AbstractA directrational exponential scheme is proposed to investigate multi-soliton solutionsof nonlinear evolution equations.

To do that the truncated Painleve expansion,bilinear differential equations of the (1+1)-dimensional Benjamin-Ono equationare presented. As a result, the multi-solitonwave interaction solutions of the equation are explicitly given and fissionphenomenon is found, which are difficult to find by other traditional methods. Thedifferent solutions are established with different dispersive relations betweenthe wave number and wave speed; a possible condition for fission is proposed. Furthermore,three-dimensional plots of some obtained wave solutions are given out to showthe properties of the explicit analytic interaction solutions. Two classes ofrational solutions to the Benjamin-Ono equation are constructed via Maplesearch which are generated from polynomial solutions to the correspondinggeneralized bilinear equation.

A conjecture is made that the two presentedclasses of rational solutions contain all rational solutions to the consideredequation. From a class of polynomial generating functions, a Maple search tellsus two classes of rational solutions to the considered KdV-like equation, alongwith some special interesting solutions.Keywords: Multi-soliton;rational solution; bilinear form; Benjamin-Ono equation.

Mathematics Subject Classification: 35K99, 35P05, 35P99. IntroductionSolitonequations connect affluent histories of exactly solvable systems constructed inmathematics, fluid physics, microphysics, cosmology, field theory, etc. Toexplain some physical phenomenon further, it becomes more and more important toseek exact solutions and interactions among solutions of nonlinear wavesolutions. Owing to the wide applications of soliton theory in mathematics, hydrodynamics,plasma physics, nuclear physics, biology, astrophysics and geophysics, thestudy on integrable models has attracted much attention of many researchers. Tofind some exact explicit soliton solutions for nonlinear integrable partial differentialequations (NLPDEs) models is one of the most key and significant tasks. Thereare many methods exist in the literature such as -expansion method 1, exp-function2, -expansion method 2, 3by which single soliton solutions can be found. But the most remarkableproperty of exactly integrable equations is the presence of exact solitonicsolutions, and the existence of one-soliton solution is not itself a specificproperty of integrable partial differential equations, many non-integrable equationsalso possesses simple localized solutions that may be called one-solitonic.

Nevertheless,there are integrable equations only, which posses exact multi-soliton solutionswhich describe entirely elastic interactions between individual solitons 4. A variety of efficient methods like theinverse scattering transformation (IST) 5, 6,Backlund transformation 7, 8, Hirota’sdirect method 9, mapping and deformationapproach 10 etc. have been well developedto find exact solutions for integrable models in which interactions betweensoliton solutions gained for integrable models are to be completely elastic. Yetsome soliton models gives completely non-elastic soliton interactions whenspecific conditions between the wave vectors and velocities are satisfied; at aspecific time, one soliton may fission to two or more solitons (soliton fissionphenomena) or two or more solitons will fusion to one soliton (soliton fusionphenomena). Actually, for numerous genuine physical models such as in organicmembrane and macromolecule material 11, ineven-clump DNA 12 and in many physicalfields like plasma physics, nuclear physics and so on 13,people have observed the same phenomena. Wazwaz 14-16investigated multiple soliton solutions such type of non-elastic phenomena.

Burgers equation and Sharma-Tasso-Olver equation are such types of model in which Wang et al. 17 found non-elasticsoliton fission and fusion phenomena with only two dispersion relations. Neyrame18 established some periodic and solitonsolutions of Benjamin-Ono equation via basic -expansion method.

In this paper,we would like to investigate non-elastic fission phenomenon of the Benjamin-Onoequation. We also would like to discuss their polynomial solutions whichgenerate rational solutions to scalar nonlinear differential equations byfocusing on the Benjamin-Ono equation. Finally, we propose a specific conditionon parameters for which the fission phenomena will occurs.2. Multi-Solitonsolution of Benjamin-Ono equation and its fission In this section, we bring to bear a direct rationalexponential approach to the Benjamin-Ono equation, (1)where H is the Hilbert transform, we consider it as free from and that is constant.

TheBO equation describes internal waves. It is a completely integrable equationthat gives N-soliton solutions.For the (1+1)-dimensional BO equation (1), there exists atruncated Painleve expansion (2)with being functions of and , the function is the equation of singularity manifold.

Inserting Eq. (2)into Eq. (1) and balancing all the coefficients of different powers of , we get and (3)Setting the value of in Eq. (2) and truncating we reach to or .

(4)Directly substituting Eq. (4) into the Eq. (1), we arrive at thebilinear form (5)For single soliton solution we first consider (6)and the corresponding potential field reads .Inserting Eq.

(6) with Eq. (4) into bilinear form Eq. (5), wecan find the value of as and thus the solution is (7)and corresponding potentialfunction is read as (8) (a) (b) Fig. 1(a) 3D surface of the single solitary wave solution Eq.

(7) of the Benjamin-Onoequation, (b) potential field Eq. (8) with .To achieve two soliton solution of Eq. (1), wejust suppose, (9)where and the correspondingpotential field reads Setting Eq. (9) with Eq. (4) into the bilinear form Eq.

(5), wegain some polynomials which are functions of the variables and Equating all thecoefficients of different power of exponential to zero, we can obtained asystem of algebraic equations in terms of and Solving the system of algebraic equations with the help of symboliccomputation system Maple-13, we attain some solution sets of the unknownparameters which are given below.Set-1: then (10)where are arbitrary constants.The corresponding potential field reads (11) (a) (b) Fig. 2 (a) 3D surface of two solitarywave fission solution Eq. (10) of the Benjamin-Ono equation, (b) potentialfield Eq.

(11) with .From the Fig. 2, which shows the fissionphenomenon between two solitary waves with the parameters choosing as , we can evidently see that, after interaction one singlesolitary waves fission to two solitary wave and decrease wave height withsmaller potential energy.

i.e., at a specific time From careful analysisof Eq. (10), Eq. (11), it is conclude that for any values of the parameters , only fission occurs. Neither elastic scattering nor fusiondoes exist. Set-2: then (12)where , and are arbitrary constants and the corresponding potential fieldreads .

This solution also has the same phenomena like solution Eq.(12).Set-3: then (13)where are arbitrary constants.

The corresponding potential field reads . (a) (b) Fig. 3 (a) 3D surface of two solitarywave solution Eq. (13) of the Benjamin-Ono equation, (b) correspondingpotential field v with .From the Fig.3 Solution Eq. (13) is completely elastic, before and after collision ofthe two salitary waves their shape and size remain same.

So, elasticscattering, no fussion and no fission phenomena does exist for any values ofthe parameters. To achieve three soliton solution of Eq. (1),we just suppose (14)where and the corresponding potential field reads .

Setting Eq. (14) with Eq. (4) into the bilinear form Eq.

(5),we gain some polynomials which are functions of the variables and Equating all thecoefficients of different power of exponential to zero, we can obtained asystem of algebraic equations in terms of and . Solving the system of algebraic equations with the aid ofsymbolic computation system Maple-13, we attain the following constraining solutionsets of the unknown parameters which are given below.Set-1: then (15)where are arbitrary constants.The corresponding potential field reads (a) (b) Fig.

4 (a) Profile of three solitarywave fusion solution Eq. (15) of the Benjamin-Ono equation, (b) correspondingpotential field with .Set-2: then (16)where ,and the corresponding potentialfield reads . (a) (b) Fig. 5 (a) Profile of three solitarywave fusion solution Eq. (16) of the Benjamin-Ono equation, (b) correspondingpotential field with .Set-3:then , (17)where ,and the corresponding potentialfield reads .Set-4: then , (18)where ,and the corresponding potentialfield reads .

Set-5: then , (19)where and and the correspondingpotential field reads .Set-6: , then (20)where are arbitrary constants and the corresponding potential fieldreads .3. RationalPolynomial function solutions of the Benjamin-Ono equation By a Maple computation on (21)we notice that such type of polynomialsolution for the bilinear equation like Eq. (5) does not allow the degree of tgreater than 1.

Which is analytically proved by Ma in the article Ref 19. He made a conjecture that this is true,namely, any polynomial solution to the bilinearequation (5) must have the degree of t not greater than 1. In this regard, wewant to consider the trial solution for the rational polynomial is of degree asfollows: , (22)with of degree one and of degree at least three.According to the bilinearform Eq. (5), we have to write Thus using Eq. (22) into the bilinearequation like Eq. (5) and solving, we achieve a class of rational polynomialsolutions of the Eq.

(1): . (23)On the other hand, if consider with of degree two and of degree four, then setting withinto the bilinear form Eq. (5) and solving coefficients ofdifferent powers of for , we getWhich are the same results of Eq. (23). Now, we can concludethat in the rational polynomial solutions does not support as a polynomial of degree more than 3.Secondly, we considerof degree two and of degree two, then setting with , into the bilinear form Eq. (5) and solving coefficients ofdifferent powers of for , we get Thus . (24) (a) (b) Fig.

6 (a) Profile of rationalpolynomial solution Eq. (23) of the Benjamin-Ono equation, (b) density plot ofthe solution with . (a) (b) Fig.

7 (a) Profile of rationalpolynomial solution Eq. (24) of the Benjamin-Ono equation, (b) density plot ofthe solution with .Remark:All of the solutions availablein this paper have been checked with the help of Maple-13 and we observe thatthey satisfy the original Benjamin-Ono equation.4. Concluding remarksWe successfully used the proposed method in theBenjamin-Ono equation and found some multi soliton solutions. From the analysiswe observed that when dispersive relation exists between the wave number andthe wave speed with any one or then fission exist in the solution. The obtained solutionsmay be significant and important for analyzing the nonlinear phenomena arisingin applied physical sciences.

We considered a generalized bilinear form of the nonlinearBenjamin-Ono (BO) differential equation, and constructed two classes of rationalsolutions to the resulting equation. We remark that it is worth checking ifthere exists a kind of Wronskian solutions to the BO equation. We alsoconjecture that the two classes of rational solutions in Eq.

(23) and Eq. (24)would contain all rational solutions to the BO equation, generated frompolynomial solutions to the generalized bilinear equation (5) under the linkEq. (4).