Generalized KP equation and Miura transformation
The author continues his elaborations of generalizations and particularization from the cannon of nonlinear differential equations. In this paper he discusses the \(1+2\)-dimensional nonlinear evolution equation \[ (u_t+u_{xxx}+6uu_x+6fu)_x+g^2u_{yy}=(f'+12f^2), \tag{1} \] where \(f\) is an arbitrary function of \(t\), \(g(t)=e^{-\int 12f \,dt},\) \(f'+12fg=0\). By specializing the values of \(f\) and \(g\) in (1), the author deduces known equations. Thus: (i) If \(f=0\), then \(g=1\) and (1) is reduced to the KP equation \[ (u_t+u_{xxx}+6uu_x)_x+u_{yy}=0. \] (ii) If \(u\) is independent of \(y\), the last equation becomes the KdV equation, namely \[u_t+u_{xxx}+6uu_x=0.\] (iii) (1) with \(f=1/12 t=g\) becomes \[ (u_t+u_{xxx}+6un_x+u/2t)_x+(1/144)u_{yy}=0. \] (iv) When \(u\) is independent of \(y\), the last equation becomes the cylindrical KdV equation: \[u_ t+u_{xxx}+6uu_ x+u/2t=0. \] (v) Finally (1) with \(f=0\), \(t=x\), \(y=T\) is reduced to the Boussinesq equation namely: \[ u_{TT}+u_{xx}+u_{xxxx}+6(uu_x)_x=0. \]
scientific article; zbMATH DE number 4109267