Modulation of Nonlinear Periodic Waves of the (N+1) Dimensional mKP Equation

Abstract

In this study, dispersive shock wave (DSW) solutions arising from the defocusing (n+1)-dimensional modified Kadomtsev–Petviashvili (mKP(d)) equation under step-like initial conditions are investigated. DSWs are considered as a class of nonlinear periodic wave solutions characterized by slowly varying modulation parameters; therefore, the analysis is carried out within the framework of Whitham modulation theory. For this purpose, a similarity transformation is applied to the (n+1)-dimensional mKP(d) equation to derive a (1+1)-dimensional variable-coefficient defocusing modified Korteweg–de Vries (nmKdV(d)) equation. The corresponding Whitham system is then derived and solved numerically. The resulting asymptotic solutions are compared with direct numerical simulations of the nmKdV(d) equation. Additionally, asymptotic and direct numerical solutions of the mKdV(d) equation are obtained to examine the effect of the variable coefficient. The findings demonstrate that the modulation approach is an effective tool for understanding the structural features of DSW solutions.

Keywords

• Dispersive Shock Waves
• Whitham Modulation Theory
• (N+1) dimensional mKP equation

(N+1) Boyutlu mKP Denklemi için Lineer Olmayan Periyodik Dalgaların Modülasyonu

Abstract

Bu çalışmada, basamak tipi başlangıç koşulları altında, defocusing (odaklanmayan) (n+1) boyutlu modifiye Kadomtsev–Petviashvili (mKP(d)) denkleminde ortaya çıkan dispersif şok dalgası (DSD) çözümleri incelenmiştir. DSD’ler, yavaş değişen modülasyon parametreleriyle tanımlanan lineer olmayan periyodik dalga çözümlerinin bir türü olarak değerlendirilmekte olup, bu nedenle analizde Whitham modülasyon teorisi kullanılmıştır. Bu amaçla, (n+1) boyutlu mKP(d) denklemine benzerlik dönüşümü uygulanarak, (1+1) boyutlu değişken katsayılı defocusing modifiye Korteweg–de Vries (nmKdV(d)) denklemi türetilmiştir. Daha sonra, ilgili denklem için Whitham sistemi türetilmiş ve sayısal olarak çözülmüştür. Elde edilen asimptotik çözümler ile nmKdV(d) denkleminin doğrudan sayısal simülasyonları karşılaştırılmıştır. Ayrıca, mKdV(d) denklemine ait asimptotik ve doğrudan sayısal çözümler yapılarak değişken katsayının etkisi araştırılmıştır. Elde edilen bulgular, modülasyon yaklaşımının DSD çözümlerinin yapısal özelliklerini anlamada güçlü bir araç olduğunu göstermektedir.

Keywords

• Dispersif Şok Dalgası
• Whitham Modülasyon Teorisi
• (N+1) Boyutlu mKP Denklemi

References

1. [1] Wan, W., Jia, S., Fleischer, J. W. 2007. Dispersive superfluid-like shock waves in nonlinear optics. Nature Physics, 3(1), 46–51.
2. [2] Vargas-Magaña, R. M., Marchant, T. R., Smyth, N. F. 2021. Numerical and analytical study of undular bores governed by the full water wave equations and bidirectional Whitham–Boussinesq equations. Physics of Fluids, 33(6), 067105.
3. [3] El, G. A., Hoefer, M. A. 2016. Dispersive shock waves and modulation theory. Physica D: Nonlinear Phenomena, 333, 11–65.
4. [4] Whitham, G. B. 1965. Non-linear dispersive waves. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 283, 238–261.
5. [5] Gurevich, A. V., Pitaevskii, L. P. 1974. Nonstationary structure of a collisionless shock wave. Soviet Physics JETP, 38, 291–297.
6. [6] Hoefer, M. A., Ablowitz, M. J. 2009. The theory of dispersive shock waves for NLS and DNLS. Physical Review A, 79(5), 053819.
7. [7] Ablowitz, M. J., Demirci, A., Ma, Y.-P. 2016. Dispersive shock waves in the Kadomtsev–Petviashvili and two dimensional Benjamin-Ono equations. Physica D: Nonlinear Phenomena, 333, 84–98.
8. [8] Ablowitz, M. J., Biondini, G., Wang, Q. 2017. Whitham modulation theory for the Kadomtsev–Petviashvili equation. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473(2204), 20160695.

9. [9] Ablowitz, M., Biondini, G., Wang, Q. 2017. Whitham modulation theory for the two-dimensional Benjamin-Ono equation. Physical Review E, 96(3), 032225.
10. [10] Demirci, A. 2020. Dispersive shock waves in three-dimensional Benjamin-Ono equation. Wave Motion, 94, 102502.
11. [11] Aslanova, G., Demirci, A., Ahmetolan, S. 2022. Modulated periodic wavetrains in the spherical Gardner equation. Wave Motion, 109, 102844.
12. [12] Özdemir, N., Demirci, A., Ahmetolan, S. 2023. Undular bores in the (3+1)-dimensional mKP equation. Physics Letters A, 483, 129051.
13. [13] Abeya, A., Biondini, G., Hoefer, M. A. 2023. Whitham modulation theory for the defocusing nonlinear Schrödinger equation in two and three spatial dimensions. Journal of Physics A: Mathematical and Theoretical, 56(10), 105302.
14. [14] Demirci, A. 2019. Özel bir başlangıç koşulu altında (N+1) boyutlu Benjamin-Ono denklemi için Whitham modülasyon teorisi. Niğde Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi, 8(3), 89–98.
15. [15] Özdemir, N., Demirci, A., Ahmetolan, S. 2024. Evolution of nonlinear periodic waves in the focusing and defocusing cylindrical modified Korteweg-de Vries equations. International Journal of Theoretical Physics, 63(12), 1–13.
16. [16] Luke, J. C. 1966. A perturbation method for nonlinear dispersive wave problems. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 292, 403–412.
17. [17] Byrd, P. F., Friedman, M. D. 1971. Handbook of Elliptic Integrals for Engineers and Scientists. Second edition, revised edn., Die Grundlehren der mathematischen Wissenschaften, vol. 67. Springer, New York–Heidelberg, 358 s.
18. [18] Ablowitz, M. J., Biondini, G., Rumanov, I. 2018. Whitham modulation theory for (2+1)-dimensional equations of Kadomtsev–Petviashvili type. Journal of Physics A: Mathematical and Theoretical, 51(21), 215501.
19. [19] Shampine, L. F. 2005. Solving hyperbolic PDEs in MATLAB. Applied Numerical Analysis and Computational Mathematics, 2(3), 346–358.
20. [20] Cox, S. M., Matthews, P. C. 2002. Exponential time differencing for stiff systems. Journal of Computational Physics, 176(2), 430–455.
21. [21] Kassam, A.-K., Trefethen, L. N. 2005. Fourth-order time-stepping for stiff PDEs. SIAM Journal on Scientific Computing, 26(4), 1214–1233.
22. [22] El, G. A., Hoefer, M. A., Shearer, M. 2017. Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws. SIAM Review, 59(1), 3–61.