Modulational instability of constant intensity waves with Wadati-Mathieu potential for polynomial law of self-phase modulation

Nirmoy Kumar Das; Srimanta Maji; Naresh Saha; Ashoke Das; Anjan Biswas

doi:10.1209/0295-5075/ae5c31

Issue		EPL Volume 154, Number 3, May 2026


Article Number		30001
Number of page(s)		7
Section		General physics
DOI		https://doi.org/10.1209/0295-5075/ae5c31
Published online		08 May 2026

EPL, 154 (2026) 30001

Modulational instability of constant intensity waves with $Mathematical equation: $\mathcal {PT}\text{-symmetric}$$ Wadati-Mathieu potential for polynomial law of self-phase modulation

Nirmoy Kumar Das¹^,2, Srimanta Maji³, Naresh Saha³ (This email address is being protected from spambots. You need JavaScript enabled to view it. ), Ashoke Das² and Anjan Biswas⁴^,5^,6^,7

¹ Department of Mathematics, Kaliyaganj College - Uttar Dinajpur, 733129, India
² Department of Mathematics, Raiganj University - Uttar Dinajpur, 733134, India
³ Department of Mathematics, School of Engineering, Dayananda Sagar University - Bengaluru, 562112, India
⁴ Department of Mathematics & Physics, Grambling State University - Grambling, LA 71245-2715, USA
⁵ Department of Mathematics, Faculty of Science, Karadeniz Technical University - Trabzon-61080, Türkiye
⁶ Department of Physics and Electronics, Khazar University - Baku, AZ-1096, Azerbaijan
⁷ Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University Medunsa-0204, Pretoria, South Africa

Received: 10 September 2025
Accepted: 7 April 2026

Abstract

Constant intensity (CI) solutions of the nonlinear Schrödinger equation with third-order dispersion (TOD), a polynomial law of self-phase modulation (cubic, quintic, and septimal nonlinearities) and $Mathematical equation: $\mathcal {PT}\text{-symmetric}$$ potential are considered. The potential is obtained by using an inverse design approach, in which the complex $Mathematical equation: $\mathcal {PT}\text{-symmetric}$$ potential is constructed to support an exact constant intensity solution. By introducing c as an asymmetry parameter, the modulational instability (MI) is mapped onto the $Mathematical equation: $(k,\beta )\text{-plane}$$ for $Mathematical equation: $c=\pm1$$ , exploring all combinations of self-focusing and self-defocusing cubic-quintic-septimal nonlinear terms. Also, the effect of the amplitude A on MI has been investigated for different values of Bloch momenta k. It has been shown that TOD, higher-order nonlinearities, and the asymmetry parameter c significantly affect MI growth and stability regions.

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