𝐹-Expansion Method and New Exact Solutions of the Schrödinger-KdV Equation

Table 1

Case	𝑃	𝑄	𝑅	𝐹(𝜉)
1	𝑚^2	−(1+𝑚^2)	1	𝑠𝑛𝜉
2	𝑚^2	−(1+𝑚^2)	1	𝑐𝑑𝜉=𝑐𝑛𝜉𝑑𝑛𝜉
3	−𝑚^2	2𝑚^2−1	1−𝑚^2	𝑐𝑛𝜉
4	−1	2−𝑚^2	𝑚^2−1	𝑑𝑛𝜉
5	1	−(1+𝑚^2)	𝑚^2	𝑛𝑠𝜉=(𝑠𝑛𝜉)^−1
6	1	−(1+𝑚^2)	𝑚^2	𝑑𝑐𝜉=𝑑𝑛𝜉𝑐𝑛𝜉
7	1−𝑚^2	2𝑚^2−1	−𝑚^2	𝑛𝑐𝜉=(𝑐𝑛𝜉)^−1
8	𝑚^2−1	2−𝑚^2	−1	𝑛𝑑𝜉=(𝑑𝑛𝜉)^−1
9	1−𝑚^2	2−𝑚^2	1	𝑠𝑐𝜉=𝑠𝑛𝜉𝑐𝑛𝜉
10	−𝑚^2(1−𝑚^2)	2𝑚^2−1	1	𝑠𝑑𝜉=𝑠𝑛𝜉𝑑𝑛𝜉
11	1	2−𝑚^2	1−𝑚^2	𝑐𝑠𝜉=𝑐𝑛𝜉𝑠𝑛𝜉
12	1	2𝑚^2−1	−𝑚^2(1−𝑚^2)	𝑑𝑠𝜉=𝑑𝑛𝜉𝑠𝑛𝜉
13	14	1−2𝑚^22	14	𝑛𝑠𝜉±𝑐𝑠𝜉
14	1−𝑚^24	1+𝑚^22	1−𝑚^24	𝑛𝑐𝜉±𝑠𝑐𝜉
15	14	𝑚^2−22	𝑚^24	𝑛𝑠𝜉±𝑑𝑠𝜉
16	𝑚^24	𝑚^2−22	𝑚^24	𝑠𝑛𝜉±𝑖𝑐𝑛𝜉
17	𝑚^24	𝑚^2−22	𝑚^24	√1−𝑚^2𝑠𝑑𝜉±𝑐𝑑𝜉
18	14	1−𝑚^22	14	𝑚𝑐𝑑𝜉±𝑖√1−𝑚^2𝑛𝑑𝜉
19	14	1−2𝑚^22	14	𝑚𝑠𝑛𝜉±𝑖𝑑𝑛𝜉
20	14	1−𝑚^22	14	√1−𝑚^2𝑠𝑐𝜉±𝑑𝑐𝜉
21	𝑚^2−14	𝑚^2+12	𝑚^2−14	𝑚𝑠𝑑𝜉±𝑛𝑑𝜉
22	𝑚^24	𝑚^2−22	14	𝑠𝑛𝜉1±𝑑𝑛𝜉
23	−14	𝑚^2+12	(1−𝑚^2)^24	𝑚𝑐𝑛𝜉±𝑑𝑛𝜉
24	(1−𝑚^2)^24	𝑚^2+12	14	𝑑𝑠𝜉±𝑐𝑠𝜉
25	𝑚^4(1−𝑚^2)2(2−𝑚^2)	2(1−𝑚^2)𝑚^2−2	1−𝑚^22(2−𝑚^2)	𝑑𝑐𝜉±√1−𝑚^2𝑛𝑐𝜉
26	𝑃>0	𝑄<0	𝑚^2𝑄^2(1+𝑚^2)^2𝑃	−𝑚^2𝑄(1+𝑚^2)𝑃𝑠𝑛−𝑄1+𝑚^2𝜉
27	𝑃<0	𝑄>0	(1−𝑚^2)𝑄^2(𝑚^2−2)^2𝑃	−𝑄(2−𝑚^2)𝑃𝑑𝑛𝑄2−𝑚^2𝜉
28	𝑃<0	𝑄>0	𝑚^2(𝑚^2−1)𝑄^2(2𝑚^2−1)^2𝑃	−𝑚^2𝑄(2𝑚^2−1)𝑃𝑐𝑛𝑄2𝑚^2𝜉−1
29	1	2−4𝑚^2	1	𝑠𝑛𝜉𝑑𝑛𝜉𝑐𝑛𝜉
30	𝑚^4	2	1	𝑠𝑛𝜉𝑐𝑛𝜉𝑑𝑛𝜉
31	1	𝑚^2+2	1−2𝑚^2+𝑚^4	𝑑𝑛𝜉𝑐𝑛𝜉𝑠𝑛𝜉
32	𝐴^2(𝑚−1)^24	𝑚^2+12+3𝑚	(𝑚−1)^24𝐴^2	𝑑𝑛𝜉𝑐𝑛𝜉𝐴(1+𝑠𝑛𝜉)(1+𝑚𝑠𝑛𝜉)
33	𝐴^2(𝑚+1)^24	𝑚^2+12−3𝑚	(𝑚+1)^24𝐴^2	𝑑𝑛𝜉𝑐𝑛𝜉𝐴(1+𝑠𝑛𝜉)(1−𝑚𝑠𝑛𝜉)
34	−4𝑚	6𝑚−𝑚^2−1	−2𝑚^3+𝑚^4+𝑚^2	𝑚𝑐𝑛𝜉𝑑𝑛𝜉𝑚𝑠𝑛^2𝜉+1
35	4𝑚	−6𝑚−𝑚^2−1	2𝑚^3+𝑚^4+𝑚^2	𝑚𝑐𝑛𝜉𝑑𝑛𝜉𝑚𝑠𝑛^2𝜉−1
36	14	1−2𝑚^22	14	𝑠𝑛𝜉1±𝑐𝑛𝜉
37	1−𝑚^24	1+𝑚^22	1−𝑚^24	𝑐𝑛𝜉1±𝑠𝑛𝜉
38	4𝑚1	2+6𝑚1−𝑚^2	2+2𝑚1−𝑚^2	𝑚^2𝑠𝑛𝜉𝑐𝑛𝜉𝑚1−𝑑𝑛^2𝜉
39	−4𝑚1	2−6𝑚1−𝑚^2	2−2𝑚1−𝑚^2	−𝑚^2𝑠𝑛𝜉𝑐𝑛𝜉𝑚1+𝑑𝑛^2𝜉
40	2−𝑚^2−2𝑚14	𝑚^22−1−3𝑚1	2−𝑚^2−2𝑚14	𝑚^2𝑠𝑛𝜉𝑐𝑛𝜉𝑠𝑛^2𝜉+(1+𝑚1)𝑑𝑛𝜉−1−𝑚1
41	2−𝑚^2+2𝑚14	𝑚^22−1+3𝑚1	2−𝑚^2+2𝑚14	𝑚^2𝑠𝑛𝜉𝑐𝑛𝜉𝑠𝑛^2𝜉+(−1+𝑚1)𝑑𝑛𝜉−1+𝑚1
42	𝐶^2𝑚^4−(𝐵^2+𝐶^2)𝑚^2+𝐵^24	𝑚^2+12	𝑚^2−14(𝐶^2𝑚^2−𝐵^2)	((𝐵^2−𝐶^2)/(𝐵^2−𝐶^2𝑚^2))+𝑠𝑛𝜉𝐵𝑐𝑛𝜉+𝐶𝑑𝑛𝜉
43	𝐵^2+𝐶^2𝑚^24	12−𝑚^2	14(𝐶^2𝑚^2+𝐵^2)	((𝐶^2𝑚^2+𝐵^2−𝐶^2)/(𝐵^2+𝐶^2𝑚^2))+𝑐𝑛𝜉𝐵𝑠𝑛𝜉+𝐶𝑑𝑛𝜉
44	𝐵^2+𝐶^24	𝑚^22−1	𝑚^44(𝐶^2+𝐵^2)	((𝐵^2+𝐶^2−𝐶^2𝑚^2)/(𝐵^2+𝐶^2))+𝑑𝑛𝜉𝐵𝑠𝑛𝜉+𝐶𝑐𝑛𝜉
45	−(𝑚^2+2𝑚+1)𝐵^2	2𝑚^2+2	2𝑚−𝑚^2−1𝐵^2	𝑚𝑠𝑛^2𝜉−1𝐵(𝑚𝑠𝑛^2𝜉+1)
46	−(𝑚^2−2𝑚+1)𝐵^2	2𝑚^2+2	−2𝑚+𝑚^2+1𝐵^2	𝑚𝑠𝑛^2𝜉+1𝐵(𝑚𝑠𝑛^2𝜉−1)