Water wave models using conformal coordinates

A review of nonlinear water wave problems in complex domains is presented. These include non-smooth highly variable bottom topographies and boundaries as for example forked channel regions. The physical and mathematical relevance of each case is discussed. The Schwarz–Christoffel mapping is the common modeling tool for these different problems. The mapping produces conformal coordinates which allow the problems’ dimension to be reduced and the weakly nonlinear partial differential system of equations to be simplified. Usually conformal mapping is adopted while preserving the Laplace operator. The main focus in the present review is to highlight some water wave problems where the underlying differential operators are not invariant under the conformal change of coordinates. The Schwarz–Christoffel mapping is used to produce a boundary fitted, curvilinear conformal coordinate system. The models reported include solitary waves on branching channels, and its reduction to nonlinear waves on graphs, as well as three dimensional surface-waves propagating over highly variable ridges. Of particular interest is nonlinear water waves on graphs, which has important applications and a scarce mathematical literature. Conformal mapping plays a crucial role regarding a systematic incorporation of forking angles into the equations, as not done before, and deducing a generalized compatibility condition at the graph nodes. •Conformal mapping is used irrespective of operator invariance.•Conformal coordinates as boundary fitted coordinates in intricate domains.•Application to nonlinear water waves on graphs.•Incorporation of forking angles and generalized compatibility condition at nodes.