ÖZEL BİR BAŞLANGIÇ KOŞULU ALTINDA (N+1) BOYUTLU BENJAMİN- ONO DENKLEMİ İÇİN WHITHAM MODÜLASYON TEORİSİ

Year 2019, Volume: 8 Issue: 3, 89 - 98, 20.12.2019

Ali Demirci

https://doi.org/10.28948/ngumuh.632051

Abstract

Bu
çalışmada, paraboloid tipi bir dalga cephesi boyunca uzanan basamak tipi bir
başlangıç koşulu için (n+1) boyutlu Benjamin- Ono denkleminin dispersif şok
dalga çözümleri incelenmiştir. Bu amaçla, (n+1) boyutlu Benjamin- Ono denklemi
uygun bir çözüm formu kullanılarak, 
(1+1) boyutlu değişken katsayılı Benjamin- Ono (nBO) tipi bir denkleme indirgenmiştir.
nBO denkleminin dispersif şok dalgası çözümünü betimleyen Whitham modülasyon
denklemleri uygun Riemann tipi değişkenler cinsinden türetilmiştir. Türetilen
bu modülasyon denklemlerinin sayısal çözümlerinden elde edilen dispersif şok
dalgası çözümleriyle, nBO denkleminin doğrudan sayısal çözümleri n=4 boyutu
için karşılaştırılmış ve aralarında iyi bir uyumun olduğu görülmüştür. (n+1)
boyutlu Benjamin- Ono denkleminin paraboloid tipi bir dalga cephesi boyunca
yayılan dispersif şok dalgası çözümünün, indirgenmiş (1+1) boyutlu nBO
denkleminin dispersif şok dalgası çözümüyle betimlenebileceği gösterilmiştir. 

Keywords

Dispersif şok dalgaları, Whitham modülasyon teorisi, Benjamin- Ono denklemi

WHITHAM MODULATION THEORY FOR (N+1) DIMENSIONAL BENJAMIN- ONO EQUATION WITH A SPECIAL INITIAL CONDITION

Abstract

Dispersive shock
waves (DSWs) in (n+1) dimensional Benjamin–Ono equation (nDBO) is considered
using step like initial data along a paraboloid front. Employing a similarity
reduction exactly reduces the study of such DSWs in (n + 1) dimensions to finding
DSW solutions of (1
+ 1) dimensional equations.
With this ansatz, the nDBO equation can be exactly reduced to a Benjamin–Ono
(nBO) type equation. Whitham modulation equations which describe DSW evolution
in the nBO equation are derived and Riemann type variables are introduced. DSWs
obtained from the numerical solutions of the corresponding Whitham systems and
direct numerical simulations of the nBO equation are compared with very good
agreement obtained. It is concluded that the (n+1) DSW behavior along self
similar parabolic fronts can be effectively described by the DSW solutions of
the reduced (1
+ 1) dimensional equations.

Keywords

Dispersive shock waves, Whitham modulation theory, , Benjamin- Ono equation

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