Gaussian solitary waves to Boussinesq equation with dual dispersion and logarithmic nonlinearity

Department of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL-35762, USA biswas.anjan@gmail.com Department of Mathematics and Statistics, College of Science, Al–Imam Mohammad Ibn Saud Islamic University, Riyadh-13318, Saudi Arabia Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria-0008, South Africa Department of Mathematics, Faculty of Science and Arts, Yozgat Bozok University, 66100 Yozgat, Turkey ekici-m@hotmail.com; abdullah.sonmezoglu@bozok.edu.tr


Introduction
There are several models that address the dynamics of shallow water waves along lake shores and sea beaches. A few of these models that carry a lot of visibility and frequented upon are the Korteweg-de Vries (KdV) equation, Kadomtsev-Petviashvili (KP) equation, Kawahara equation, Boussinesq equation (BE), Benjamin-Bona-Mahoney equation, and others. All of these models have been studied with algebraic forms of nonlinearity. The current practice is to study these models with logarithmic nonlinearity; a trend that was first introduced by Wazwaz during 2014 [20]. Thus, KdV equation, KP equation, and Boussinesq equations have all been studied for logarithmic nonlinearities using a variety of rich mathematical schemes such as soliton perturbation theory, semi-inverse variational method, traveling wave hypothesis, and several others. In fact, there exists a plethora of mathematical techniques that are applied to several nonlinear evolution equations in fluid c Vilnius University, 2018 dynamics, nonlinear optics, nuclear physics, and other areas to address them . This paper studies the dynamics of Gaussian solitary waves due to BE with logarithmic nonlinear form by the aid of another powerful mathematical principle, namely, the extended trial equation algorithm. The next couple of sections details the scheme that yields Gaussian solitary waves to the BE.

Governing equation
BE with logarithmic nonlinearity and dual dispersion reads as follows [2,21]: This dynamical model was introduced by Wazwaz [21]. In Eq. (1), q(x, t) represents the wave profile, where the independent variables x and t represent spatial and temporal coordinates, respectively. The first two terms in Eq. (1) constitute the wave operator. The coefficient of a is the logarithmic nonlinear term. The coefficients of b 1 and b 2 are dispersion terms, where, in particular, the coefficient of b 2 gives the spatio-temporal dispersion.

Mathematical analysis
To secure Gaussons or solitary wave solutions to Eq. (1), the starting hypothesis is where s = x − vt.
In Eq. (2), v represents the speed of the wave, and the functional form of g will give the solitary wave solution. Substituting hypothesis (2) into (1) and integrating twice yields where g = d 2 g/ds 2 . The integration constant is taken to be zero, both times, since the search is for a localized solitary wave solution.
To obtain a closed form analytic solution, we employ a transformation formula .

Conclusions
This paper secured Gaussian solitary wave solutions to BE that is considered with logarithmic nonlinearity and dual dispersion. The powerful extended trial function method is the integration scheme that has been implemented to retrieve the solitary wave solutions that are also referred to as Gaussons in this context. These solutions appear with constraint conditions that guarantee their existence.
The results of this manuscript are new and are being reported for the first time in this paper. In regards to the physical meaning of the model, this represents shallow water wave dynamics along lake shores and beaches. This is generalized model to the regular Boussinesq equation that is known. Upon carrying out the Taylor series expansion of the logarithmic function about q = 1 and retaining till the first term, the regular Boussinesq equation, with drifting term, falls out. Thus the model of study incorporates all of the previously established results.
The results of this paper paves way to carry out further research in this avenue. Later, perturbation terms will be included in this model and thus the perturbed BE will be addressed using this integration scheme as well as various other integration algorithms. These will be Lie symmetry analysis, Kudryashov's method, modified simple equation method, and several others. Additionally, this model will be studied with time-dependent coefficients along with stochastic perturbation terms. The results of those research will be reported in the future.