Exotic modulated signals in a nonlinear electrical transmission line: Modulated peak solitary wave and gray compacton

Abstract

The dynamics of modulated signals in a nonlinear discrete electrical transmission line with the intersite nonlinearity is interpreted in terms of the extended nonlinear Schrödinger-type (ENLS) equation with nonlinear dispersion introduced by Yemele and Kenmogne (2009) [10]. We show that this ENLS equation may simplify as iA t + PA xx + Q∣ A∣ 2 A = ir 1∣ A∣ 2 A x + r 23 A ∗( A 2) xx + r 3 A(∣ A 2∣) xx and exhibits two branches of non smooth solutions according to the sign of the quantity μ 2 = ( 16 r 23 Q - r 1 2 ) / [ 64 r 23 ( r 23 + r 3 ) ] : a branch which contains a peak solitary wave when μ 2 > 0 and another containing gray compacton for μ 2 < 0. Exact analytical expressions for these solutions are derived as well as their properties, namely existence and stability. The exactness of this analytical analysis is confirmed by numerical simulations performed both on the ENLS equation and on the exact equations of the network. These solutions may have important applications in communication systems where solitons are used to codify data.