Amplitude Modulation of Weakly Nonlinear Electrostatic Solitary Waves in Ultrarelativistic Degenerated Semiconductor Quantum Plasma

The amplitude modulation of weakly nonlinear electrostatic solitary waves is investigated using standard quantum hydrodynamic model with multiple scale perturbation technique by deriving a nonlinear Schrödinger equation for ultrarelativistic degenerate semiconductor quantum plasma. Due to nonlinear development of amplitude of electrostatic solitary waves, stable and unstable wave packets are generating and it is numerically analyzed that critical wave number separating stable and unstable wave packets is significantly dependent upon ultrarelativistic parameter, quantum Bohm potential, degenerate pressure, hole-electron mass ratio, and hole-electron temperature ratio. It is seen that critical wave number is varying in a different semiconductor material. Further instability growth characteristic is also examined for different semiconductors. In linear analysis, a dispersion relation has been evaluated analytically and the nature of two wave modes (slow frequency and fast frequency) is studied by varying different plasma parameters.