NEW MULTI-SOLITON SOLUTIONS OF WHITHAM-BROER-KAUP SHALLOW-WATER-WAVE EQUATIONS

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1 Zha S. et al.: New Multi-Solito Solutios o Whitham-Broer-Kaup... THERMAL SCIENCE: Year 7 Vol. Suppl. pp. S37-S44 S37 NEW MULTI-SOLITON SOLUTIONS OF WHITHAM-BROER-KAUP SHALLOW-WATER-WAVE EQUATIONS by She ZHANG a* Miyi LIU a ad Bo XU b a School o Mathematics ad Physics Bohai Uiversity Jizhou Chia b School o Educatio ad Sports Bohai Uiversity Jizhou Chia Oriial scietiic paper I this paper ew ad more eeral Whitham-Broer-Kaup equatios which ca describe the propaatio o shallow-water waves are exactly solved i the ramework o Hirota s biliear method ad ew multi-solito solutios are obtaied. To be speciic the Whitham-Broer-Kaup equatios are irst reduced ito Ablowitz-Kaup-Newell-Seur equatios. With the help o this equatios biliear orms o the Whitham-Broer-Kaup equatios are the derived. Based o the derived biliear orms ew oe-solito solutios two-solito solutios three-solito solutios ad the uiorm ormulae o -solito solutios are ially obtaied. It is show that adopti the biliear orms without loss o eerality play a key role i obtaii these ew multi-solito solutios. Key words: Whitham-Broer-Kaup equatios Ablowitz-Kaup-Newell-Seur equatios Hirota s biliear method Biliear orms solito solutio Itroductio No-liear PDE are ote used to describe some o-liear pheomea o the real world ivolved i may ields rom physics to bioloy ecoomics chemistry mechaics luid dyamics eieeri etc. Usually researchers resort to solutios o such o-liear PDE or more isiht ito these physical pheomea. Solito is such a kid o o-liear pheomeo which ot oly ca be observed i ature but also ca be produced throuh experimet. As poited out by Drazi ad Johso [] it is ot easy to ive a comprehesive ad precise deiitio o a solito. However oe ca associate the term with ay solutio o o-liear PDE which: represets a wave o permaet orm is localized so that it decays or approaches a costat at iiity ad ca udero a stro iteractio with other solitos preservi its idetity. With the developmet o solito theory idi solito solutios [-6] o o-liear PDE has become oe o the most exciti ad extremely active areas o research. I 97 Hirota proposed a direct method [7] or costructi multi-solito solutios o o-liear PDE. Sice put orward by Hirota Hirota s biliear method has developed to a systematic method [8] or multi-solito solutios [9-9]. I this paper we shall exted Hirota s biliear method to ew ad more eeral Whitham-Broer-Kaup (WBK) equatios with arbitrary costat coeiciets γ ( i = 6) []: i u uu v u = () t γ x γ x γ3 xx * Correspodi author szhachia@6.com

2 S38 Zha S. et al.: New Multi-Solito Solutios o Whitham-Broer-Kaup... THERMAL SCIENCE: Year 7 Vol. Suppl. pp. S37-S44 v u v uv v u = () t γ4 x γ4 x γ5 xx γ6 xxx or costructi ew multi-solito solutios. It should be oted that eqs. () ad () are more eeral tha the ollowi kow WBK model or the dispersive lo waves i shallow water [7-9]: u uu v γ u = (3) t x x xx vt ( uv) x βuxxx γvxx = (4) Besides i we select appropriate values o γ i ( i = 6) the eqs. () ad () ive some other kow o-liear PDE such as the approximate equatios or lo water waves [] the Boussiesq-Burers equatios []. I [] some symmetries ad similarity reductios o eqs. () ad () are obtaied. Recetly eqs. (3) ad (4) have attached much attetio ad may exact solutios like those i [3-5] have bee costructed. It is worth metioi that Li et al. [7 8] ad Wa et al. [9] obtaied multi-solito solutios i terms o double Wroskia determiat. As ar as we kow there are o multi-solito solutios ad other solutios o eqs. () ad () have bee reported i literature. Biliear orms I order to derive the biliear orms coveietly we reduce eqs. () ad () i advace. Theorem. I let A x u = a A γ aγ γ 3 Ax A xx v = a AB a γ γ A A where a is a arbitrary costat A ad B are udetermied smooth ormatios o x ad t the the WBK eqs. () ad () reduce ito the AKNS equatios: At aγ ( AB Axx ) = Bt aγ ( AB Axx ) = (6) uder the costraits: a γ γ4 = γ γ5 = γ3 γ 6 = (7) 4γ Proo. Supposi that: u = a(l A) x v = b(l A) xx cab (8) where a b ad c are costats to be determied ad the substituti eqs. (8) ito eqs. () ad () we arrive at eqs. (6) uder the costraits (7). We iish the proo o Theorem. Theorem. Uder the costraits (7) the WBK eqs. () ad possess the biliear orms: Dt = aγ D x (Dx h) (9) (5) γ h Dth = a D xh (Dx h) ()

3 Zha S. et al.: New Multi-Solito Solutios o Whitham-Broer-Kaup... THERMAL SCIENCE: Year 7 Vol. Suppl. pp. S37-S44 S39 where = () xt = () xt h= hxt () D x ad D t are Hirota s dieretial operators [8]. Proo. Starti rom eqs. (6) we suppose that: h A = B = () Usi Hirota s biliear dieretial operators ad eqs. () we ca re-write eqs. (6) as eqs. (9) ad (). Thus the proo o Theorem is ed. Multi-solito solutios Geerally speaki it is diiculty i usi the biliear orms (9) ad () to costruct multi-solito solutios o eqs. () ad (). Usually oe assumes D x h = to use a special case [3 6-9] o eqs. (9) ad () or the multi-solito solutios. This is ot the starti poit o this paper. Without loss o eerality we shall costruct ew multi-solito solutios by employi the biliear orms (9) ad () with D x h. Theorem 3. Uder the costraits (7) the WBK eqs. () ad () possesses the uiorm ormulae o -solito solutios determied by: with u x x = a v= a γ h aγ γ a 3 x x xx xx γ γ () h µ j ( ξj θ j l ) µµ j l Ajl t j= j< l = e e (3) µ = µ j ( ξj θ j l ) µµ j l Ajl t j= j< l = e e µ = ξ = ω t kx ξ j j j j k = 4 sih θ j j µ j ( ξ j l ) µµ j l Ajl j= j< l = e (4) µ = ωj aγ sih θ j = ( j = ) (5) θ j θl sih Ajl e = ( j < l ) (6) θ j θl sih where is a costat parameter ad ξ i a arbitrary costat. The summatio Σ µ = reers to all possible combiatios o each µ i = or i =. Proo. We irst itroduce a parameter which is idepedet with x ad t so that: D x h = a (7) the the biliear orms (9) ad () become: Dt aγdx = h Dt aγdxh = h Further taki the trasormatios [9]: (8)

4 S4 Zha S. et al.: New Multi-Solito Solutios o Whitham-Broer-Kaup... THERMAL SCIENCE: Year 7 Vol. Suppl. pp. S37-S44 = e t = t h= e h (9) ad usi eqs. (8) ad (9) we covert eqs. (9) ad () ito the biliear orms o ad h (here we still write them as ad h or coveiece): aγ Dt Dx = aγ h Dt Dx = D x = h a () I what ollows usi eqs. () we costruct multi-solito solutios o eqs. () ad (). To costruct oe-solito solutios we itroduce a parameter ε ad expad ad h: ( ) j ( j) = ε ε ε j ( j) = ε ε ε () ( ) j ( j) h= h εh ε h ε h () Substituti eqs. () ad () ito eqs. () ad the collecti all the coeiciets with same order o ε we et a system o dieretial equatios (SDE): () () t aγ xx = () () ht aγ hxx = aγ = D aγ D () () () () t xx t x h aγ h = D aγ D h () () () () t xx t x () () h = (3) () () () () () () xx aγ = D aγ D (4) = h h (5) () () () () () () t xx t x h aγ h = D aγ D h h () () () () () () t xx t x = D h h h () () () () () () () () () () () xx x aγ = D aγ D (3) (3) () (3) () () () () t xx t x h aγ h = D aγ D h h h (3) (3) () (3) () () () () t xx t x =D h h h h (3) (3) () () () (3) () () () () (3) () (3) () () xx x ad so orth. From eqs. (3) we have: () () (6) (7) (8) (9) (3) = h = (3) Substituti eq. (3) ito eqs. (4) ad (5) we ca see that:

5 Zha S. et al.: New Multi-Solito Solutios o Whitham-Broer-Kaup... THERMAL SCIENCE: Year 7 Vol. Suppl. pp. S37-S44 S4 () ξ () ξ θ = e e () ξ θ = h = e (33) () ξ = ωt kx ξ k = sih θ ω = sih aγ θ (34) satisy eqs. (4) ad (5). () (3) () (3) () (3) I = = h = h = = = = the eqs. (33) ad (34) satisy all the other equatios i previous SDE. Thus eqs. () ad () are trucated. Letti ε = yields: ξ ξ = ( θ e ) e ξ = ( θ h e ) = (35) Usi eqs. (5) (6) () (9) ad (35) we obtai oe-solito solutios o eqs. () ad (): u ad suppose that: x x = a v= a γ h aγ γ a 3 x x xx xx γ γ To costruct two-solito solutios o eqs. () ad () we select: () ξ ξ () ξ = (e e ) ( e θ ξ e θ ) () ξ = ( e θ ξ h e θ ) aγ ( ) = () () () () Dt Dx () () () () Dt aγ Dx h h = (36) = (37) () () () () () () () () D (38) h h = (39) x It is easy to see that eqs. (6)-(8) (38) ad (39) have solutios: () ξ ξ A () ξ ξ θ θ A = 4e e () ξ ξ θ θ A = h = e (4) Substituti eqs. (37) (4) ad (4) ito eqs. (9)-(3) we have: (3) (3) (3) (4) (4) (4) = = h = = = h = = (4) I this case eqs. (7) ad (8) have solutios: (e e ) 4e ξ ξ ξ ξ A = t ξ θ ξ θ ξ ξ θ θ A = e (e e ) 4e (4) t ξ θ ξ θ ξ ξ θ θ A h = e (e e ) 4e (43) We thereore obtai two-solito solutios o eqs. () ad (): u x x = a v= a γ h aγ γ a 3 x x xx xx γ γ (44)

6 S4 Zha S. et al.: New Multi-Solito Solutios o Whitham-Broer-Kaup... THERMAL SCIENCE: Year 7 Vol. Suppl. pp. S37-S44 Similiarly three-solito solutios o eqs. () ad () are obtaied: with u 3x 3 3x 3 = a 3 3 v= a γ h aγ γ a x 3x 3xx 3xx γ 3 γ (45) t ξ θ ξ θ ξ3 θ3 ξ ξ θ θ A ξ ξ3 θ θ3 A3 3 = e (e e e ) 4e 4e 4e 8e ξ ξ3 θ θ3 A3 ξ ξ ξ3 θ θ θ3 A A3 A3 (46) t ξ θ ξ θ ξ3 θ3 ξ ξ θ θ A ξ ξ3 θ θ3 A3 h3 = e (e e e ) 4e 4e ξ ξ3 θ θ3 A3 ξ ξ ξ3 θ θ θ3 A A3 A3 4e 8e (47) 3 A 3 A3 3 A3 3 A A3 A3 3 (e e ξ e ) 4e ξ ξ ξ ξ ξ ξ ξ ξ = 4e 4e 8e (48) I selecti: () ξ ξ ξ () ξ θ ξ θ ξ θ = (e e e ) h = (e e e ) (49) () ξ θ ξ θ ξθ = (e e e ) (5) by iductio we ca ially reach the -solito solutios () determied by eqs. (3)-(6) o eqs. () ad (). Thus we iish the proo o Theorem 3. Coclusio I summary we have biliearized the WBK eqs. () ad () ad obtaied ew oe-solito solutios (36) two-solito solutios (44) three-solito solutios (45) ad the uiorm ormulae o -solito solutios () throuh Hirota s biliear method. I the procedure o extedi Hirota s biliear method to eqs. () ad () oe o the key steps is taki the trasormatios (5) to reduce eqs. () ad () to the AKNS eq. (6) which provide with coveiece or the biliear orms (9) ad () o eqs. () ad (). Recetly ractioal-order dieretial calculus ad its applicatios have attached much attetio [6-9]. How to costruct multi-solito solutios o o-liear PDE with ractioal derivatives is worthy o study. Ackowledmet This work was supported by the Natural Sciece Foudatio o Chia (5475) ad the Natural Sciece Foudatio o Liaoi Provice o Chia (7547). Nomeclature a b c costats [ ] D t D x Hirota s dieretial operators [ ] e base o atural loarithms [ ] i j atural umbers [ ] k j costat [ ] l atural umbers [ ] t time [s] x displacemet [m] Greek symbols β γ costats [ ] γ i costat [ ] θ j θ l costats [ ] μ j μ l iteers [ ] ξ j costat [ ω j costat [ ]

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8 S44 Zha S. et al.: New Multi-Solito Solutios o Whitham-Broer-Kaup... THERMAL SCIENCE: Year 7 Vol. Suppl. pp. S37-S44 [7] Ya X. J. et al. O Exact Traveli-Wave Solutios or Local Fractioal Kortewe-de Vries Equatio Chaos 6 (6) 8 ID 843 [8] Ya X. J. et al. Exact Travelli Wave Solutios or the Local Fractioal Two-Dimesioal Burers-Type Equatios Computers ad Mathematics with Applicatios 73 (7) pp. 3- [9] Ya X. J. et al. O a Fractal LC-Electric Circuit Modeled by Local Fractioal Calculus Commuicatios i No-Liear Sciece ad Numerical Simulatio 47 (7) 6 pp. -6 Paper submitted: April 7 Paper revised: May 7 7 Paper accepted: May Society o Thermal Eieers o Serbia Published by the Viča Istitute o Nuclear Scieces Belrade Serbia. This is a ope access article distributed uder the CC BY-NC-ND 4. terms ad coditios