Chaotic, fractal, and coherent solutions for a new integrable system of equations in 2+1 dimensions

Abstract

A new integrable and nonlinear system of partial differential equations in 2+1 dimensions is obtained by means of an asymptotically exact reduction method based on Fourier expansion and spatio-temporal rescaling. We find interacting coherent excitations such as the soliton, dromion, lump, ring soliton, breather, and instanton solutions. The interaction between the localized solutions are completely elastic because they pass through each other and preserve their shapes and velocities, the only change being a phase shift. Moreover, the arbitrariness of the functions included in the general solution implies that lower dimensional chaotic patterns such as chaotic-chaotic patterns, periodic-chaotic patterns, chaotic-soliton patterns, and chaotic-dromion patterns can appear in the solution. In a similar way, fractal dromion patterns and stochastic fractal excitations also exist for appropriate choices of the initial conditions.