International Journal of Differential Equations

Volume 2018, Article ID 1716571, 10 pages

https://doi.org/10.1155/2018/1716571

## Nonlinear Evolution of Benjamin-Bona-Mahony Wave Packet due to an Instability of a Pair of Modulations

^{1}Dynamic Application and Optimization Research Group, Syiah Kuala University, Banda Aceh 23111, Indonesia^{2}Mathematics Graduate Study Program, Syiah Kuala University, Banda Aceh 23111, Indonesia^{3}Department of Physics, Syiah Kuala University, Banda Aceh 23111, Indonesia^{4}Department of Mathematics, Syiah Kuala University, Banda Aceh 23111, Indonesia

Correspondence should be addressed to Marwan Ramli; di.ca.haiysnu@htam.nawram

Received 1 January 2018; Accepted 19 April 2018; Published 3 June 2018

Academic Editor: Masaaki Tamagawa

Copyright © 2018 Vera Halfiani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

This research is motivated by the need of generating extreme waves (also called freak, rogue, or giant waves) in a typical tank in a hydrodynamic laboratory. An extreme wave is defined as a wave which has significant height more than twice the height of measured wave train [1, 2]. There are some physical mechanisms which cause the occurrence of the extreme wave. In [3], it was explained that it can occur due to wave-current interaction, geometrical or spatial focusing, focusing due to dispersion, focusing due to modulational instability, or soliton collision. An extreme wave can take place in deep water, shallow water, and at coastal area. It was reported that the wave appeared often at coastal area and in shallow water [4]. The extreme wave is a major threat to the ships and other marine structures because it can cause severe damage and loss when they are hit by the wave [5–11]. However, the occurrence of the extreme wave is unpredictable. The generation of an extreme wave in the laboratory is important to test the endurance of the ships or marine structures before they are installed and operated in the real environment. Many studies have been conducted to investigate the behavior of the extreme wave analytically and numerically [4, 12–28].

While ships and marine structures are typically built and operate on the offshore, the occurrence of the extreme wave in deep water is not as easily understood as the one in shallow water and coastal area since the wave on the surface of deep water may not be significantly affected by the topography and current. This fact has given a rise to the theory of modulational instability. The instability causes growing modulation of the wave envelope. The phenomenon happens as a wave nonlinearity behavior. This nonlinear event occurs when a wave is perturbed by other waves which cause envelope modulations that evolve into steep waves corresponding to nonlinear focusing of wave energy [3]. Modulational instability may be an appropriate mechanism to generate extreme waves in a wave tank. In a wave tank which initially contains still water, signals are given to the wavemakers generating waves to propagate downstream along the tank. Owing to the nonlinearity effect, the waves deform during their propagation (see [29–32]).

Extreme wave behavior due to modulational instability has been fiercely studied involving various wave equations. The evolution of Korteweg-de Vries (KdV) equation’s solution in wave packet form was investigated in [33]. In the study, the KdV wave packet which is initially in the bichromatic signals experiences instability deforming the wave and causing an amplitude amplification. Bichromatic wave is formed from a superposition of two monochromatic waves which have same amplitudes and slightly different frequencies [27, 30]. This result was obtained by deriving the solution of the KdV equation in the form of asymptotic series up to the third order. KdV’s solution in higher order, up to the fifth order, was conducted in [34] and the result suggests that the higher the order is, the higher the amplitude amplification is. However, these two waves’ maximum amplitudes are far lower than the amplitude of a wave generated by HUBRIS model although the extreme positions at which the maximum peaking happens conform. HUBRIS model is numerical wave model which was developed based on the Laplace equation (full water equation) [27]. Bichromatic signal deformation was also investigated by employing Boussinesq equation in [24]. The resulting wave give matching amplitude to the wave of HUBRIS model. Study on Benjamin-Bona-Mahony (BBM) model’s wave packet with bichromatic signal is presented in [35]. The results suggest that the waves experience instability during its propagation. Moreover, if the frequencies in the bichromatic signal are closer to each other then the amplitude amplification tends to be higher; also higher initial amplitude may cause higher amplification on the amplitude.

Modulational instability also occurs when a monochromatic signal is perturbed by a pair of side bands of smaller amplitudes and slightly different frequencies. These waves form trichromatic wave also known as Benjamin-Feir type wave. Nonlinear evolution of wave packet with trichromatic signal based on KdV equation was studied in [18]. Investigating the wave packet evolution can also be carried out through the envelope evolution. In many studies, it has been derived that the envelope of the KdV wave packet is in the NLS equation form [36–40]. One of NLS’ exact solutions, Soliton on Finite Background (SFB), shows an instability during its propagation. SFB itself is a best approximation to the Benjamin-Feir type wave. KdV wave packet with its NLS envelope has been applied to study the occurrence of extreme waves in many areas, not only in the fluid dynamic but also in the fields of plasma physics, quantum electronic, and optics [41–45]. BBM wave packet envelope had been also analyzed in [46]. The study found that envelope evolves satisfying the NLS equation.

BBM equation was firstly proposed as a revision of classic KdV equation which has dispersion relation contradicting to its physical wave behavior [47]. The classic KdV equation only works for small values of wavenumber. When the wavenumber values are large, the phase velocity of the KdV waves becomes negative, which indicates that the waves change their direction of propagation. This behavior is against the original assumption of forward-travelling waves. BBM equation yields more reasonable dispersion relation for any value of wavenumber. A family of KdV equation with exact dispersion relation was derived in [48]. The exact dispersion relation describes positive phase velocity for any value of wave number and this finding has corrected the shortcoming of the classic KdV. This exact KdV equation has been applied in some studies of extreme wave generation. BBM and KdV equations are originally derived from the same equation which is Laplace equation. These equations both describe shallow water wave propagation and yield soliton solution. However, the two equations in all likelihood illustrate waves with distinct characteristics.

In this study, we are interested in observing the evolution of BBM wave packet in the extreme wave context. The waves will be investigated through its envelope which is in NLS equation form. The SFB solution will be applied to analyze the instability which may trigger an extreme wave. The amplitude amplification will be measured through the waves’ Maximal Temporal Amplitude (MTA), a quantity measuring the highest elevation at every spatial position in the domain during the observation time. It is also interesting to compare the results to the KdV equation waves which have been actively studied especially in extreme wave area.

In the next section, we will briefly review the BBM wave packet envelope which is adopted from the study in [46]. Also we will derive the elevation equation based on the wave packet and the obtained envelope equation. Section 3 contains explanation of two quantities which importantly describe the resulting wave characteristics, Amplitude Amplification Factor (AAF) and Maximal Temporal Amplitude (MTA). The resulting waves will be presented in Section 4 which is divided into two parts; the first part explains the modulational instability and phase singularity phenomenon and the second part presents the result comparison from BBM and KdV wave packet evolution. This paper is closed by presenting conclusions in Section 5.

#### 2. Envelope Equation of BBM Wave Packet

We consider propagating waves developed from regularized long wave equation, Benjamin-Bona-Mahony equation [47]; where is wave elevation, is spatial variable and is time variable. The linear form of BBM equation (1) which describes dispersive effect to the equation yields dispersion relation stating the relation between the wavenumber and frequency ; the dispersion relation reads . The BBM equation presented in (1) is in the nondimensional form with the scaling factors for the variables as well as the corresponding frequency and wavenumber are given as , , , , , where , , , , represent the laboratory/physical variables and and are water depth and gravitational acceleration, respectively.

The deformation of the wave packet can be investigated through its envelope. An ansatz depicting the solution of (1) as a wave packet is taken, where represents the envelope equation of the waves, and represent spatial and time variables, respectively, and denotes the complex conjugate. Envelope waves tend to move in slower velocity than the carrier waves. Therefore, and are called slow variables transformed from the fast variables and by the following relations, where is a small real number. Through these fast-to-slow variables transformation and applying the asymptotic expansion method, envelope equation is derived and it is found that it evolves satisfying the so-called Nonlinear Schrodinger (NLS) equation where Equation (4) is called spatial-NLS because it describes the change of wave envelope on every spatial position. The derivation of NLS equation from BBM equation can be thoroughly studied in [46].

NLS equation has many solutions. The second important solution is Soliton on Finite Background (SFB). This solution explains an interaction between a soliton and its background which have different frequencies along its propagation on the spatial domain [41]. SFB can also be represented as nonlinear interaction between a monochromatic signal perturbed by a pair of modulation waves having small frequency difference from the monochromatic’s one [40, 41]. The interaction triggers growing instability with a certain rate. The SFB reads [41, 49] where and , ; is modulation frequency. SFB reaches its maximum amplitude at (see [50]). The frequency causes an amplitude amplification if the value in the interval (see [49]). It is convenient to rewrite the interval as .

Let describes the slowly varying envelope and satisfies the NLS equation (4); then the physical wave profile given by the surface elevation (2) can be written in real valued function where and and are defined as stated in the transformation (3).

#### 3. Amplitude Amplification and Maximal Temporal Amplitude

The instability which happens to the SFB wave causes a growing amplitude during its propagation. Let be the maximum amplitude at an extreme position ; hence Since and , it is obtained thatIn (10), is a decreasing function with respect to and proportional to the initial amplitude at a far distance before the extreme position . Through this relation a certain value for the maximum amplitude can be achieved by setting an appropriate value to parameter .

Amplitude Amplification Factor (AAF) can be employed to measure the magnitude of the amplitude peaking. AAF is defined as the ratio of the maximum amplitude at the extreme position to the amplitude of the monochromatic wave at the far distance before it [33, 34, 40]. Therefore, we can state the AAF as Equation (11) explains that when , meaning that the amplitude does not experience amplification. However, when , the ; the amplitude of initially monochromatic wave will increase as much as three times at the extreme position. This value also represents the maximum amplification the wave can achieve for any given initial amplitude .

During the propagation, the wave elevation will change at every position. To investigate how the wave deforms along its propagation along the spatial domain, we apply a quantity called Maximal Temporal Amplitude (MTA). MTA measures the maximum elevation of the wave signal at every position [24, 40, 50]. MTA is defined as MTA is invariant to time. This function can be applied to determine the amplitude amplification globally by comparing the maximum MTA with the MTA at the initial position of the wave.

#### 4. Results and Discussion

##### 4.1. Modulational Instability and Phase Singularity

Wave peaking occurs due to focusing phenomenon when waves experience modulational instability which is also known as Benjamin-Feir Instability [49]. For the NLS equation case, focusing can happen depending on the value of the coefficients of the dispersive and the nonlinear terms. If the values of both and are of the same sign then both terms give balance effect resulting in the formation of solitary wave envelope [37]. Figure 1 depicts the graphs of and with respect to the wavenumber . For positive real valued , the dispersive coefficient is a negative function while the nonlinear coefficient is positive on and negative on . Therefore, we can choose a value for in the interval which triggers the focusing phenomenon.