The Multi-Soliton Solutions to The KdV Equation by Hirota Method

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1 Progrss Appld Mhcs Vol. 8, o., 4, pp. -5 OI:.968/69 ISS 95-5X [Pr] ISS [Ol].cscd..cscd.org Th Mul-Solo Soluos o Th KdV Equo y Hro Mhod MA L [],* [] pr of Mhcs Sccs, zhou Uvrsy, zhou, Ch. *Corrspodg uhor. Addrss: E-l: 9579@qq.co Rcvd: rury, 4; ccpd: Spr 6, 4/Pulshd ol: Ocor 6, 4 Asrc:Th Hro lr hod s usd o solv h KdV odl. As rsul, h c prsso of ul-solo soluos of h KdV quo s od. Ky ords: olr prl dffrl quos; Th KdV quo; Hro lr hod; Mul-solo soluo M, L. X. 4. Th Mul-Solo Soluos o Th KdV Equo y Hro Mhod. Progrss Appld Mhcs, 8, -5. Avll fro hp://.cscd./d.php/p/rcl/ v/69 OI: hp://d.do.org/.968/69. ITROUCTIO Th vsgo of c soluos of olr v quos plys por rol h sudy of olr prl dffrl quo [PE]. I s ll o h hr r fly y soluos for vry olr quo, such s h h-fuco hod [-], h Jco llpc fuco pso hod [-4], h -pso hod [5-8], s-cos hod [9-], h hoogous lc hod [-], d h Hro lr hod [4-7]. Aog h, h Hro lr hod s o of h os ffcvly hods for cosrucg c soluos o PEs. I hs ppr, h Hro lr hod s usd o solv h KdV quo. Our hs ppr s o vsg ul-solo soluos of h KdV quo y h Hro lr hod. W cosdr h h KdV quo [8] u uu uu.

2 MA L/Progrss Appld Mhcs, 8, 4. THE HIROTA BILIEAR METHO f h Hro lr opror hs h for ' ', ],, ' ' [ f g f g, hr, r ogv gr. Hc g f g f - g f, g f g z f - g f, g f g f- g f fg. I prculr f f f f -f. 4 L, hc, 5, 6, SOLUTIO O THE KV EQUATIO W cosdr h grlzd KdV Equo,l u, hv. 9 By grd o sd of h Equo 9 h rspc o, drv. Ag, L, us orul o 8 d rrg, drv 4 d. L h coffc of r zro, hv quos, ] [ Ag, l, y Equo, drv h codos 4 Blo, y d of h codos 4, cosdr h spcl cs of Equo. Cs h, h, so h quo co u uu u. 5

3 Th Mul-Solo Soluos o Th KdV Equo y Hro Mhod W cosdr h Equo 5 d gv h c prsso of ul-solo soluos of h Equo 5. By Equo 5, o h Hro quo rsly, suppos cos, r h soluos, susu o Equo 5, hv. Th h dsprso rlo of h Equos 5 Scodly, l θ hr, r rrry cos. Also l -. Susu o Equo 6, hv L h coffc of ε s zro, h.. 5, θ,., ε,. [ ε, ε l [ l ε, ],, ] ε, l,. l, >. l o,. 7 Wh, h rduco Equo 7 s. 8 Suppos h hr,, -d 5, hv h sgl solo soluo o h quos 5 s l u. 9 Wh, h rduco Equo 7 s... l

4 MA L/Progrss Appld Mhcs, 8, 4 o ] [ ] [ [ <. Suppos h hr,,,, ] [ ] [, hv h doul solo soluo o h Equos 5 s l u. W suppos h - solo soluo o h Equos 5 s, µ µ η µ µ. Cs h, h 6, so h Equo co u 6uu uu. Rr: h, h Equo s h usul KdV quo u 6uu uu. W cosdr h Equo d gv h c prsso of ul-solo soluos of h Equo. By Equo 6, o h Hro quo....

5 Th Mul-Solo Soluos o Th KdV Equo y Hro Mhod So c rp cs d drv h sgl solo soluo, h doul solo soluo d h - solo soluo o h Equos. Cs h, h 4, so h quo co u 4uu uu. 4 Cs 4 h 4, h, so h quo co u uu uu. 5 Cs 5 h 6, h, so h Equo co u uu uu. 6 Cs 6 h, h, so h quo co u uu uu. 7 Rr: h, h Equo 7 s h usul KdV quo u uu uu. 8 Cs 7 h, h, so h Equo co u uu uu. 9 Rr: h, h Equo 9 s h usul KdV quo u uu uu. COCLUSIO I hs ppr, hv usd h Hro lr hod o solv h KdV odl. I s sgfc o osrv h codo 4 h h Hro lr hod c usd o o soluos. REERECES [] Prs, E. J., & uffy, B. R A uod h-fucohod for fdg solry v soluos o olr voluo quos. Copur Physcs Coucos, 98, 88-. [], E.. Edd h-fuco hod d s pplcos o olr quos. Physcs Lrs A, 774-5, -8. [] Lu, S., u, Z., Lu, S., & Zho, Q.. Jco llpc fuco pso hod d prodc v soluos of olr v quos. Physcs Lrs A, 89-, [4] u, Z., Lu, S., Lu, S., & Zho, Q.. Jco llpc fuco pso d prodc soluos of olr v quos. Physcs Lrs A, 9-, [5] Wg, M., & Zhou, Y.. Th prodc v soluos for h Kl-ordo-Schr odgr quos. Physcs Lrs A, 8-, [6] Zhou, Y., Wg, M., & T. Mo, T. 4. Th prodc v soluos d solry v soluos for clss of olr prl dffrl quos. Physcs Lrs A, -, [7] Wg, M., & L, X. 5. Edd -pso hod d prodc v soluos for h grlzd Zhrov quos. Physcs Lrs A, 4-, [8] Wg, M., & L, X. 5. Applcos of -pso o prodc v soluos for Hlo plud quo. Chos, Solos d rcls, 45, [9] Wzz, A.-M.. A sudy o olr dsprsv prl dffrl quos of copc d ocopc soluos. Appld Mhcs d Copuo, 5-, [] Wzz, A.-M.. A cosruco of copc d ocopc soluos for olr dsprsv quos of v ordr. Appld Mhcs d Copuo, 5-, [] Wg, M Solry v soluos for vr Bousssq quos. Physcs Lrs A, 99-4,

6 MA L/Progrss Appld Mhcs, 8, 4 [] Wg, M Ec soluos for copoud KdV-Burgrs quo. Physcs Lrs A, 5-6, [] Wg, M., Zhou, Y., & L, Z. B Applco of hoogous lc hod o c soluos of olr voluoquos hcl physcs. Physcs Lrs A, 6, [4] Hro, R., & Ssu, J olr voluo quos grd fro h Bäclud rsforo for h Bousssq qu-o. Prog Thor Phys, 57, [5] Hro, R. 97. Ec soluo of h KdV quo forulpl collsos of solos. Phys Rv L, 7, [6] Hu, X. B., & Clrso, P. A Rol soluos of dffrl-dffrc KdV quo,h Tod quo d h dscr KdV quo. J Phys A: Mh, 8, [7] Khr, A. H., Hss, M. M., & Tsh, R. S. 7. Co-odl v soluos for clss of ffh-ordr KdV quos. Mhcs d Copurs Sulo, 7, -6. [8] Wzz, A. M. 7. Alyc sudy o h g-rlzd ffh-ordr KdV quo: solos d prodc soluos. Coucos olr Scc d urcl Sulo,,

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