear equation (11). Via the variable transformations *x*_{1}=*x*, *x*_{2}=i*y*, and *x*_{3}=−*t*, the bilinear equation (11) is reduced to the bilinear form (7) of the potential YTSF equation (3).■

In the following three sections, we consider special cases for ${m}_{ij}^{\mathrm{(}n\mathrm{)}}$ in (9) with *N*=1, *n*_{1}=1 as fundamental lump solution, *N*=1, *n*_{1}=2, 3, ℒ as *n*_{1}-order-lump solutions, *n*_{1}=1, *N*>2 as *N*-lump solutions, respectively. For simplicity, we normalise *c*_{i}_{0}=1 and *c*_{ij}=0 (*i*·*j*≠0) without the loss of generality.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.