Irregular modulation of non-linear Alfvén/Beltrami wave coupled with an ion-sound wave

• A singularity of a system of differential equations may produce intrinsic solutions.
• In the recently formulated non-linear model of Alfvén wave, we find a singularity.
• The singularity produces an irregular modulation with an arbitrary number of pulses.
• This occurs only to a fully non-linear Alfvén wave inherited from the Hall MHD.
• Previous models that linearize the Hall term overlooked such a phenomenon.

A singularity of a system of differential equations may produce “intrinsic” solutions that are independent of initial or boundary conditions—such solutions represent “irregular behavior” uncontrolled by external conditions. In the recently formulated non-linear model of Alfvén/Beltrami waves [Commum Nonlinear Sci Numer Simulat 17 (2012) 2223], we find a singularity occurring at the resonance of the Alfvén velocity and sound velocity, from which pulses bifurcate irregularly. By assuming a stationary waveform, we obtain a sufficient number of constants of motion to reduce the system of coupled ordinary differential equations (ODEs) into a single separable ODE that is readily integrated. However, there is a singularity in the separable equation that breaks the Lipschitz continuity, allowing irregular solutions to bifurcate. Apart from the singularity, we obtain solitary wave solutions and oscillatory solutions depending on control parameters (constants of motion).