> 2006 International Symposium on Electrohydrodynamics -Buenos Aires-4 ttl -6 ,h December 2006 < Chaos In Non Lineal Alfven Waves Using the DNLS Equation Sergio A. Elaskar, Gonzalo Sanchez-Arriaga and Juan R. Sanmartin Abstract— The electro-dynamical tethers emit waves in structured denominated Alfven wings. The Derivative Non- lineal Schrodinger Equation (DNLS) possesses the capacity to describe the propagation of circularly polarized Alfven waves of finite amplitude in cold plasmas. The DNLS equation is truncated to explore the coherent, wealdy nonlinear, cubic coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. In this article is presented a theoretical and numerical analysis when the growth rate of the unstable wave is next to zero considering two damping models: Landau and resistive. The DNLS equation presents a chaotic dynamics when is consider only three wave truncation. The evolution to chaos possesses three routes: hard transition, period-doubling and intermittence of type I. I. INTRODUCTION The interaction of arr\electro-dynamical tether with the ionosphere and the terrestrial magnetic field is a source of electromagnetic waves [lf2]. The Alfven waves generated by a conductive body submerged in plasma were predicted by Drell [2] and theses waves were observed for the first time in the Jupiter magnetosphere [3]. An electro-dynamical tether emits waves in structured denominated Alfven wings. It is possible to approach the Alfven wings structure differentiating two regions: 1 - near of the tether (close field), 2 - far away of the tether (distant field). In the close field, the waves have highest intensity and the non-lineal effects are important. The study of the Alfven waves evolution near of the tether can be realized using the Derivative Non-lineal Schrodinger Equation (DNLS). The DNLS equation describes the parallel or almost parallel propagation of the circularly polarized Alfven waves with respect to a non-perturbed magnetic field [4]. This equation has been successful in the understanding of the Alfven waves propagation because numerical simulations and empirical observations, in the space environment, have been explained as the DNLS solutions [5-7]. This work was supported by Ministerio de Ciencia y Tecnologia of Spain under Grant ESP2004-01511, by CONICET under Grant PIP 5692, by FONCYT - Universidad Nacional de Rio Cuarto under Grant PICTO-UNRC-2005 N° 30339, and by Universidad Nacional de Cordoba, under Grant of SECYT. S. A. Elaskar is with Departamento de Aeronautica, Universidad Nacional de Cordoba and CONICET, Av. Velez Sarfield 1611, Cordoba (5000), Argentina, (e-mail: selaskar(o>,efh.uncor, edu). G. Sanchez-Arriaga is with Escuela Tecnica Superior de Ingenieros Aeronauticos, Universidad Politecnica de Madrid. Plaza Cardenal Cisneros 3, Madrid (28040), Espafia. J. R. Sanmartin is with Escuela Tecnica Superior de Ingenieros Aeronauticos, Universidad Politecnica de Madrid. Plaza Cardenal Cisneros 3, Madrid (28040), Espafia. II. DNLS EQUATION The DNLS equation can be obtained from two fluid magnetogasdynamics equations considering neutral plasma and rejecting the electrons inertia and the current displacement [8]. If the direction of the magnetic field without perturbing, B 0 , is coincident with the z-axis, the DNLS can be expressed in the following form [4,8,9]: 5(f) a<t>_,_ / a 2 4> ~ f ' ,|2 ^ . + _!_ ± ±.+ dt dz 2dz 2 dz, <1> W + y<|) =0 (1) <{>, t, z represent dimensionless variables and fields: <1> = (B x ±i B y )/B 0 ; Q t z/V A ->z and Q, t -» / . Qi is the ionic cyclotron frequency, V A is the Alfven velocity. The upper and lower signs in Eq.(l) correspond to left and right handed polarized waves (LH and RH) respectively. The last term of Eq.(l) is a damping/growth linear operator [10]. The DNLS equation belongs to the solitones theory and it includes as limits cases the following equations: "Korteweg- Vries, KdV", "Modified Korteweg-Vries, MKdV" and "Non-linear Schrodinger Equation, NLS". DNLS has been analyzed by means of three techniques: search of exact solutions [11], numerical integration [12-13] and reduction to an ordinary differential equations system. This last technique has been carried out in two different ways: supposing stationary travelling waves [14] and by means of a finite number of modes [ 15 -16]. Ill TRUNCATION OF THE DNLS EQUATION Because the order of the resulting system depends directly on the modes number, it is important to select the minimum number of waves that reproduces correctly the numerical solutions and empirical data. Numerical integration of the DNLS equation suggests the existence of three dominant modes with a resonance condition [17-18]: 2kj = k 2 + k 3 . It is considered, in this work that an approximate solution of the Eq.(l) consists of three travelling waves satisfying the resonance condition: ;=3 * = ^L a J (t)e ^ i(t)+i(kjZ '" i>J,)] (2) y=i aj, \\jj are real numbers. The lineal dispersion relation is coj = kj+kj 2 /2. Introducing the resonance conditions in Eq.(l) and considering only the components of kj, k 2 and k 3 (the other components don't possess influence for long times [10]) the following system is obtained [16]: a x = Taj- r a x a 2 2 sin$ a 2 =-y 2 a 2 +r a 2 a \ 5/n P * = ( Y 2 -y 3 )'' + (l-'' 2 )«i 2 sinP $=v-2a? _ (l + r 2 )co5p 2r 2 2V[ K (3a) (3b) (3c) (3d)