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Nonlinear evolution of bichromatic waves based on the fifth order solution of benjamin bona mahony equation

This research concerns on the influence of higher order solution to nonlinear evolution of bichromatic waves based on Benjamin Bona Mahony (BBM) equation. Asymptotic method was applied to determine the solution expanded to the fifth order. In previous researches, this method was applied on the same equation to find its solution up to the third order and it was obtained that the waves underwent deformation and amplitude amplification affected by the occurrence of a pair of side bands during their propagation. In this study, the effect of the higher order solution on the increasing amplitude was observed. A quantity called as Maximal Temporal Amplitude (MTA) was measured to find the maximum position where the highest amplitude occurred and the Amplitude Amplification Factor (AAF). MTA gives information about the waves’ highest elevation at every spatial position during the observation time. The analysis includes comparison between the third and fifth order solutions in their contribution to the rise of the amplitude and the maximum position.


Introduction
Waves generally occur on the surface of water bodies open to the atmosphere. The waves are a manifestation of forces acting on fluids which tend to violate the gravitation and surface tension trying to preserve the surface state [1]. Many unexpected phenomena that occur in the ocean and become a threat to offshore activities, one of which is the phenomenon of extreme waves. Waves are considered extreme if the wave height is more than 2.2 times the height of the significant waveform [2]. There are several types of causes for the formation of waves such as wind, earth plate activity, tides and currents. But waves often occur due to wind, especially if strong winds occur. These conditions can cause high waves that hamper marine transportation activities and offshore facilities. According to Dean and Dalrymple [1], the highest wave ever observed was 34 m and had attacked countless ships. This shows that extreme waves are dangerous for sea transportation and offshore facilities.
Surface wave behavior can be described in partial differential equations, namely the Korteweg de Vries (KdV), Boussinesq, Kadomtsev-Petviashvili (KP), Benjamin Bona Mahony (BBM) equations, and other equations which are all in nonlinear form. These equations are a simplification of Laplace full water equation. Simplification is done by investigating the direction of the wave propagation under review.
The KdV equation is a wave equation that describes one-way wave propagation on shallow water surfaces with small wave amplitudes and large wavelengths. The KdV equation was discovered in 1895, this equation can be applied to explain some physics problems, including studies of hydromagnetic waves, internal waves, acoustic-ion waves, and plasma physics [4]. However, the KdV equation has a weakness, the KdV equation is only able to model waves with small wave numbers so that the resulting waves have large wavelengths, whereas it is expected that in wave generation experiments in hydrodynamics laboratories, waves must have short wavelengths. Thus to obtain waves with dispersive properties in accordance with these equations, a very long water tank and a very long time are needed [5].
To correct this weakness, the BBM Equation was developed by Benjamin, Bona and Mahony in 1972. The BBM Equation was an improvement on the weakness of the KdV equation in modeling waves that have large wave numbers. The BBM Equation form has a dispersion relation and short wave properties that can explain its physical phenomena better than the KdV [6] equation.
Most of wave equations are presented in nonlinear form which are often difficult solve. One method of solving nonlinear equation is asymptotic method [8]. According to Mashuri and Marwan [9], the asymptotic method can be applied to determine the approximate solution of the KdV equation where the amplitude can be expanded using a power series of the third order. The asymptotic expansion method was also applied by Ramli [10] to determine the solution of Boussinesq equation in studying extreme waves. This method was also applied by Mashuri et. al. [11] to determine the solution of Kadomsevt Petviashvili equation which describes the two dimensions of the KdV equation.
Marwan [12] conducted a research on the KdV Equation using asymptotic power series up to the third order side band to observe the maximum height of the amplitude that can be produced, it turned out that the amplitudes of the side bands in the third order term exceeded the ones of second order term. However the height of the amplitude was lower than the results of experiments using the HUBRIS software which was developed from full water equation. This is allegedly due to inadequate expansion of the power series. Therefore Ramli et. al [13] conducted a study of the KdV Equation to find a solution up to fifth order side bands using asymptotic expansion and to compare the amplitude heights of third-order solutions with fifth-order solutions. Afriadi et. al [14] continued the research to find a fifth-order solution and determine the maximum position. However, the amplitude obtained from the fifth order solution was not able to match the experimental results using the HUBRIS software. Because the KdV equation has not been able to match the HUBRIS experimental results in reaching extreme positions, Halfiani and Ramli [15] conducted research on the BBM equation up to third-order side band terms to find solutions and obtained that waves experiences deformation and amplification amplitude caused by the appearance of a pair of side bands in propagation.
In this study, the effect of a higher order on increasing wave amplitude using the BBM equation is observed. The solution of the BBM equation is calculated using the asymptotic expansion method of the fifth order. Finally, the comparison of the results and the third-order BBM solution is presented.

Benjamin Bona Mahony (BBM) Equation
The BBM equation is a wave equation that models a one directional long wave propagation that has a small amplitude on the surface. The equation reads [6] with the dispersion relation as follows [7].
where ω is the frequency and k is the wave number. η t + η x describes wave propagation on shallow water surfaces that have small amplitudes and large wavelengths. Propagation occurs in one direction and moves in the positive x direction. The term ηη x describes the nonlinear effect affecting wave propagation and η xxt describes the dispersive nature of the wave. From the dispersion relation phase velocity is obtained: and a group velocity: which has a boundary value to the value of k and is close to zero for a large value of k. This means that for large scale k, BBM waves tend not to propagate. In other words, the model has desirable properties related to short wavelengths and large wave numbers.

The Fifth Order Solution of BBM Equation and Extreme Positions of BBM Bichromatic signals
The asymptotic method emerged in the early 1800s because of the desire to develop formulas for evaluating specific functions. The asymptotic expansion method is a method for finding an approximate solution to a problem of differential equations, where the function is expressed in power series of a small parameter value [8]. The smaller the value of the parameter chosen, the more accurate the approach obtained.
To solve the BBM equation using the asymptotic expansion method, η(x, t) will be expanded to the fifth order and written in the following form The wave number (k) is also expanded and written in the following form where ε is a small valued parameter. Equations 6 and 7 will be substituted into the BBM Equation 5 and separated by the order of ε. By collecting the polynomial coefficients for each order of ε, it is obtained first order: where a is the amplitude, θ p = k p x − ω p t, k is the wave number, ω is the frequency and c.c is the complex conjugate. Equation 9 shows the form of bichromatic signals. By substituting the ansatz into the linear form of the BBM equation 9, we get the following dispersion relations.