Soliton

equations connect affluent histories of exactly solvable systems constructed in

mathematics, fluid physics, microphysics, cosmology, field theory, etc. To

explain some physical phenomenon further, it becomes more and more important to

seek exact solutions and interactions among solutions of nonlinear wave

solutions. Owing to the wide applications of soliton theory in mathematics, hydrodynamics,

plasma physics, nuclear physics, biology, astrophysics and geophysics, the

study on integrable models has attracted much attention of many researchers. To

find some exact explicit soliton solutions for nonlinear integrable partial differential

equations (NLPDEs) models is one of the most key and significant tasks. There

are many methods exist in the literature such as -expansion method 1, exp-function

2, -expansion method 2, 3

by which single soliton solutions can be found. But the most remarkable

property of exactly integrable equations is the presence of exact solitonic

solutions, and the existence of one-soliton solution is not itself a specific

property of integrable partial differential equations, many non-integrable equations

also possesses simple localized solutions that may be called one-solitonic. Nevertheless,

there are integrable equations only, which posses exact multi-soliton solutions

which describe entirely elastic interactions between individual solitons 4. A variety of efficient methods like the

inverse scattering transformation (IST) 5, 6,

Backlund transformation 7, 8, Hirota’s

direct method 9, mapping and deformation

approach 10 etc. have been well developed

to find exact solutions for integrable models in which interactions between

soliton solutions gained for integrable models are to be completely elastic. Yet

some soliton models gives completely non-elastic soliton interactions when

specific conditions between the wave vectors and velocities are satisfied; at a

specific time, one soliton may fission to two or more solitons (soliton fission

phenomena) or two or more solitons will fusion to one soliton (soliton fusion

phenomena). Actually, for numerous genuine physical models such as in organic

membrane and macromolecule material 11, in

even-clump DNA 12 and in many physical

fields like plasma physics, nuclear physics and so on 13,

people have observed the same phenomena. Wazwaz 14-16

investigated 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-elastic

soliton fission and fusion phenomena with only two dispersion relations. Neyrame

18 established some periodic and soliton

solutions of Benjamin-Ono equation via basic -expansion method.

In this paper,

we would like to investigate non-elastic fission phenomenon of the Benjamin-Ono

equation. We also would like to discuss their polynomial solutions which

generate rational solutions to scalar nonlinear differential equations by

focusing on the Benjamin-Ono equation. Finally, we propose a specific condition

on parameters for which the fission phenomena will occurs.