The truncation model of the derivative nonlinear Schrödinger equation

Abstract

The derivative nonlinear Schrödinger (DNLS) equation is explored using a truncation model with three resonant traveling waves. In the conservative case, the system derives from a time-independent Hamiltonian function with only one degree of freedom and the solutions can be written using elliptic functions. In spite of its low dimensional order, the truncation model preserves some features from the DNLS equation. In particular, the modulational instability criterion fits with the existence of two hyperbolic fixed points joined by a heteroclinic orbit that forces the exchange of energy between the three waves. On the other hand, numerical integrations of the DNLS equation show that the truncation model fails when wave energy is increased or left-hand polarized modulational unstable modes are present. When dissipative and growth terms are added the system exhibits a very complex dynamics including appearance of several attractors, period doubling bifurcations leading to chaos as well as other nonlinear phenomenon. In this case, the validity of the truncation model depends on the strength of the dissipation and the kind of attractor investigated.

Instability saturation by the oscillating two-stream instability in a weakly relativistic plasma

“…Miranda et al 29 studied the nonlinear wave-wave interactions in space plasmas and reported the phenomenon of intermittent chaos. Chaos in a three-waves truncation model of the derivative NLSE for nonlinear Alfv en waves has been studied by Sanmartin et al 30 and Sanchez-Arriaga et al [31][32][33] Mori and Janssen 34 derived an expression relating the kurtosis of the sea surface elevation and the occurrence probability of freak waves generated via nonlinear four-wave modulational interactions. Their statistical theory is in good agreement with observational data of extreme wave events.…”

Alfvén waves in space and astrophysical dusty plasmas

“…Low-dimensional chaos of DNLS can also be studied by considering Alfvén modulational wave coupling processes (Sánchez-Arriaga et al, 2009;Miranda et al, 2012). Sánchez-Arriaga et al (2009) studied a truncation model of DNLS with three resonant traveling waves.…”

“…The aforementioned studies on Alfvén chaos elucidate how Alfvén intermittency appears in space and astrophysical plasmas, such as the magnetic intermittent turbulence observed in the solar wind (Koga et al, 2007;Sahraoui, 2008;Chian and Miranda, 2009;Chian and Muñoz, 2011), due to nonlinear dynamical phenomena such as the type-I intermittency induced by a saddle-node bifurcation, the crisisinduced intermittency induced by an interior crisis, and the extrinsic intermittency induced by a noise (Hada et al, 1990;Chian et al, 1998Chian et al, , 2006Chian et al, , 2007Rempel et al, 2006Rempel et al, , 2008Sánchez-Arriaga et al, 2009;Miranda et al, 2012).…”

Chaotic saddles in nonlinear modulational interactions in a plasma

“…Chaos in a three-wave truncation model of the derivative nonlinear Schrödinger equation for nonlinear Alfvén waves has been studied by Sanmartin et al 16 and Sánchez-Arriaga et al 17,18,19 . Mori and Janssen 20 derived an expression relating the kurtosis of the sea surface elevation and the occurrence probability of freak waves generated via nonlinear four-wave modulational interactions.…”