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

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.

Anahtar Kelimeler:

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

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

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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