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.

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