In this section, we construct a MLVS solution which is based on the corresponding Bäcklund transformation for the MSS equation (1)–(2). It is well known that Bäcklund transformation can be obtained by using the truncated Painlevé expansion usually. See (20)–(21) with (5)–(6). So we begin to discuss the problem from the following Bäcklund transformation:

$${p}_{i}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{p}_{i0}}{f}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\mathrm{0,}\text{\hspace{0.17em}}i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{1,}\text{\hspace{0.17em}}\mathrm{2,}\text{\hspace{0.17em}}\cdots \mathrm{,}\text{\hspace{0.17em}}N\mathrm{,}\text{\hspace{1em}(10)}$$(10)

$$u\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{3}{2\beta}{\mathrm{(}\text{ln}\text{\hspace{0.17em}}f\mathrm{)}}_{xx}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{u}_{2}\mathrm{(}x\mathrm{,}\text{\hspace{0.17em}}t\mathrm{}\mathrm{)}\mathrm{,}\text{\hspace{1em}(11)}$$(11)

where $$\left\{\underset{N}{\underbrace{\mathrm{0,}\text{\hspace{0.17em}}\mathrm{0,}\text{\hspace{0.17em}}\cdots \mathrm{,}\text{\hspace{0.17em}}0}}\mathrm{,}\text{\hspace{0.17em}}{u}_{2}\mathrm{(}x\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\right\}$$ is a seed solution of (1)–(2) and *p*_{i0} ≡ *p*_{i0}(*x*, *y*, *t*), *f* ≡ *f*(*x*, *y*, *t*) need be determined. To consider furthermore, by taking the prior variable separation ansatz

$$f\text{\hspace{0.17em}}=\text{\hspace{0.17em}}F\mathrm{(}x\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}G\mathrm{(}y\mathrm{,}\text{\hspace{0.17em}}t\mathrm{}\mathrm{)}\mathrm{,}$$

we have

$$\sum _{i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{N}}{\displaystyle \sum _{j\text{\hspace{0.17em}}=\text{\hspace{0.17em}}i}^{N}}{a}_{ij}{p}_{i0}{p}_{j0}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\frac{3}{2\beta}{F}_{x}{G}_{y}\mathrm{,$$

from (2). So, we can let

$${p}_{i0}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{H}_{i}\mathrm{(}y\mathrm{)}\sqrt{{F}_{x}{G}_{y}}\mathrm{,}\text{\hspace{0.17em}\hspace{0.17em}}i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{1,}\text{\hspace{0.17em}}\mathrm{2,}\text{\hspace{0.17em}}\cdots \mathrm{,}\text{\hspace{0.17em}}N\mathrm{,}$$

and this restrict equation is reduced to

$$\sum _{i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{N}}{\displaystyle \sum _{j\text{\hspace{0.17em}}=\text{\hspace{0.17em}}i}^{N}}{a}_{ij}{H}_{i}\mathrm{(}y\mathrm{)}{H}_{j}\mathrm{(}y\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\frac{3}{2\beta}\mathrm{.}\text{\hspace{1em}(12)$$(12)

Then, a direct computation gives

$$\begin{array}{c}-2{F}_{t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2{G}_{t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4A\beta {u}_{2}{F}_{x}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{3}{2}A\frac{{F}_{xx}^{2}}{{F}_{x}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2A{F}_{xxx}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\mathrm{(}F\text{\hspace{0.17em}}+\text{\hspace{0.17em}}G\mathrm{)}\\ [2A\beta {u}_{2x}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{{F}_{xt}}{{F}_{x}}+\text{\hspace{0.17em}}\frac{{G}_{yt}}{{G}_{y}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2A\beta {u}_{2}\frac{{F}_{xx}}{{F}_{x}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{3}{4}A{\mathrm{(}\frac{{F}_{xxx}}{{F}_{x}}\mathrm{)}}^{3}\\ -\text{\hspace{0.17em}}\frac{3}{2}A\frac{{F}_{xx}{F}_{xxx}}{{F}_{x}^{2}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}A\frac{{F}_{4x}}{{F}_{x}}]\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0\end{array}$$

from (1). For solving this equation, we change it to the following form,

$$\begin{array}{l}\left[-2\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{F\text{\hspace{0.17em}}+\text{\hspace{0.17em}}G}{{G}_{y}}{\partial}_{y}\right]{G}_{t}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\left[-2\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{F\text{\hspace{0.17em}}+\text{\hspace{0.17em}}G}{{F}_{x}}{\partial}_{x}\right]\\ \text{\hspace{1em}}\mathrm{(}{F}_{t}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2A\beta {u}_{2}{F}_{x}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}A{F}_{xxx}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3}{4}A\frac{{F}_{xx}^{2}}{{F}_{x}}\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0.\end{array}$$

Because *F*, *u*_{2} are only functions of {*x*, *t*} and *G* is a function of {*y*, *t*}, this equation can be solved by the following variable separated equations:

$${G}_{t}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{c}_{0}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{c}_{1}G\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{c}_{2}{G}^{2}\mathrm{,}\text{\hspace{1em}(13)}$$(13)

$${F}_{t}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2A\beta {u}_{2}{F}_{x}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}A{F}_{xxx}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3}{4}A\frac{{F}_{xx}^{2}}{{F}_{x}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-{c}_{0}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{c}_{1}F\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{c}_{2}{F}^{2}\text{\hspace{1em}(14)}$$(14)

with the arbitrary functions *c*_{0} ≡ *c*_{0}(*t*), *c*_{1} ≡ *c*_{1}(*t*) and *c*_{2} ≡ *c*_{2}(*t*). To solve the Riccati equation (13) is also quite easy because of the arbitrariness of the functions *c*_{0}, *c*_{1}, and *c*_{2}. The result reads

$$G\mathrm{(}y\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}\mathrm{=}\text{\hspace{0.17em}}\frac{{C}_{0}\mathrm{(}t\mathrm{)}}{{C}_{1}\mathrm{(}t\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}K\mathrm{(}y\mathrm{)}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{C}_{2}\mathrm{(}t\mathrm{}\mathrm{)}\mathrm{,}\text{\hspace{1em}(15)}$$(15)

where *C*_{0} ≡ *C*_{0}(*t*), *C*_{1} ≡ *C*_{1}(*t*), *C*_{2} ≡ *C*_{2}(*t*), and *K*(*y*) can all be considered as arbitrary functions of the indicated variables while *c*_{0}, *c*_{1}, and *c*_{2} are related to *C*_{0}, *C*_{1}, and *C*_{2} by

$$\begin{array}{c}{c}_{0}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{{C}_{0}}\mathrm{(}{C}_{0}{{C}^{\prime}}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{C}_{2}{{C}^{\prime}}_{0}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{C}_{2}^{2}{{C}^{\prime}}_{1}\mathrm{)}\mathrm{,}\text{\hspace{0.17em}}\\ {c}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{{C}_{0}}\mathrm{(}{{C}^{\prime}}_{0}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2{C}_{2}{{C}^{\prime}}_{1}\mathrm{)}\mathrm{,}\text{\hspace{0.17em}}\\ {c}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\frac{{{C}^{\prime}}_{1}}{{C}_{0}}\mathrm{.}\end{array}$$

For (14), we may treat it alternatively. Namely, we consider *F* is an arbitrary function while the function *u*_{2} can be determined (14), that is

$${u}_{2}\mathrm{(}x\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{-{c}_{0}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{c}_{1}F\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{c}_{2}{F}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{F}_{t}}{2A\beta {F}_{x}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{1}{2\beta}\frac{{F}_{xxx}}{{F}_{x}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{3}{8\beta}{\mathrm{(}\frac{{F}_{xx}}{{F}_{x}}\mathrm{)}}^{2}\mathrm{.}$$

Therefore, the MSS equation (1)–(2) has an exact solution

$${p}_{i}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{H}_{i}\mathrm{(}y\mathrm{)}\frac{\sqrt{{F}_{x}{G}_{y}}}{F\text{\hspace{0.17em}}+\text{\hspace{0.17em}}G}\mathrm{,}\text{\hspace{0.17em}\hspace{0.17em}}i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{1,}\text{\hspace{0.17em}}\mathrm{2,}\text{\hspace{0.17em}}\cdots \mathrm{,}\text{\hspace{0.17em}}N\mathrm{,}\text{\hspace{1em}(16)}$$(16)

$$\begin{array}{c}u\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{3}{2\beta}\left[\frac{{F}_{xx}}{F\text{\hspace{0.17em}}+\text{\hspace{0.17em}}G}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\mathrm{(}\frac{{F}_{x}}{F\text{\hspace{0.17em}}+\text{\hspace{0.17em}}G}\mathrm{)}}^{2}\right]\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{{c}_{0}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{c}_{1}F\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{c}_{2}{F}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{F}_{t}}{2A\beta {F}_{x}}\text{\hspace{0.17em}}\\ -\text{\hspace{0.17em}}\frac{1}{2\beta}\frac{{F}_{xxx}}{{F}_{x}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{3}{8\beta}{\mathrm{(}\frac{{F}_{xx}}{{F}_{x}}\mathrm{)}}^{2}\mathrm{,}\end{array}\text{\hspace{1em}(17)}$$(17)

where *F*(*x*, *t*) is an arbitrary function, *G*(*y*, *t*) is determined by (15) and *H*_{i}(*y*), (*i*=1, 2, …, *N*) need to satisfy the restrict equation (12).

For the quantity $$U\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle {\sum}_{i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{N}}{\displaystyle {\sum}_{j\text{\hspace{0.17em}}=\text{\hspace{0.17em}}i}^{N}}{a}_{ij}{p}_{i}{p}_{j},$$ we have

$$\sum _{i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{N}}{\displaystyle \sum _{j\text{\hspace{0.17em}}=\text{\hspace{0.17em}}i}^{N}}{a}_{ij}{p}_{i}{p}_{j}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\frac{3}{2\beta}\frac{{F}_{x}{G}_{y}}{{\mathrm{(}F\text{\hspace{0.17em}}+\text{\hspace{0.17em}}G\mathrm{)}}^{2}$$

with *F* being an arbitrary function of {*x*, *t*} and *G* being (15). It is interesting that this expression is valid for many (2+1)-dimensional integrable equations, such as the DS equation, the NNV equation, the dispersive long wave equation, the Broer–Kaup–Kupershmidt (BKK) equation, the general (*M*+*N*)-component AKNS system, and the symmetric sine–Gordon equation. If we select the functions *F* and *G* appropriately, we can obtain many kinds of new coherent structures, and some of them are listed as follows.

Since the pioneering work of Camassa and Holm (CH) [16], a special type of (1+1)-dimensional weak solutions has attracted the attention of scientists. These types of solitary waves are called peakons because they are discontinuous at their crest, and the collisions among them are completely elastic. Furthermore, Rosenau and Hyman [17] introduced a class of (1+1)-dimensional solitary waves with compact support (called compacton) in fully nonlinear KdV equation K(m, n) for understanding the role of nonlinear dispersion. Although many soliton equations, such as the CH equation [18] have been extended to (2+1)-dimensions in several ways, one does not know anything on the (2+1)-dimensional peakons and compactons which are localized in all directions. In fact, the entrance of the arbitrary functions *F* and *G* in (1) tells that the (2+1)-dimensional peakons and compactons can exist by selecting the arbitrary functions as some suitable piecewise continuous functions [8]. Thus, if setting

$$F\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{M}}\{\begin{array}{ll}{P}_{i}\mathrm{(}x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{e}_{i}t\mathrm{}\mathrm{)}\mathrm{,}\hfill & x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{e}_{i}t\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}\mathrm{0,}\hfill \\ -\text{\hspace{0.17em}}{P}_{i}\mathrm{(}-\text{\hspace{0.17em}}x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{e}_{i}t\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2{P}_{i}\mathrm{(}0\mathrm{)}\mathrm{,}\hfill & x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{e}_{i}t\text{\hspace{0.17em}}>\text{\hspace{0.17em}}\mathrm{0,}\hfill \end{array}\text{\hspace{1em}(18)}$$(18)

where *P*_{i}(*ξ*)=*P*_{i}(*x* + *e*_{i}), *i*=1, 2, …, *M* are differentiable functions and possess the boundary conditions *P*_{i}(±∞)=*E*_{i}, *i*=1, 2, …, *M* with *E*_{i} being constants and

$$G\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{j\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{N}}\{\begin{array}{ll}\mathrm{0,}\hfill & y\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{h}_{j}t\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}{y}_{1j}\mathrm{,}\hfill \\ {Q}_{j}\mathrm{(}y\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{h}_{j}t\mathrm{)}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{Q}_{j}\mathrm{(}{y}_{1j}\mathrm{}\mathrm{)}\mathrm{,}\hfill & {y}_{1j}\text{\hspace{0.17em}}<\text{\hspace{0.17em}}y\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{h}_{j}t\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}{y}_{2j}\mathrm{,}\hfill \\ {Q}_{j}\mathrm{(}{y}_{2j}\mathrm{)}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{Q}_{j}\mathrm{(}{y}_{1j}\mathrm{}\mathrm{)}\mathrm{,}\hfill & y\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{h}_{j}t\text{\hspace{0.17em}}>\text{\hspace{0.17em}}{y}_{2j}\mathrm{,}\hfill \end{array}\text{\hspace{1em}(19)}$$(19)

where *Q*_{j}, *j*=1, 2, …, *N* are differentiable functions and satisfy the conditions *Q*_{jy}|_{y=y1j}=*Q*_{jy}|_{y}=_{y2j}=0, *j*=1, 2, …, *N*, we have a new mixed coherent structure which possesses peakons at *x*-axis and compactons at *y*-axis.

In Figures 1–5, we plot the interaction property between two peakon-compacton structures for the quantity $$U\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1000\frac{{F}_{x}{G}_{y}}{{\mathrm{(}F\text{\hspace{0.17em}}+\text{\hspace{0.17em}}G\mathrm{)}}^{2}}$$ with

Figure 1: Two peakon-compacton structures at t=−3.

Figure 2: Interaction at t=−1.

Figure 3: Interaction at t=0.

Figure 4: Interaction at t=1.

Figure 5: Position is exchanged after interaction at t=3.

$$\begin{array}{c}F\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\{\begin{array}{ll}-\text{ln}\left[\text{tanh}\mathrm{(}\frac{1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}t}{2}\mathrm{)}\right]\mathrm{,}\hfill & x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}t\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}\mathrm{0,}\hfill \\ \text{ln}\left[\text{tanh}\mathrm{(}\frac{1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}t}{2}\mathrm{)}\right]\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2\text{ln}\left[\text{tanh}\mathrm{(}\frac{1}{2}\mathrm{)}\right]\mathrm{,}\hfill & x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}t\text{\hspace{0.17em}}>\text{\hspace{0.17em}}\mathrm{0,}\hfill \end{array}\\ +\lambda \{\begin{array}{ll}-\text{ln}\left[\text{tanh}\mathrm{(}\frac{1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}x\text{\hspace{0.17em}}-2t}{2}\mathrm{)}\right]\mathrm{,}\hfill & x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2t\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}\mathrm{0,}\hfill \\ \text{ln}\left[\text{tanh}\mathrm{(}\frac{1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2t}{2}\mathrm{)}\right]\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2\text{ln}\left[\text{tanh}\mathrm{(}\frac{1}{2}\mathrm{)}\right]\mathrm{,}\hfill & x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2t\text{\hspace{0.17em}}>\text{\hspace{0.17em}}\mathrm{0,}\hfill \end{array}\end{array}\text{\hspace{1em}(20)}$$(20)

$$G\text{\hspace{0.17em}}=\text{\hspace{0.17em}}20\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\{\begin{array}{ll}\mathrm{0,}\hfill & y\text{\hspace{0.17em}}-\text{\hspace{0.17em}}t\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}-\text{}\text{\hspace{0.17em}}\frac{\pi}{2}\mathrm{,}\hfill \\ \text{sin}\mathrm{(}y\text{\hspace{0.17em}}-\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\mathrm{1,}\hfill & -\frac{\pi}{2}\text{\hspace{0.17em}}<\text{\hspace{0.17em}}y\text{\hspace{0.17em}}-\text{\hspace{0.17em}}t\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}\frac{\pi}{2}\mathrm{,}\hfill \\ \mathrm{2,}\hfill & y\text{\hspace{0.17em}}-\text{\hspace{0.17em}}t\text{\hspace{0.17em}}>\text{\hspace{0.17em}}\frac{\pi}{2}\mathrm{,}\hfill \end{array}\text{\hspace{1em}(21)}$$(21)

where $$\mathrm{|}\lambda \mathrm{|}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{2}$$ means the amplitude of the right structure in Figure 1 and if *λ*>0, this structure is a bright soliton.

In Figure 6, for simplification, we only plot the four peakon-compacton structures for the quantity $$U\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1000\frac{{F}_{x}{G}_{y}}{{\mathrm{(}F\text{\hspace{0.17em}}+\text{\hspace{0.17em}}G\mathrm{)}}^{2}}$$ with the above *F* and

Figure 6: Four peakon-compacton structures at t=−3.

$$\begin{array}{c}G\text{\hspace{0.17em}}=\text{\hspace{0.17em}}20\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\{\begin{array}{ll}\mathrm{0,}\hfill & y\text{\hspace{0.17em}}-\text{\hspace{0.17em}}t\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{\pi}{2}\mathrm{,}\hfill \\ \text{sin}\mathrm{(}y\text{\hspace{0.17em}}-\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\mathrm{1,}\hfill & -\frac{\pi}{2}\text{\hspace{0.17em}}<\text{\hspace{0.17em}}y\text{\hspace{0.17em}}-\text{\hspace{0.17em}}t\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}\frac{\pi}{2}\mathrm{,}\hfill \\ \mathrm{2,}\hfill & y\text{\hspace{0.17em}}-\text{\hspace{0.17em}}t\text{\hspace{0.17em}}>\text{\hspace{0.17em}}\frac{\pi}{2}\mathrm{,}\hfill \end{array}\\ \text{\hspace{0.17em}}+\text{\hspace{0.17em}}\{\begin{array}{ll}\mathrm{0,}\hfill & y\text{\hspace{0.17em}}-\text{\hspace{0.17em}}5t\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{\pi}{2}\mathrm{,}\hfill \\ 2\text{sin}\mathrm{(}y\text{\hspace{0.17em}}-\text{\hspace{0.17em}}5t\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\mathrm{2,}\hfill & -\frac{\pi}{2}\text{\hspace{0.17em}}<\text{\hspace{0.17em}}y\text{\hspace{0.17em}}-\text{\hspace{0.17em}}5t\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}\frac{\pi}{2}\mathrm{,}\hfill \\ \mathrm{4,}\hfill & y\text{\hspace{0.17em}}-\text{\hspace{0.17em}}5t\text{\hspace{0.17em}}>\text{\hspace{0.17em}}\frac{\pi}{2}\mathrm{.}\hfill \end{array}\end{array}\text{\hspace{1em}(22)}$$(22)

## 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.