Wave Modulations in the Nonlinear Biinductance Transmission Line

Pattern selection and modulational instability in the one-dimensional modified complex Ginzburg–Landau equation

“…The linear stability of the continuous wave (cw) solution (3) can be investigated by seeking a solution in the form [22] wðx;…”

“…The Bejamin-Feir instability, originally established for periodic wave trains (Stokes waves in surface water theory) [6], which is actually a generic process in physics and well understood in the context of the non-linear Schrö dinger equation [7], has been extended to the one-dimensional (1D) complex Ginzburg-Landau (CGL) equation which is a generic equation describes dissipative systems above the point of bifurcation [8]. The CGL equation plays an important role in many branches of physics, including fluid dynamics [2], non-linear optics [9][10][11][12][13][14], laser physics [15][16][17][18], theory of phase transitions [19][20][21], non-linear transmission line [22,23] and in the stick-slip motion [24]. The with asymptotic group velocity negative), sources (propagating hole with asymptotic group velocity positive), periodic unbounded solution [25][26][27][28][29][30], vacuum, periodic and quasi-periodic solutions [31], slowly varying fully non-linear wavetrains [32], and a transition to chaos [8,[33][34][35][36].…”

Soliton-like excitations in the one-dimensional electrical transmission line

“…For this purpose, we use the semi-discrete approximation [5,6] to obtain short-wavelength envelope solitons. This approach allows us to properly treat the carrier with its discrete character and to describe the envelope in the continuum approximation.…”

“…In relation (3.5), the group velocity is expressed as 6) and the real k c denotes the critical value of the wave number of the signal. Therefore, at o ε 3/2 , Eq.…”

“…Therefore, several soliton perturbation theories have been developed to study the effect of small perturbations on integrable equations. In these theories, the reductive perturbation method [5,6] is well known. Within this method, we have the semi-discrete approximation, which consists of considering the continuum approximation to describe the envelope of the signal and to treat the carrier wave with its discrete character.…”