The modulational instability (MI) phenomenon in the nonlinear Schrödinger equation (NLSE) extended by two different nonlinear dispersion terms and the gradient term is investigated. We find that the possibility of instability of plane waves depends on the sign of the nonlinear dispersion parameters with regard to the linear dispersion coefficient. In contrast to the basic NLSE, the system may exhibit instability in the defocusing media for amplitude exceeding a critical value depending on the magnitude of the nonlinear dispersion. An additional feature, namely the higher order or the infinite gain band, absent in the NLSE case, may appear and in which MI induces the birth of the nonlinear localized wave (NLW) of different carrier wave numbers. The result of the qualitative investigations of the system's dynamics indicates the existence of the NLW, such as peak, bright, dark, and compact dark solitary waves which can be well predicted by the MI criteria. In addition the nonlinear dispersion induces the existence of a pair of bright-dark solitary waves which is usually exhibited by the coupled NLSEs only, and the pairs of peak-dark and compact dark-bright solitary waves.

We investigate the compactlike pulse signal propagation in a two-dimensional nonlinear electrical transmission network with the intersite circuit elements (both in the propagation and transverse directions) acting as nonlinear resistances. Model equations for the circuit are derived and can reduce from the continuum limit approximation to a two-dimensional nonlinear Burgers equation governing the propagation of the small amplitude signals in the network. This equation has only the mass as conserved quantity and can admit as solutions cusp and compactlike pulse solitary waves, with width independent of the amplitude, according to the sign of the product of its nonlinearity coefficients. In particular, we show that only the compactlike pulse signal may propagate depending on the choice of the realistic physical parameters of the network, and next we study the dissipative effects on the pulse dynamics. The exactness of the analytical analysis is confirmed by numerical simulations which show a good agreement with results predicted by the Rosenau and Hyman $K(2,2)$ equation.

A modified Colpitts oscillator (MCO) associated with a nonlinear transmission line (NLTL) with intersite nonlinearity is introduced as a self-sustained generator of a train of modulated dark signals with compact shape. Equations of state describing the dynamics of the MCO part are derived and the stationary state is obtained. Using the Routh-Hurwitz criterion, the result of a stability analysis indicates the existence of a limit cycle in certain parameter regimes and there the oscillation of the circuit delivers pulselike electrical signals. The train of generated signals is then transformed into a train of compact modulated dark voltage solitons by the NLTL. The exactness of this analytical analysis is confirmed by numerical simulations performed on the circuit equations. Finally, simulations show the capacity of this circuit to work as a generator of compactlike dark voltage solitons. The performance of the generator, namely, the pulse width and the repetition rate, is controlled by the magnitude of the characteristic parameters of the electronic components of the device.