Analytical and Numerical Computations of Multi-Solitons in the Korteweg-de Vries (KdV) Equation

In this paper, an analytical and numerical computation of multi-solitons in Korteweg-de Vries (KdV) equation is presented. The KdV equation, which is classic of all model equations of nonlinear waves in the soliton phenomena, is described. In the analytical computation, the multi-solitons in KdV equation are computed symbolically using computer symbolic manipulator—Wolfram Mathematica via Hirota method because of the lengthy algebraic computation in the method. For the numerical computation, Crank-Nicolson implicit scheme is used to obtain numerical algorithm for the KdV equation. The simulations of solitons in MATLAB as well as results concerning collision or interactions between solitons are presented. Comparing the analytical and numerical solutions, it is observed that the results are identically equal with little ripples in solitons after a collision in the numerical simulations; however there is no significant effect to cause a change in their properties. This supports the existence of solitons solutions and the theoretical assertion that solitons indeed collide with one another and come out without change of properties or identities.


Introduction
The study of nonlinear evolution models which describes a large variety of physical phenomena is found to have two fascinating manifestations of opposite nature: chaos that are the apparent randomness in the behavior of perfectly deterministic systems and special kind of solitary waves called solitons. Soliton theory gives us various significant instances of nonlinear systems behaving in a persistent, quasi-linear pattern. Solitons are therefore, a consequence of a dynamic balance between dispersion and nonlinear effects in any nonlinear evolution models. They are waves of permanent form that preserve their shape while traveling over long distances. The permanent speed and form of a soliton is however not the only special property; but it is said of its special characteristic that, solitons maintain their shape and speed after collisions with other solitons, summing up in the following two basic properties; One, propagating without change of its characteristics (shape, size, velocity etc.), Two, localized waves (stable against mutual collisions and retaining their identities:). The first is a solitary wave condition acknowledged in hydrodynamics since the nineteenth century.
The second implies that the wave has the property of a particle [1] [2] [3]. [4] and [5] revealed that solitons in modeling physical phenomena arise in a wide range of areas such as shallow and deep water waves, optics communications, Bose-Einstein condensates and Biological models. They are universal in nature, and can be found in different classifications: from water waves, sound waves and charge-density waves to matter waves and electromagnetic waves [6].
Solitons can be found in a variety of materials including Plasmas, Josephson junction, Polyacetylene (Proteins and DNA) molecules among others and are used for many different purposes. They may carry electrical charges in some substances and when charged, solitons travel through certain polymer chains, which tend to curve. [7] predicted that this property may one day be used in applications such as artificial muscles and also showed that solitons can be used to transfer large amounts of information over large distances with little or no errors in the signal.
Interactions or collision between solitons is perhaps the most captivating features of soliton phenomena. A critical observation was however made in [8] that solitons preservation of identities after collisions is basically suggested by numerical simulations. But in detailed analysis of the results of such numerical simulation, some ripples can be observed after a collision and it therefore seems that the original identity is not completely recovered; it is most imperative to find exact solutions of nonlinear evolution models that admit soliton solutions for proper scrutiny of collisions. [9] in the same vein, studied in contrast, "overtaking interactions" and "head-on collisions" between two electrostatic solitons in Korteweg-de Vries (KdV) equation using an approximate method, and revealed that though numerical method offers valuable insights it still limits the weight of validity of the study of soliton collisions.
In this paper we focus on the existence of more than one soliton solution called multi-solitons, since this enables us to study solitons collision or interactions especially for their said preserved behavior. We look at the famous Korte-

Method: Computation of Multi-Solitons in Korteweg-de Vries Equation
One of the most famous time evolution models and perhaps the simplest nonlinear system is the third order Korteweg-de Vries (KdV) equation, hereafter abbreviated as KdV equation, given in a general form as = is a function of two variables which represent the amplitude of the wave at position x, and time t, and α, β are arbitrary. The Equation (2.0) is nonlinear because of the product shown in the second term and of third order for the reason that the third derivative is highest. The KdV Equation (2.0) arises in many physical situations. [10] noted that it very well may be used to portray the investigation of shallow-water waves, gas dynamics, Anharmonic nonlinear grids, Hydro-attractive and Icon-acoustic waves in cool plasma, for instance.
The physical derivation of this model is given in [11].

Solution of the KdV Equation by Hirota Method
We solve the KdV Equation (2.0) in particular form on the infinite line, stage by stage using Hirota method.
where , m n are nonnegative integers and ( ) More generally we denote some sort of combination of the Hirota D-operator as a polynomial of D-operator ( ) where the subscripts of the functions f and g define the order of the partial derivatives with respect to x. Thus, D operates on a product of two functions like the Leibniz rule, except for the crucial sign difference.
Thus, from Equation (2.6) we have: Now replacing g with f to have the same function in (2.4) we have that: Similarly, from (2.9) we have that ( ) ( ) which is the Hirota bilinear form of KdV Equation (2.1).

Application of the Hirota Perturbation: The supplementary function
( ) (2.13) is expressed as: where 1 2 , , , n f f f  , represent simple exponential functions. We now insert (2.14) into Equation (2.13) so that The coefficient of like powers of  in (2.15) can be equated to zero to obtain the following sets of equations, We make use of the scheme (2.16) to obtain appropriate dispersion relations and coupling coefficients in the KdV Equation is a solution of Equation (2.13), then ( )

The Vacuum Soliton Solution
What we need to find now is a truncated supplementary function

The Single Soliton Solution
The single soliton solution [2] [12] is given as, where k is the wave number, c is wave velocity, and q is the initial point of propagation,

The Two-Soliton Solution
To obtain the two-soliton solution, the supplementary function f in (2.14) is truncated after the third term ( ) To find 1 f and 2 f , use is made of a two-term form of 1 e v f = that is usual- Since two-soliton solution is built from single soliton, and one principle is that for integrable systems one must be able to combine any pair of single-soliton built on top of the same vacuum. Now, using the perturbation scheme (2.16), we have: Thus, it follows that, i.e. the dispersion relation is obtained as: Now, we obtain 2 f from the perturbation scheme (2.16), as

The Three-Soliton Solution
This process is similar to the two-soliton problem except that here we need to find 1 2 , f f and 3 f to form the supplementary function truncated from (2.14) And since we are seeking for a three-soliton solution, we use three-term forms of 1 f and 2 f that were successful for the two-soliton case, i.e. where 12 13 , c c and 23 c are coupling constants yet to be determined in terms of 1 2 , k k and 3 k .
And we shall deal with finding 3 f in the computation process, similar to the 2 f in the two-soliton case.

Numerical Solution of the KdV Equation Using Crank-Nicolson Method
Based on the Crank Nicolson scheme [13], we proposed to evaluate each term in the Equation (2.0) not only at time level 1 2 n   +     , but also at spatial location at the midpoint of every subinterval) so as to ensure the first order time derivative and third order space derivative (dispersive term) terms are both suitably centered. This ensures that we will be employing a cell-edge grid, but that the spatial finite differences in the scheme will be cell centered [12].
To carefully deal with the nonlinear term x ww β , we assume that the leading coefficient w β is known so as to escape the nonlinearity in the system of algebraic equations that will be obtained in the scheme. We designate it by ŵ β and decide how to suitably estimate its value. Reviewing again that we have to calculate at time level

Estimation in the Nonlinear Term using Predictor-Corrector Technique
We use a predictor-corrector technique [14] to properly center Crank-Nicolson in time between n t and

Discussion
In Figure 1, Figures 5-8       It is also noticed that during interactions of solitons in all the simulations carried out, the amplitude of the supposed single soliton pulse formed by two or more solitons is observed to be smaller than the amplitude of the larger soliton in the interactions. For instance, in Figure 14, the larger soliton amplitude is 10 units while the amplitude of single-soliton pulse formed is 7.5 units in Figure 16, but regains their heights or amplitudes after interactions. This similarly can also be seen in Figures 10-13 and Figures 5-8.
Again there is a phase shift after interactions, since the smaller soliton that is in front becomes behind and the larger or taller one becomes further head as opposed to linear waves. This confirms that solitons do not obey superposition principle but interact nonlinearly with each other. That notwithstanding we again checked to prove if superposition of two linear waves will form a soliton but this was not possible. Testing initial profile to be superposition of two linear waves, only a blowup solution appeared, justifying the nonlinearity in KdV equation and all models that admit this kind of solution are nonlinear in nature, balanced with dispersive term. The simulations results were observed with little ripples after collision particularly in the numerical computation after critical inspections as can be seen in figures16 and 18 but it has no significant effects to cause change in them.

Conclusions
In this paper, we have performed several computations both analytical and numerical. We obtained exact solutions of KdV equation via Hirota Method for one, two and three solitons. In order to ease computation process we wrote some We also uphold that in physical application, the study of Multi-Soliton solution in KdV equation gives us a clue why data receivers' sets such as Radio sets get access to (signal transmitted easily during tuning) data and information transmission centers say Radio Stations with very high mast antennas faster than those with lower mast antennas since solitons of higher amplitude travel faster than solitons of shorter amplitude. A bit of noise of such data transmission can be perceived when different signal pulses transmitted collide but after, fade away as the tall signal pulse moves faster separating itself from the short signal pulse both of which however do not dissipate energy and continue with their identity and this is typical of all data transmission processes.