### Soliton equation. We also would like to discuss

Solitonequations 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.