A Method of Detecting Solitons among Geophysical Sig- Nals

A method of detecting solitons and determine their parameters based on the scattering problem solution for the relevant nonlinear equation is developed. As an example the Derivative Nonlinear Schrödinger (DNLS) equation has been considered. The integral reflection coefficient, which should rapidly drop when a signal is close to N-soliton profile, has been used as a soliton detector. Application of this technique to numerically simulated signals shows that it is more efficient than standard Fourier transform and can be used as a practical tool for the analysis of outputs from nonlinear systems. Introduction: Solitons in geophysical media Nonlinear waves and solitons are commonly observed in various geophysical media: the interplanetary space [Ovenden et al., 1983], near-Earth plasma [Patel and Dasgupta, 1987; Baumgärtel, 1999], atmosphere [Shen, 1966; Pelinovsky and Romanova, 1977; Petviashvili and Pokhotelov, 1992], Earth's crust [Lund, 1983]. Solitons are the basic structural elements of developed turbulence, because a disturbance with finite amplitude in a nonlinear medium commonly evolves to the soliton state. The modern theory predicts and has mathematical tools to describe N-soliton structures and soliton turbulence gas [Gurevich et al., 2000; Mazur et al., 2002]. The detection of soliton component and determination of its properties demands elaboration of special nonlinear methods of signal analysis. Standard methods of spectral analysis based on the Fourier transform (FT) fit well the detection of linear waves, but they are not very effective for the examination of highly structured space plasma turbulence. The simplest approach is based on the determination of the statistical relationships between amplitudes, duration, velocity, etc. of the observed signal ensemble. Then, the comparison with the theoretically predicted relationships for a given soliton class may be used as a simple observational test for its identification [Guglielmi et al., 1978]. However, the above simple statistical method of the soliton identification requires an analysis of substantial number of signals under the same external conditions. The method described in this paper can be applied to a single event. The proposed method is based on the idea of Hada et al. [1993] who suggested to apply the scattering transform (ST) to a complex time series of analyzed data instead of FT. We have built an effective numerical algorithm to implement the soliton transform. Below we give a short description of this algorithm, comprising calculations of discrete data of the scattering problem (otherwise, soliton parameters) and variation of spatial scale. Derivative nonlinear Schrödinger equation The derivative nonlinear Schrödinger (DNLS) equation 2 (1) (| | ) 0 t xx x b ib b b + + = may describe the nonlinear circularly polarized Alfvén wave x z b b ib = + , propagating along x -axis. Multisoliton solution ( , ) N b x t of the equation (1) can be derived via elementary functions, although even under 2 N = the relevant formula is too cumbersome. Onesoliton solution has the form: 2 0 ( , ) ( ) exp[ ( ( ) 4 | | )], sol b x t a i t ξ φ φ ξ λ = + where 2 2 0 0 8 4 ( ), , | | cosh(4 ) i r i r x x t t a λ ξ λ λ λξ λ = = -2 3arctan tanh(2 ) | | , i r i r λ φ λ ξ λξ λ λ = + − ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ i.e. one-soliton solution is determined by two independent real parameters , r i λ λ . When the complex eigenvalue r i i λ λ λ = + has been determined all the physical parameters of searched soliton can be found. The found eigenvalues enable one to determine with explicit formulas the “physical” parameters of solitons, such as amplitude A , non-linear component of velocity V , characteristic length L and duration T : 2 8(| | ), 4 , A V r r λ λ λ = + = -1 -1 (4 ) , (16 | | ) . L T i r i λ λ λ = =