Similarity reductions and new nonlinear exact solutions for the 2D incompressible Euler equations

• The Clarkson–Kruskal reduction method is applied to reduce the 2D incompressible Euler equations.
• A system of completely solvable ordinary equations is obtained.
• Several novel nonlinear exact solutions with respect to the variables x and y are found.

For the 2D and 3D Euler equations, their existing exact solutions are often in linear form with respect to variables x, y, z. In this paper, the Clarkson–Kruskal reduction method is applied to reduce the 2D incompressible Euler equations to a system of completely solvable ordinary equations, from which several novel nonlinear exact solutions with respect to the variables x and y are found.