Int. J. Appl. Comput. Math (2018) 4:100
Soliton and Exact Solutions for the KdV–BBM Type
Equations by tanh–coth and Transformed Rational
· D. Vinodh
© Springer (India) Private Ltd., part of Springer Nature 2018
Abstract In this work, we study the nonlinear partial differential equations such as KdV–
BBM, mKdV–BBM, generalized KdV–BBM and potential KdV–BBM equations. We apply
the tanh–coth and transformed rational function methods for these model equations to obtain
soliton, kink, periodic, rational and travelling wave solutions with the help of Mathematica.
Keywords KdV–BBM equation · MKdV–BBM equation · Generalized KdV–BBM
equation· Potential KdV–BBM equation · tanh–coth method · Transformed rational function
method · Solitary wave solutions
The study of nonlinear partial differential equation (NLPDE) problems arise in various ﬁelds
of mathematics, physics, ﬂuid dynamics, optics, engineering, quantum and plasma physics.
Initially, the study of solitary waves has been discovered experimentally by John Scott Rus-
sell . Korteweg and de Vries  are derived analytically a NLPDE, which become well
known as the KdV equation.
This well known nonlinear and dispersive long wave equation is integrable and contains
an inﬁnite number of conserved quantities. The delicate balance between nonlinearity and
dispersion produces the stable solitary waves. Interaction of solitary waves and the recurrence
of initial states are numerically investigated by Zabusky and Kruskal . They discovered
that, solitary waves interact mutually each other and after collision retaining its original
shape, amplitude and maintain its constant speed. These resultant solitary waves are called
Department of Mathematics, Madurai Kamaraj Univerisy, Madurai, Tamil Nadu 625021, India