This article discusses the evolution of Benjamin-Bona-Mahony (BBM) wave packet's envelope. The envelope equation is derived by applying the asymptotic series up to the third order and choosing appropriate fast-to-slow variable transformations which eliminate the resonance terms that occurred. It is obtained that the envelope evolves satisfying the Nonlinear Schrodinger (NLS) equation. The evolution of NLS envelope is investigated through its exact solution, Soliton on Finite Background, which undergoes modulational instability during its propagation. The resulting wave may experience phase singularity indicated by wave splitting and merging and causing amplification on its amplitude. Some parameter values take part in triggering this phenomenon. The amplitude amplification can be analyzed by employing Maximal Temporal Amplitude (MTA) which is a quantity measuring the maximum wave elevation at each spatial position during the observation time. Wavenumber value affects the extreme position of the wave but not the amplitude amplification. Meanwhile, modulational frequency value affects both terms. Comparison of the evolution of the BBM wave packet to the previous results obtained from KdV equation gives interesting outputs regarding the extreme position and the maximum wave peaking.
Modulation instability is one of the consequences of the water medium's inclination. It causes surface water waves to run into phenomena of splitting and merging in their propagation. An increase in wave amplitude follows this phenomenon, which can encourage the appearance of extreme waves. It is known that the Benjamin Bona Mahony (BBM) wave has modulation instability in its propagation, with the envelope evolving by the equation Nonlinear Schrodinger (NLS) equation dynamic. One of the NLS equation solution is known as Soliton on Finite Background (SFB). SFB is a continuation of the Benjamin-Feir nonlinear terms. Here, the probe of the BBM wave dynamics is conducted by transforming the complex amplitudes form of SFB variable into the polar form of displaced phase-amplitude. It was done to observe changes in the amplitude of the wave in a complex plane with phases that depend only on position. The description of the dynamics of the SFB can be illustrated through Argand diagrams. It was found that the modulation frequency affects the SFB phase: the smaller the modulation frequency, the higher the phase angle. Also, it is found that the phenomenon of SFB phase singularity occurs in extreme waves for certain frequency modulation intervals.
Displaced phase-amplitude variable in polar form. This variable is used to investigate changes in amplitude in complex fields with phases that depend only on the position in the propagation. Soliton on Finite Background (SFB) which is an exact solution of Nonlinear Schrodinger (NLS) equation has been widely used in investigating wave propagation dynamics so that it is the basic for the proposed displaced phase-amplitude. Using displaced phase-amplitude, the results obtained can be described in Argand Diagrams. Wave equation used as a model is the Benjamin Bona Mahony (BBM) equation where the envelope of this wave evolves following the NLS equation. This wave is unidirectional long wave on the surface and has low amplitude characteristics. Step by step to obtain a SFB solution that contains displaced phase-amplitude described and displayed in an argand diagram. In additional, the envelope graph is given.
This research discusses stationary discrete nonlinear Schrödinger equation with cubic-quintic nonlinearity. This equation is interesting to study because it has a unique solution known as a soliton. This solution has a fixed profile and speed when propagating and in the context of applications in the optical field, soliton can also be engineered as a carrier of information that can propagate on media with very long distances without experiencing significant interference. This paper only focuses on on-site type soliton (soliton that peak in the middle on one site). The method of determining solution on stationary discrete nonlinear Schrödinger equation with cubic-quintic nonlinearity is divided into two cases. The first case for the value of parameter C is zero and the soliton solution is determined analytically. In this case the soliton solution can be stated explicitly, therefore the soliton solution will be displayed and also the boundaries on the parameters that make the solution in the form of on-site soliton. The second case for the value of parameter C is not zero and the soliton solution is determined using a numerical approach namely Trust Region Dogleg Method. In this case the soliton solution cannot be stated explicitly, therefore only boundaries of the parameters that make the solution in the form of on-site soliton will be displayed.
Unidirectional surface water waves used in this study were obtained from the Benjamin Bona Mahony (BBM) equation. The BBM wave research is carried out by observing the envelope of BBM wave groups that involve following the Nonlinear Schrodinger (NLS) Equation. Soliton on Finite Background (SFB) is an excellent NLS solution for describing the dynamics of wave propagation. It is known that there are modulation instability and phase singularity phenomena which can significantly affect the amplitude of the wave in its propagation. Hamiltonian contains potential energy which can explain the singularity and amplitude dynamics of the phase. Hamiltonian equations using the displaced phase-amplitude variables have been obtained and are presented in this article.
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