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Asymptotic analysis of the 2-dimensional soliton solutions for the Nizhnik–Veselov–Novikov equations

In this paper we present a direct approach to determining a class of solutions, the asymptotic analysis of the dromion solutions, and their asymptotic properties of the Nizhnik–Veselov–Novikov equations by means of Pfaffians. The form of the solution obtained allows a detailed asymptotic analysis of the dromion solutions and compact expression for the phase shifts and changes of amplitude as a result of interaction of the dromions to be determined. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2014) 38: 278 – 296 ¨ ITAK ˙ c TUB ⃝ doi:10.3906/mat-1212-13 Asymptotic analysis of the 2-dimensional soliton solutions for the Nizhnik–Veselov–Novikov equations ∗ ¨ Metin UNAL Department of Mathematics, U¸sak University, U¸sak, Turkey Received: 10.12.2012 • Accepted: 28.08.2013 • Published Online: 27.01.2014 • Printed: 24.02.2014 Abstract: In this paper we present a direct approach to determining a class of solutions, the asymptotic analysis of the dromion solutions, and their asymptotic properties of the Nizhnik–Veselov–Novikov equations by means of Pfaffians. The form of the solution obtained allows a detailed asymptotic analysis of the dromion solutions and compact expression for the phase shifts and changes of amplitude as a result of interaction of the dromions to be determined. Key words: Soliton, dromion 1. Introduction In recent years the generalizations of integrable (1+1)-dimensional equations to (2+1) dimensions have been widely studied. The integrable generalization of the nonlinear Schr¨odinger (NLS) equation is the Davey– Stewartson (DS) equations [5]. The generalization of the Korteweg–de-Vries (KdV) equation has 2 possibilities, which are the Kadometsev–Petviashvili (KP) equations [11] and the Nizhnik–Veselov–Novikov (NVN) equations [14]. The NVN equations are Ut = Uxxx + Uyyy + 3(Φxx U )x + 3(Φyy U )y , (1) U = Φxy . (2) These generalizations, the DS and NVN equations, have 2-dimensional localized hump solutions that decay exponentially in all directions, which are called 2-dimensional solitons or dromions. The KP equation does not have such solutions. The word dromion comes from the Greek word dromos, which means track, and it has been given [6] to these objects because they are located at the intersection of plane waves, which can be thought to form tracks. These 2-dimensional solitons, like the well-known solutions in (1 + 1) .