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

Year 2025, Volume: 29 Issue: 3, 610 - 620, 25.12.2025

Neşe Özdemir

https://doi.org/10.19113/sdufenbed.1714782

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

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