Exact solutions of nonlinear diffusion reaction equation with quadratic, cubic and quartic nonlinearities

A Study of (2+1)-Dimensional Konopelchenko-Dubrovsky (KD) System: Closed-Form Solutions, Solitary Waves, Bifurcation Analysis and Quasi-Periodic Solution

“…In this research article, the study of (2 + 1)-dimensional Konopelchenko-Dubrovsky (KD) model ( 1) is investigated using GERF (generalized exponential rational function) technique. The primary goal of this research study is to construct abundant exact analytical closed-form solutions for the system (1). Moreover, we exhibit the dynamics of solitary wave profiles of some soliton solutions in three dimensional and two dimensional graphics in numerical simulations, and hence we believe that the evolutionary profile dynamics of generated exact closed-form solutions are very impressive and advantageous for physical phenomena.…”

“…Finding the exact solutions for NLEE is an essential task as NLEE describes numerous phenomenon in nonlinear dynamics, engineering, optical fibre, plasma physics, fluid mechanics, natural sciences, etc. A large number of researchers and mathematicians have developed various effective techniques for computing exact solutions of NLPDEs (nonlinear partial differential equations), for instance, tanh function method [1], Hirota's bilinear method [2,3], the Jacobi elliptic function expansion method [4], Kudryashov method [5], the G ′ G -expansion method [6], Darboux transformation method [7], the Backlund transformation method [8], the inverse scattering method [9], Lie-symmetry analysis [10], multiple exp-function method, and many others. Among these techniques, GERF method [11][12][13][14] is very effective, robust and straightforward approach for finding the abundant exact soliton-form solutions of various NLPDEs.…”

Global existence for vector valued fractional reaction-diffusion equations

“…As an example, we explore the scalar system where the nonlinearity is given by F (u) = (1 + ia)u − (1 + ib)|u| 2 u with a,b ∈ R (see [12], [35] and [10]). For particular nonlinearities exact solutions are known, for example, in [20] was studied the existence of scalar traveling waves for the quadratic, cubic and quartic cases by the tanh method. We also explore a FitzHugh Nagumo pattern formation system in R 2 and a population dynamic system in a Banach space.…”

Topological and non-topological soliton solution of the 1 + 3 dimensional Gross-Pitaevskii equation with quadratic potential term

“…In fact, nowadays, there are various modern methods of integrability that are used to integrate these di erent kinds of nonlinear evolution equations. Some of these common methods of integrability are F -expansion method, Projective Ricatti equation method, Lie symmetry analysis method, G 0 =G method of integrability, He's semi-inverse variational principle, tanh-coth method, and many more [10][11][12][13][14][15][16][17]. However, one needs to be careful in applying these methods of integrability as it could lead to incorrect results.…”