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In this article, we derive rogue wave (RW) solutions of a fifth-ordernonlinear Schr\"odinger equation over a double-periodic wave background.Choosing the elliptic functions (combinations of $cn$, $dn$ and $sn$) as seedsolutions in the first iteration of Darboux transformation and utilizing thenonlinearization of Lax pair procedure, we create the double-periodic wavebackground for the fifth-order nonlinear Schr\"odinger equation. By introducingthe second linearly independent solution, we generate the RW solutions on thecreated background for three different eigenvalues. We demonstrate thedifferences that occur in the appearance of RWs due to the lower-order andhigher-order dispersions terms. We examine the derived solution in detail forcertain system and elliptic modulus parameters values and highlight someinteresting features that we obtain from our studies. We also calculate thegrowth rate for instability of double-periodic solutions under different valuesof elliptic modulus parameter.

Rogue waves with two different double-periodic wave backgrounds and  their modulational instabilities of a fifth-order nonlinear Schr\"odinger  equation