Solitons for the modified $(2 + 1)$ -dimensional Konopelchenko–Dubrovsky equations

Lyu, Xiumei; Gu, Wei

doi:10.1186/s13662-019-2370-3

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 1 Different solutions of the sub-equation

From: Solitons for the modified $(2 + 1)$-dimensional Konopelchenko–Dubrovsky equations

No.	z(ξ)	$c_{3}$	$c_{2}$	$c_{1}$
1	sn(ξ), $cd(\xi )=\frac{cn(\xi )}{dn(\xi )}$	$2r^{2} $	$-(r^{2}+1)$	1
2	cn(ξ)	$-2r^{2} $	$2r^{2}-1$	$1-r^{2}$
3	dn(ξ)	−2	$2-r^{2}$	$r^{2}-1$
4	$nc(\xi )=\frac{1}{cn(\xi )}$	$2(1-r^{2}) $	$2r^{2}-1$	$-r^{2}$
5	$ns(\xi )=\frac{1}{sn(\xi )}, dc(\xi )=\frac{dn(\xi )}{cn(\xi )}$	2	$-(r^{2}+1)$	$r^{2}$
6	$nd(\xi )=\frac{1}{dn(\xi )}$	$2(r^{2}-1) $	$2-r^{2}$	-1
7	$cs(\xi )= \frac{cn(\xi )}{sn(\xi )}$	2	$2-r^{2}$	$1-r^{2}$
8	$sc(\xi )=\frac{sn(\xi )}{cn(\xi )}$	$2(1-r^{2}) $	$2-r^{2}$	1
9	$sd(\xi )=\frac{sn(\xi )}{dn(\xi )}$	$2r^{2}(r^{2}-1)$	$2r^{2}-1$	1
10	$ds(\xi )=\frac{dn(\xi )}{sn(\xi )}$	2	$2r^{2}-1$	$r^{4}-r^{2}$
11	rcn(ξ)±dn(ξ)	$-\frac{1}{2}$	$\frac{r^{2}+1}{2}$	$-\frac{(1-r^{2})^{2} }{4}$
12	$\frac{1}{sn(\xi )}\pm \frac{cn(\xi )}{sn(\xi )}$	$\frac{1}{2}$	$\frac{-2r^{2}+1}{2}$	$\frac{1}{4}$
13	$\frac{1}{cn(\xi )}\pm \frac{sn(\xi )}{cn(\xi )}$	$\frac{1-r^{2}}{2}$	$\frac{r^{2}+1}{2}$	$\frac{1-r^{2}}{4}$
14	$\frac{1}{sn(\xi )}\pm \frac{dn(\xi )}{sn(\xi )}$	$\frac{1}{2}$	$\frac{r^{2}-2}{2}$	$\frac{r^{4}}{4}$
15	$sn(\xi )\pm i cn(\xi ),\frac{dn(\xi )}{\sqrt{1-r^{2}}sn(\xi )\pm cn(\xi )}$	$\frac{r^{2}}{2} $	$\frac{r^{2}-2}{2}$	$\frac{r^{2}}{4}$
16	$rsn(\xi )\pm i dn(\xi ),\frac{sn(\xi )}{1\pm cn(\xi )}$	$\frac{1}{2}$	$\frac{1-2r^{2}}{2}$	$\frac{1}{4}$
17	$\frac{sn(\xi )}{1\pm dn(\xi )}$	$\frac{r^{2}}{2}$	$\frac{r^{2}-2}{2}$	$\frac{1}{4}$
18	$\frac{dn(\xi )}{1\pm rsn(\xi )}$	$\frac{r^{2}-1}{2}$	$\frac{r^{2}+1}{2}$	$\frac{r^{2}-1}{4}$
19	$\frac{cn(\xi )}{1\pm sn(\xi )}$	$\frac{1-r^{2}}{2}$	$\frac{r^{2}+1}{2}$	$\frac{-r^{2}+1}{4}$
20	$\frac{sn(\xi )}{dn(\xi )\pm cn(\xi )}$	$\frac{(1-r^{2})^{2}}{2}$	$\frac{r^{2}+1}{2}$	$\frac{1}{4}$
21	$\frac{cn(\xi )}{\sqrt{1-r^{2}}\pm dn(\xi )}$	$\frac{r^{4}}{2} $	$\frac{r^{2}-2}{2}$	$\frac{1}{4}$

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No.	z(ξ)	\(c_{3}\)	\(c_{2}\)	\(c_{1}\)
1	sn(ξ), \(cd(\xi )=\frac{cn(\xi )}{dn(\xi )}\)	\(2r^{2} \)	\(-(r^{2}+1)\)	1
2	cn(ξ)	\(-2r^{2} \)	\(2r^{2}-1\)	\(1-r^{2}\)
3	dn(ξ)	−2	\(2-r^{2}\)	\(r^{2}-1\)
4	\(nc(\xi )=\frac{1}{cn(\xi )}\)	\(2(1-r^{2}) \)	\(2r^{2}-1\)	\(-r^{2}\)
5	\(ns(\xi )=\frac{1}{sn(\xi )}, dc(\xi )=\frac{dn(\xi )}{cn(\xi )}\)	2	\(-(r^{2}+1)\)	\(r^{2}\)
6	\(nd(\xi )=\frac{1}{dn(\xi )}\)	\(2(r^{2}-1) \)	\(2-r^{2}\)	-1
7	\(cs(\xi )= \frac{cn(\xi )}{sn(\xi )}\)	2	\(2-r^{2}\)	\(1-r^{2}\)
8	\(sc(\xi )=\frac{sn(\xi )}{cn(\xi )}\)	\(2(1-r^{2}) \)	\(2-r^{2}\)	1
9	\(sd(\xi )=\frac{sn(\xi )}{dn(\xi )}\)	\(2r^{2}(r^{2}-1)\)	\(2r^{2}-1\)	1
10	\(ds(\xi )=\frac{dn(\xi )}{sn(\xi )}\)	2	\(2r^{2}-1\)	\(r^{4}-r^{2}\)
11	rcn(ξ)±dn(ξ)	\(-\frac{1}{2}\)	\(\frac{r^{2}+1}{2}\)	\(-\frac{(1-r^{2})^{2} }{4}\)
12	\(\frac{1}{sn(\xi )}\pm \frac{cn(\xi )}{sn(\xi )}\)	\(\frac{1}{2}\)	\(\frac{-2r^{2}+1}{2}\)	\(\frac{1}{4}\)
13	\(\frac{1}{cn(\xi )}\pm \frac{sn(\xi )}{cn(\xi )}\)	\(\frac{1-r^{2}}{2}\)	\(\frac{r^{2}+1}{2}\)	\(\frac{1-r^{2}}{4}\)
14	\(\frac{1}{sn(\xi )}\pm \frac{dn(\xi )}{sn(\xi )}\)	\(\frac{1}{2}\)	\(\frac{r^{2}-2}{2}\)	\(\frac{r^{4}}{4}\)
15	\(sn(\xi )\pm i cn(\xi ),\frac{dn(\xi )}{\sqrt{1-r^{2}}sn(\xi )\pm cn(\xi )}\)	\(\frac{r^{2}}{2} \)	\(\frac{r^{2}-2}{2}\)	\(\frac{r^{2}}{4}\)
16	\(rsn(\xi )\pm i dn(\xi ),\frac{sn(\xi )}{1\pm cn(\xi )}\)	\(\frac{1}{2}\)	\(\frac{1-2r^{2}}{2}\)	\(\frac{1}{4}\)
17	\(\frac{sn(\xi )}{1\pm dn(\xi )}\)	\(\frac{r^{2}}{2}\)	\(\frac{r^{2}-2}{2}\)	\(\frac{1}{4}\)
18	\(\frac{dn(\xi )}{1\pm rsn(\xi )}\)	\(\frac{r^{2}-1}{2}\)	\(\frac{r^{2}+1}{2}\)	\(\frac{r^{2}-1}{4}\)
19	\(\frac{cn(\xi )}{1\pm sn(\xi )}\)	\(\frac{1-r^{2}}{2}\)	\(\frac{r^{2}+1}{2}\)	\(\frac{-r^{2}+1}{4}\)
20	\(\frac{sn(\xi )}{dn(\xi )\pm cn(\xi )}\)	\(\frac{(1-r^{2})^{2}}{2}\)	\(\frac{r^{2}+1}{2}\)	\(\frac{1}{4}\)
21	\(\frac{cn(\xi )}{\sqrt{1-r^{2}}\pm dn(\xi )}\)	\(\frac{r^{4}}{2} \)	\(\frac{r^{2}-2}{2}\)	\(\frac{1}{4}\)