Research Projects

The study of propagation of waves in different mediums is always an interesting topic of research. The wave interactions do open a new avenue pointing to the behaviour of the solutions in the longer term. Recently, the study of wave propagation in a liquid medium with gas bubbles has caught a lot of attention. Already, a lot of work consisting of linear wave propagation is studied in different dimensions and order. But, most interesting is the study of the nonlinear phenomenon in wave propagation. The presence of the nonlinear terms leads to certain PDES upto order four which cannot be solved analytically. Moreover, when various other phenomena such as the viscosity, heat transfer in the medium is being considered the model becomes quite difficult. The only plausible solution seems to be using certain numerical methods. A ray of hope seems to appear when the long weakly nonlinear waves are considered which seems to lead to certain equations in (1+3)- dimension which looks promising. The equation can be infused with various parameters with respect to various physical phenomena. Now, the main question is the integrability and solvability of this equation. The (1+3)- dimensional equation is claimed to be not integrable in nature as it does not pass the Painleve test. But, still certain soliton type solutions, hyperbolic functions and periodic solutions were computed for this equation. Now, this forms the main motivation of our work. Our main objective is to actually establish the integrability of the (1+3)- dimensional equation using the integrability of its reduced equation which is obtained through the reductions of point symmetries. Next is to look for the nonlocal symmetry of this equation. This may sound a bit absurd but the derived nonlocal symmetry can be used to prolong the given equation and hence the nonlocal symmetry becomes a local symmetry of the prolonged equation and hence the exact solutions can be derived from them for the original equation. Next in line is the computation of nonclassical symmetries. The computation of these nonclassical symmetries leads to solving certain nonlinear equations and hence leads to solutions which are not mentioned anywhere in literature. Overall, the work is primarily focussed on computation of new closed form solutions and to discuss their propagation properties. Recently, these equations were observed to be highly beneficial in drug delivery. The shock wave solution does play an important role in corroborating the fact that the medication induced gas bubble reaches the targeted area in the human body. It will be also interesting to observe the impact of other types of solutions on the propagation of gas bubbles. Finally, the deductions were made to look at the behaviour of the solutions in a longer term around the equilibrium points and closed orbits.