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The Nonlinear Schrödinger’s Equation
Published in Anjan Biswas, Swapan Konar, Introduction to non-Kerr Law Optical Solitons, 2006
The Hamiltonian is one of the most fundamental notions in mechanics and, more generally, in the theory of conservative dynamical systems with finite or even infinite degrees of freedom. The Hamiltonian formalism has turned out to be one of the most universal in the theory of integrable system and nonlinear waves in general. In the case of nonintegrable systems, the Hamiltonian exists whenever the system is conservative and it is useful for stability analysis. The most useful approach in the soliton theory of conservative nonintegrable Hamiltonian system is a representation on the plane of conserved quantities, namely the Hamiltonian-versus-energy diagrams [35]. In a two-parameter family of solitons, the Hamiltonian-momentum-energy diagrams are quite informative.
Residual symmetry and Bäcklund transformations of model equations for shallow water waves
Published in Waves in Random and Complex Media, 2021
Jinxi Fei, Quanyong Zhu, Zhengyi Ma
Nonlinear partial differential equations have wide applications in the field of physical science, engineering and other applied disciplines, e.g. nonlinear optics [1–5], fluid flows [6–9], plasma physics [10–13], excitable media [14,15], etc. Experience has shown that the Painlevé analysis [16,17] may provide a unified description of integrable behavior in nonlinear partial differential equations, meanwhile, providing an efficient method for evaluating the integrability of particular nonlinear problems. Recently, it is found that the truncated Painlevé expansion is a Bäcklund transformation (BT) [18,19] and the residual of the expansion is the nonlocal symmetry with respect to this BT, this lead Lou to his so-called residual symmetry [20,21]. The residual symmetry can be used to obtain finite transformations and discover novel symmetry reduction [22,23] solutions by means of certain localization procedure. An effective way is to enlarge the original system to localize the nonlocal symmetries and then explore novel solutions by the standard Lie symmetry [24,25] approach. The essential idea of BT is to construct new solutions of integrable systems from seed solutions, which indicates that there exist underlying symmetries related to these transformations for the system under survey. This paper deals with the BT of model equations for shallow water waves, namely, the Boussinesq-Burgers (B-B) system, by localizing the residual symmetry and then generalize the method to explore the nth BT. Theorems 2.1–3.2, stated in Sections 2 and 3, are the main results of this paper.
Higher dimensional localized and periodic wave dynamics in an integrable (2+1)-dimensional deep water oceanic wave model
Published in Waves in Random and Complex Media, 2023
Sudhir Singh, A. Mukherjee, K. Sakkaravarthi, K. Murugesan
Integrable systems show wider variety of nonlinear coherent structures including the much celebrated solitons which attract huge attention not only due to their remarkable stability but also due to their intriguing collision dynamics. This enhances the understanding and consequences of different interactions in physics, such as the interaction between large amplitude waves along with wave–wave and wave–particle interactions. It is well known that the occurrence/existence of solitons is bound only to integrable nonlinear systems, while solitary waves (not solitons) appear in certain non-integrable as well as nearly-integrable models. Also, one has to note that a higher dimensional extension of an integrable one-dimensional model can become non-integrable and its dynamics will be restricted with a lesser number of free parameters [1–3]. In view of the ever increasing interest on higher dimensional integrable nonlinear systems, the following new integrable (2+1) dimensional nonlinear evolution equation was proposed by Anjan Kundu, Abhik Mukherjee, and Tapan Naskar to describe the dynamics two-dimensional oceanic rogue waves and two-dimensional ion-acoustic wave in a magnetized plasma [4,5]: where is the envelope of nonlinear wave and asterisk corresponds to the complex conjugation. Here the first and second terms represent the temporal (t) evolution and the dispersion of the wave along two spatial (x and y) dimensions. This equation was first derived in Ref. [4] to model two-dimensional oceanic rogue waves and later it was named as Kundu–Mukherjee–Naskar (KMN) equation [6–11]. Importantly, it can be seen that the nonlinear terms are different from the conventional Kerr-type nonlinearity arising in the celebrated nonlinear Schrödinger (NLS) and other related/generalized models. Physically, it can be viewed as current-like nonlinearity arising from chirality as well as to study the phenomena of bending of light beams [12]. Recently, the above KMN Equation (1) was shown to govern the dynamics of optical wave or soliton along an Erbium-doped coherently excited resonant waveguides [13]. Also, KMN Equation (1) possesses a rich dynamical characteristic similar to the standard NLS equation, and hence it can also be treated as a (2+1)-dimensional integrable generalization of the NLS equation. Due to these reasons, it has attracted considerable interest among the researchers including us for the present investigation, and several reports are available in the literature.