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

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Abstract

In this work, we investigate the instability of single- and double-periodic waves of a fourth-order nonlinear Schrödinger 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 Schrödinger 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.

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Funding

NS wishes to thank MoE-RUSA 2.0 Physical Sciences, Government of India, for sponsoring a Fellowship to carry out this work. SR acknowledges MoE-RUSA 2.0 Physical Sciences, Government of India, for providing financial support to achieve the research aim. MS acknowledges MoE-RUSA 2.0 Physical Sciences, Government of India, for sponsoring this research work.

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Author notes

  1. N. Sinthuja, S. Rajasekar and M. Senthilvelan have contributed equally to this work.

Authors and Affiliations

  1. Department of Nonlinear Dynamics, Bharathidasan University, Tiruchirappalli, Tamilnadu, 620024, India

    N. Sinthuja & M. Senthilvelan

  2. Department of Physics, Bharathidasan University, Tiruchirappalli, Tamilnadu, 620024, India

    S. Rajasekar

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  1. N. Sinthuja
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All the authors have contributed equally to the research and to the writing up of the paper.

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Correspondence to M. Senthilvelan.

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Sinthuja, N., Rajasekar, S. & Senthilvelan, M. Instability of single- and double-periodic waves in the fourth-order nonlinear Schrödinger equation. Nonlinear Dyn 111, 16497–16513 (2023). https://doi.org/10.1007/s11071-023-08722-1

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