New Similarity Reductions of Boussinesq-Type Equations

Abstract

In this paper new similarity reductions are presented for several physically interesting partial differential equations, including the Boussinesq and modified Boussinesq equations

$$ {x_{tt}} + u{u_{xx}} + u_x^2 + {u_{xxxx}} = 0, $$

(1)

$$ {u_t} - {u_t}{u_{xx}} - \frac{1}{2}u_x^2{u_{xx}} + {u_{xxxx}} = 0, $$

(2)

, which are completely integrable soliton equations [1,2], and the SRLW equation.

$$ {{\text{u}}_{{\text{tt}}}}{\text{ + }}{{\text{u}}_{{\text{xx}}}}{\text{ + u}}{{\text{u}}_{{\text{xt}}}}{\text{ + }}{{\text{u}}_{\text{x}}}{{\text{u}}_{\text{t}}}{\text{ + }}{{\text{u}}_{{\text{xxtt}}}}{\text{ = 0,}} $$

(3)

(3)