Instability of single- and double-periodic waves in the fourth-order nonlinear Schrodinger equation

In this work, we investigate the instability of single- and double-periodic waves of a fourth-order nonlinear Schrodinger equation, which describes the propagation of ultra-short pulses in a high-speed, long-distance optical fiber transmission system. The single- and double-periodic solutions of this fourth-order nonlinear Schrodinger equation are derived in terms of Jacobian elliptic functions such as dn, cn and sn. From the spectral problem, we compute Lax and stability spectrum for different values of elliptic modulus parameter. We then calculate the instability rate of single- and double-periodic waves for different values of elliptic modulus and system parameters. We also highlight certain novel features come out from our studies. In the case of single-periodic waves, the instability rate for the dn periodic wave is larger when compared to the cn periodic one. Also, our results reveal that the instability growth rate is higher for the single-periodic waves when compared to the double-periodic waves. Further, the width and height (maximal instability rate) of the instability rate of double-periodic waves increase when we increase the system parameter value. This, in effect, leads to faster evolution of the periodic waves (single and double) with a higher growth rate.