Integrable two-dimensional generalisation of the sine- and sinh-Gordon equations

A two-dimensional generalisation of the sine- and the sinh-Gordon equations, which one refers to as the shine-Gordon equations, is obtained and solved through the inverse spectral transform (IST) method. The Backlund transformation and nonlinear superposition formula are constructed and explicit wave solitons are given. It is shown also that a slightly different procedure furnishes an IST-solvable extension in 2+1 dimensions of the dispersive long-wave equation.