Solutions and Conservation Laws of a (2+1)-Dimensional Boussinesq Equation

Abstract

We study a nonlinear evolution partial differential equation, namely, the (2+1)-dimensional Boussinesq equation. For the first time Lie symmetry method together with simplest equation method is used to find the exact solutions of the (2+1)-dimensional Boussinesq equation. Furthermore, the new conservation theorem due to Ibragimov will be utilized to construct the conservation laws of the (2+1)-dimensional Boussinesq equation.

Symmetry analysis and exact solutions of extended Kadomtsev-Petviashvili equation in fluids

“…Conservation laws [13][14][15][16][17][18] play a significant role in finding the solutions of PDEs. Noether's theorem helps us with an innovation formula for deriving conservation laws by appealing to the symmetries of the action.…”

Regarding New Wave Patterns of the Newly Extended Nonlinear (2+1)-Dimensional Boussinesq Equation with Fourth Order

“…Investigating deeper into properties of this model, some powerful methods [35,36] have been applied successfully. One model based on this equation, namely, the extended nonlinear (2+1)-dimensional Boussinesq equation [37,38] defined by…”

Gaussons

“…which describes the propagation of gravity waves on the surface of water; in particular it describes the head-on collision of an oblique wave ( Johnson, 1996;Wazwaz, 2010Wazwaz, , 2012Wazwaz, , 2013Moleleki and Khalique, 2013). The (2+1)-dimensional BE (2) combines the two-way propagation of the classical BE (1) with the dependence on a second spatial variable as that occurs in the two-dimensional Kadomtsev-Petvviashvil (KP) equation ( Johnson, 1996;Wazwaz, 2010Wazwaz, , 2011Moleleki and Khalique, 2013). It is to be noted that the (2+1)-dimensional equation may arise in a different form by:…”

“…It is to be noted that the standard BE is integrable and admits multiple soliton solutions (Bona et al, 2002;Wazwaz, 2007Wazwaz, , 2008. However, the (2+1)-dimensional BEs (2) and ( 3), and the (3+1)-dimensional BE (4) are not integrable because these equations do not pass the Painlevé test, and each gives at most two soliton solutions ( Johnson, 1996;Wazwaz, 2006Wazwaz, , 2008Wazwaz, , 2009Moleleki and Khalique, 2013). Wazwaz (2012) used the Hirota's bilinear method to emphasize the existence of two soliton solutions only for each of the last two equations.…”