Sinh-Poisson Equation

The sinh-Poisson equation is the nonlinear elliptic partial differential equation

(1)

where is a constant parameter and is the Laplacian (Ting et al. 1987; Zwillinger 1997, p. 135). In two-dimensional Cartesian coordinates, . Since , the equation can also be written as

(2)

The constant function is a solution. Since , if is a solution then so is . In particular, homogeneous boundary conditions preserve this symmetry. Since as , replacing by its first-order linear approximation gives the Helmholtz differential equation

(3)

The Laplacian form distinguishes the sinh-Poisson equation from the sinh-Gordon equation, which in the convention used here has .

Solutions of the form satisfy an ordinary differential equation. Multiplication by and integration give

			(4)
			(5)

where is a constant. For real and , these solutions are periodic functions that range between the two roots of , and the resulting quadrature is an elliptic integral.

The nonlinear boundary value problem need not have a unique solution. For homogeneous Dirichlet boundary conditions on a square, McDonald (1974) computed six distinct solutions for the same value of . Exact solutions on a square, as well as doubly periodic functions satisfying the equation, can be expressed using Jacobi elliptic functions. One class has the form , where denotes the inverse hyperbolic tangent, and and are one-variable functions whose parameters are constrained by the equation (Ting et al. 1987, Chow et al. 2003).