Integrable properties and similarity reductions of the sine-laplace equation from an inviscid incompressible fluid with symbolic computation

Several integrable properties of the Sine-Laplace equation arising from an inviscid incompressible fluid are symbolically presented, including the Lax pair, auto-Bäcklund transformation, nonlinear superposition formula, bilinear form, and static N-soliton solution. Furthermore, with symbolic computation, two similarity reductions for the Sine-Laplace equation are derived by virtue of the classical Lie group method of infinitesimal transformations. One reduces to the third Painlevé equation and the other to a known ordinary differential equation. Sample static solutions are discussed and pictured.