Solitary solution of modified bad and good Boussinesq equation by using of tanh and extended tanh methods (Q384868)

The authors investigate the modified ``bad'' and ``good'' Boussinesq equations, respectively given by \[ u_{tt}-u_{xx}-u_{xxxx}-3(u^2)_{xx}+3(u^2u_x)_x=0 \] and \[ u_{tt}-u_{xx}+u_{xxxx}-3(u^2)_{xx}+3(u^2u_x)_x=0. \] Motivations for the investigations stem from nonlinear waves applications in nonlinear optics, plasma physics, and fluid mechanics. More precisely, solitons solutions are obtained by first assuming a traveling waves ansatz and then solving the resulting ordinary differential equations by either a reduction of order technique or the standard and extended tanh methods. The latter assume finite expansions of the unknown function \(u\) in terms of \(\mathrm{tanh}\) functions. Various kink, anti-kink, compacton and periodic solutions are obtained. Conditions for their formation as well as certain two-dimensional plots are also given. Better connections and detailed discussions with similar solutions appearing in the existing literature would have been expected.