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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.

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 [

[

Interactions or collision between solitons is perhaps the most captivating features of soliton phenomena. A critical observation was however made in [

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 Korteweg-de Vries (KdV) equation which has soliton solutions, and we describe the context in which this model arises, and solve it both analytically and numerically and discuss properties of the multi-soliton solutions. Korteweg-de Vries (KdV) equation has been extensively solved for single soliton solution using analytical, semi-analytical methods and, multi-solitons for numerical methods. Our motivation here is to apply dissimilar computational techniques to produce its multi-soliton solutions for scrutiny.

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

w t ± β w w x + α w x x x = 0 , α , β ∈ ℝ (2.0)

where w = w ( x , t ) 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. [

We solve the KdV Equation (2.0) in particular form on the infinite line, stage by stage using Hirota method.

w t ± 6 w w x + w x x x = 0 , − ∞ < x < ∞ , 0 < t < ∞ (2.1)

with initial condition w ( x , 0 ) = g ( x ) .

The solution of the KdV Equation (2.1) is a travelling wave which has permanent form, occurs due to a balance of its dispersive term ( w x x x ) and its nonlinear term ( 6 w w x ). This idea of balancing dispersion and nonlinearity is typical of any nonlinear evolution equation that admits soliton solution. The constant coefficients behind each of the terms are unimportant and we express the equation this way for historical reasons, particularly, the factor 6 is just a scaling factor to make the solutions (solitons or solitary waves) easier to be described via Hirota Method [

Bilinearization: First the KdV Equation (2.1) is rewritten as

w t + 3 ( w 2 ) x + w x x x = 0 (2.2)

The supplementary function (the Cole-Hopf transformation) given as

w ( x , t ) = 2 ( ln f ) x x = 2 ( f f x x − f x 2 f 2 ) (2.3)

is used to transform (2.1) into a bilinear KdV equation. Thus, taking time derivative of w ( x , t ) in (2.3), we integrate once with respect to x and substitute in (2.2), after some algebraic manipulations, we obtain the bilinear form of the KdV equation (2.1) as,

2 ( f f x t − f x f t + 3 f x x 2 − 4 f x f x x x + f f x x x x ) = 0 (2.4)

The Hirota bilinear form: To put the bilinear KdV Equation (2.4) into Hirota bilinear form in terms of the D-derivative operator as defined as follows, we assume the Hirota differential operator D x , which is a binary operator defined on ordered pairs of functions f ( x ) and g ( x ) , as follows

D x n f ⋅ g = ( ∂ ∂ x 1 − ∂ ∂ x 2 ) n f ( x 1 ) g ( x 2 ) | x 2 = x 1 = x (2.5)

then

D t m D x n ( f ⋅ g ) = ( ∂ ∂ t 1 − ∂ ∂ t 2 ) m ( ∂ ∂ x 1 − ∂ ∂ x 2 ) n f ( t 1 , x 1 ) g ( t 2 , x 2 ) | x = x 2 = x 1 , t = t 1 = t 2 (2.6)

where m , n are nonnegative integers and

( ∂ ∂ x 1 − ∂ ∂ x 2 ) n = ∑ r = 0 n ( − 1 ) r ( n r ) ∂ n − r ∂ x 1 ⋅ ∂ r ∂ x 2 (2.7)

where

( n r ) = n ! r ! ( n − r ) ! , 0 ≤ r ≤ n (2.8)

More generally we denote some sort of combination of the Hirota D-operator as a polynomial of D-operator P ( D ) i.e.

P ( D ) { f ⋅ g } = D x n { f ⋅ g } = ∑ r = 0 n ( − 1 ) r ( n r ) f ( n − r ) x ⋅ g r x = f n x g − n f ( n − 1 ) x g x + n ( n − 1 ) 2 f ( n − 2 ) x g 2 x − ⋯ + ( − 1 ) n f g n x (2.9)

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:

D x D t ( f ⋅ g ) = [ f x t g − f t g x ] − [ f x g t − f g x t ] (2.10)

Now replacing g with f to have the same function in (2.4) we have that:

D x D t ( f ⋅ f ) = 2 ( f f x t − f x f t ) (2.11)

Similarly, from (2.9) we have that

D x 4 ( f ⋅ f ) = 2 ( f f x x x x − 4 f x f x x x + 3 f x x 2 ) (2.12)

Hence Equation (2.4) becomes

( D x D t + D x 4 ) { f ⋅ f } = P ( D ) { f ⋅ f } = 0 (2.13)

which is the Hirota bilinear form of KdV Equation (2.1).

Application of the Hirota Perturbation: The supplementary function f = f ( x , t ) , in (2.13) is expressed as:

f ( x , t ) = 1 + ϵ 1 f 1 + ϵ 2 f 2 + ϵ 3 f 3 + ϵ 4 f 4 + ⋯ + ϵ n f n = 1 + ϵ n ∑ i = 1 n f i (2.14)

where f 1 , f 2 , ⋯ , f n , represent simple exponential functions.

We now insert (2.14) into Equation (2.13) so that

P ( D ) { f ⋅ f } = P ( D ) ( 1 ⋅ 1 ) + ϵ P ( D ) ( f 1 ⋅ 1 + 1 ⋅ f 1 ) + ϵ 2 P ( D ) ( f 2 ⋅ 1 + f 1 ⋅ f 1 + 1 ⋅ f 2 ) + ϵ 3 P ( D ) ( f 3 ⋅ 1 + f 2 ⋅ f 1 + f 1 ⋅ f 2 + 1 ⋅ f 3 ) + ⋯ + ϵ n P ( D ) ( ∑ i = 0 n f i ⋅ f n − i ) = 0 (2.15)

The coefficient of like powers of ϵ in (2.15) can be equated to zero to obtain the following sets of equations,

Ο ( ϵ n ) : P ( D ) ( ∑ i = 0 n f i ⋅ f n − i ) = 0 , ( f 0 = 1 ) (2.16)

We make use of the scheme (2.16) to obtain appropriate dispersion relations and coupling coefficients in the KdV Equation (2.13). Thus, if f ( x , t ) in (2.14) is a solution of Equation (2.13), then w ( x , t ) in Equation (2.3) is a soliton solution to the KdV Equation (2.1).

What we need to find now is a truncated supplementary function f ( x , t ) in (2.14) that satisfies (2.13) which, when inserted into Equation (2.3), will yield w ( x , t ) , which is the solution of KdV Equation (2.1). From (2.14), if we try f ( x , t ) = 1 and substitute this into Equation (2.3), we get the trivial solution w ( x , t ) = 0 . This solution is called the vacuum or zero soliton upon which multi-solitons solutions are obtained or propagated. This shows that a soliton can travel in a vacuum.

The single soliton solution [

w ( x , t ) = 1 2 k 2 sech 2 [ 1 2 k ( x − c t ) + 1 2 q ] (2.17)

where k is the wave number, c is wave velocity, and q is the initial point of propagation, c = k 2 is the dispersion relation. A wolfram Mathematica program is used to compute the single-soliton solution and its simulation. The profile of the single soliton at q = 0 , t = 0 , and k = 2 is shown in

To obtain the two-soliton solution, the supplementary function f in (2.14) is truncated after the third term

f ( x , t ) = 1 + ϵ f 1 + ϵ 2 f 2 , f i = 0 , i ≥ 3 (2.18)

To find f 1 and f 2 , use is made of a two-term form of f 1 = e v that is usually used for construction of the single-soliton case, i.e. f 1 = e v 1 + e v 2 where v 1 = k 1 ( x − c 1 t ) + q 1 , v 2 = k 2 ( x − c 2 t ) + q 2 . 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:

P ( D ) ( 1 ⋅ f 1 ) + P ( D ) ( f 1 ⋅ 1 ) = P ( ∂ ) f 1 + P ( ∂ ) f 1 = 2 P ( ∂ ) f 1 = 0 (2.19)

Thus, it follows that,

2 P ( ∂ ) f 1 = 2 ( ∂ x ∂ t + ∂ x 4 ) { e v 1 + e v 2 } = 2 ( ∂ x ∂ t + ∂ x 4 ) { e v 1 } + 2 ( ∂ x ∂ t + ∂ x 4 ) { e v 2 } = 0 (2.20)

This implies

( 2 k 1 4 − 2 c 1 k 1 2 ) e v 1 = 0 and ( 2 k 2 4 − 2 c 2 k 2 2 ) e v 2 = 0 (2.21)

i.e. the dispersion relation is obtained as:

c i = k i 2 , i = 1 , 2 (2.22)

Now, we obtain f 2 from the perturbation scheme (2.16), as

P ( D ) ( 1 ⋅ f 2 + f 2 ⋅ 1 ) + P ( D ) ( f 1 ⋅ f 1 ) = 2 P ( ∂ ) f 2 + P ( D ) ( f 1 ⋅ f 1 ) = 0 (2.23)

Thus,

2 P ( ∂ ) f 2 + P ( D ) { ( e v 1 + e v 2 ) ⋅ ( e v 1 + e v 2 ) } = 0 (2.24)

or,

2 P ( ∂ ) f 2 − 6 k 1 k 2 ( k 1 − k 2 ) 2 e v 1 + v 2 = 0 (2.25)

Clearly, Equation (2.25) holds true, if and only if f 2 is of the form

f 2 = c 12 e v 1 + v 2 (2.26)

where c 12 is a coupling constant yet to be determined.

Now substituting (2.26) back into (2.25) we have that

2 c 12 { − ( k 1 + k 2 ) ( k 1 c 1 + k 2 c 2 ) + ( k 1 + k 2 ) 4 } e v 1 + v 2 − 6 k 1 k 2 ( k 1 − k 2 ) 2 e v 1 + v 2 = 0 (2.27)

Substituting (2.22) in (2.27) and simplifying we have the coupling constant c 12 as:

c 12 = ( k 1 − k 2 ) 2 ( k 1 + k 2 ) 2 (2.28)

Thus, f 1 and f 2 are determined as simple exponential functions in supplementary function in (2.18), without loss of generality, we set ϵ = 1 , and use (2.18) into (2.3) in constructing the two-soliton solution as

w ( x , t ) = 2 ( k 1 2 e v 1 + k 2 2 e v 2 + c 12 ( k 1 + k 2 ) 2 e v 1 + v 2 ) ( 1 + e v 1 + e v 2 + c 12 e v 1 + v 2 ) − 2 ( k 1 e v 1 + k 2 e v 2 + c 12 ( k 1 + k 2 ) e v 1 + v 2 ) 2 ( 1 + e v 1 + e v 2 + c 12 e v 1 + v 2 ) 2 (2.29)

where

v i = k i ( x − c i t ) + q i , i = 1 , 2 and c 12 = ( k 1 − k 2 ) 2 ( k 1 + k 2 ) 2 (2.30)

We now have a fully defined two-soliton solution for KdV Equation (2.2), A Wolfram Mathematica program was again used for computation and simulation of the two-soliton solution obtained in Equation (2.29). A study of this solution is given in Figures 3-8 where k 1 = 1 , k 2 = 1.5 in Equation (2.29). It is clearly seen that the Solitons move left to right with the taller faster overtaking the shorter-slower soliton.

This process is similar to the two-soliton problem except that here we need to find f 1 , f 2 and f 3 to form the supplementary function truncated from (2.14)

f ( x , t ) = 1 + ϵ f 1 + ϵ 2 f 2 + ϵ 3 f 3 , f i = 0 , i ≥ 4 (2.31)

And since we are seeking for a three-soliton solution, we use three-term forms of f 1 and f 2 that were successful for the two-soliton case, i.e.

f 1 = e v 1 + e v 2 + e v 3 (2.32)

and

f 2 = c 12 e v 1 + v 2 + c 13 e v 1 + v 3 + c 23 e v 2 + v 3 (2.33)

where c 12 , c 13 and c 23 are coupling constants yet to be determined in terms of k 1 , k 2 and k 3 .

And we shall deal with finding f 3 in the computation process, similar to the f 2 in the two-soliton case.

Again, using the perturbation scheme (2.16), we have:

P ( D ) ( 1 ⋅ f 1 + f 1 ⋅ 1 ) = P ( D ) ( 1 ⋅ f 1 ) + P ( D ) ( f 1 ⋅ 1 ) = 2 P ( ∂ ) f 1 = 0 (2.34)

Thus,

( k 1 4 − k 1 2 c 1 ) e v 1 + ( k 2 4 − k 2 2 c 2 ) e v 2 + ( k 3 4 − k 3 2 c 3 ) e v 3 = 0 (2.35)

and from (2.35) we obtain the dispersive relation:

c i = k i 2 , i = 1 , 2 , 3 (2.36)

Also

P ( D ) ( 1 ⋅ f 2 + f 2 ⋅ 1 ) + P ( D ) ( f 1 ⋅ f 1 ) = 2 P ( ∂ ) f 2 + P ( D ) ( f 1 ⋅ f 1 ) = 0 (2.37)

which implies,

2 P ( ∂ ) f 2 = ( 12 c 12 k 1 2 k 2 2 + 6 c 12 k 1 3 k 2 + 6 c 12 k 1 k 2 3 ) e v 1 + v 2 + ( 12 c 13 k 1 2 k 3 2 + 6 c 13 k 1 k 3 3 + 6 c 13 k 1 3 k 3 ) e v 1 + v 3 + ( 12 c 23 k 2 2 k 3 2 + 6 c 23 k 2 k 3 3 + 6 c 23 k 2 3 k 3 ) e v 2 + v 3 (2.38)

and,

P ( D ) ( f 1 ⋅ f 1 ) = ( 12 k 2 2 k 1 2 − 6 k 2 k 1 3 − 6 k 2 3 k 1 ) e v 1 + v 2 + ( 12 k 1 2 k 3 2 − 6 k 3 3 k 1 − 6 k 3 k 1 3 ) e v 1 + v 3 + ( 12 k 2 2 k 3 2 − 6 k 2 k 3 3 − 6 k 2 3 k 3 ) e v 2 + v 3 (2.39)

Adding Equations (2.38) and (2.39), and simplifying we have

c 12 = ( k 1 − k 2 ) 2 ( k 1 + k 2 ) 2 , c 13 = ( k 1 − k 3 ) 2 ( k 1 + k 3 ) 2 , c 23 = ( k 2 − k 3 ) 2 ( k 2 + k 3 ) 2 (2.40)

We see here that all the coupling constants for f 2 are determined, we now have to find f 3 .

Again

P ( D ) ( 1 ⋅ f 3 + f 3 ⋅ 1 ) + P ( D ) ( f 1 ⋅ f 2 + f 2 ⋅ f 1 ) = 0 (2.41)

Thus,

P ( D ) ( 1 ⋅ f 3 + f 3 ⋅ 1 ) − 6 ( k 1 − k 2 ) 2 ( k 1 − k 3 ) 2 ( k 2 − k 3 ) 2 ( k 1 + k 2 + k 3 ) ( k 1 + k 2 ) ( k 1 + k 3 ) ( k 2 + k 3 ) e v 1 + v 2 + v 3 = 0 (2.42)

Clearly, Equation (2.42) holds if and only if, f 3 is of the form:

f 3 = B ⋅ e v 1 + v 2 + v 3 (2.43)

Substituting (2.43) in the first term of Equation (2.42) we have

P ( D ) ( 1 ⋅ f 3 + f 3 ⋅ 1 ) = 6 B ( k 1 + k 2 ) ( k 1 + k 3 ) ( k 2 + k 3 ) ( k 1 + k 2 + k 3 ) e v 1 + v 2 + v 3 (2.44)

Substituting (2.46) back into (2.44) and simplifying, the constant B is determined as:

B = ( k 1 − k 2 ) 2 ( k 1 − k 3 ) 2 ( k 2 − k 3 ) 2 ( k 1 + k 2 ) 2 ( k 1 + k 3 ) 2 ( k 2 + k 3 ) 2 = c 12 ⋅ c 13 ⋅ c 23 (2.45)

Thus, we have successfully obtained f 1 , f 2 and f 3 so that the supplementary function (2.31) by setting ϵ = 1 becomes:

f ( x , t ) = 1 + e v 1 + e v 2 + e v 3 + c 12 e v 1 + v 2 + c 13 e v 1 + v 3 + c 23 e v 2 + v 3 + B e v 1 + v 2 + v 2 (2.46)

Therefore substituting (2.46) in (2.3) we obtain the three-soliton solution as

w = 2 ( f f x x − f x 2 ) f 2 = 2 N ( x , t ) D ( x , t ) (2.47)

where,

N ( x , t ) = e v 1 + v 2 [ 2 ( k 1 − k 2 ) 2 + 2 ( k 1 − k 2 ) 2 c 13 c 23 e 2 v 1 + c 12 k 1 2 e v 2 + c 12 k 2 2 e v 1 ] + e v 1 + v 3 [ 2 ( k 1 − k 3 ) 2 + 2 ( k 1 − k 3 ) 2 c 12 c 23 e 2 v 2 + c 13 k 1 2 e v 3 + c 13 k 3 2 e v 1 ] + e v 2 + v 3 [ 2 ( k 2 − k 3 ) 2 + 2 ( k 2 − k 3 ) 2 c 12 c 23 e 2 v 1 + c 23 k 2 2 e v 3 + c 23 k 3 2 e v 2 ] + k 1 2 e v 1 + k 2 2 e v 2 + k 3 2 e v 3 + B e v 1 + v 2 + v 3 [ c 12 k 3 2 e v 1 + v 2 + c 13 k 2 2 e v 1 + v 3 + c 23 k 1 2 e v 2 + v 3 ] + e v 1 + v 2 + v 3 [ c 12 ( k 1 2 + k 2 2 + k 3 2 + 2 k 1 k 2 − 2 k 1 k 3 − 2 k 2 k 3 ) + c 13 ( k 1 2 + k 2 2 + k 3 2 + 2 k 1 k 3 − 2 k 1 k 2 − 2 k 2 k 3 ) + c 23 ( k 1 2 + k 2 2 + k 3 2 + 2 k 2 k 3 − 2 k 1 k 2 − 2 k 1 k 3 ) + B ( k 1 2 + k 2 2 + k 3 2 + 2 k 2 k 3 − 2 k 1 k 2 − 2 k 1 k 3 ) ] (2.48)

and

D ( x , t ) = ( 1 + e v 1 + e v 2 + e v 3 + c 12 e v 1 + v 2 + c 13 e v 1 + v 3 + c 23 e v 2 + v 3 + B e v 1 + v 2 + v 2 ) 2 (2.49)

and v i = k 1 ( x − c i t ) + q 1 , = 1 , 2 , 3 ; c i j = ( k i − k j ) 2 ( k i + k j ) 2 , i , j = 1 , 2 , 3 , i < j and B = c 12 ⋅ c 13 ⋅ c 23

Now we have a three-soliton solution for KdV Equation (2.1), a Wolfram Mathematica program was again used for computation and simulation of the three-solitons solution obtained in Equation (2.49). A study of this solution is given in Figures 9-13 respectively, with k 1 = 1 , k 2 = 2 , k 3 = 3 in equation (2.49). The solitons move left to right with the taller faster overtaking the shorter slower soliton.

Based on the Crank Nicolson scheme [

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 [

w t = w i n + 1 − w i n 2 τ + w i + 1 n + 1 − w i + 1 n 2 τ (2.50)

w x x x = w i + 2 n − 3 w i + 1 n + 3 w i n − w i − 1 n 2 h 3 + w i + 2 n + 1 − 3 w i + 1 n + 1 + 3 w i n + 1 − w i − 1 n + 1 2 h 3 (2.51)

To carefully deal with the nonlinear term β w w x , we assume that the leading coefficient β w 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 β w ^ and decide how to suitably estimate its value. Reviewing again that we have to

calculate at time level ( n + 1 2 ) and space area ( i + 1 2 ) , the nonlinear term of the Equation (2.0) becomes:

β w w x = β w ^ i + 1 + β w ^ i 4 ( w i + 1 n + 1 − w i n + 1 h + w i + 1 n − w i n h ) (2.52)

We note here that the derivatives in (2.52) are centered in time at ( n + 1 2 )

while the w ^ term is not. This will be estimated using predictor-corrector technique. Thus the KdV Equation (2.0) becomes a system of algebraic equations:

w i n + 1 − w i n 2 τ + w i + 1 n + 1 − w i + 1 n 2 τ + β w ^ i + 1 + β w ^ i 4 h ( w i + 1 n + 1 − w i n + 1 + w i + 1 n − w i n ) + α ( w i + 2 n + 1 − 3 w i + 1 n + 1 + 3 w i n + 1 − w i − 1 n + 1 2 h 3 + w i + 2 n − 3 w i + 1 n + 3 w i n − w i − 1 n 2 h 3 ) = 0 (2.53)

Estimation in the Nonlinear Term using Predictor-Corrector TechniqueWe use a predictor-corrector technique [

procedure: The Crank Nicolson is applied two times in each time step. In the initial step (the predictor step) we essentially substitute w ^ for w n , the present estimated value of w, and call the subsequent new value (after Crank Nicolson is applied) w ^ n + 1 , the predicted future value. In the second step we join this

predicted value with current value to approximately build w ^ n + 1 2 using w n + 1 2 = 1 2 ( w n + w n + 1 ) , then reconstruct iteratively Crank Nicolson again.

Thus, the Algorithm for the Crank-Nicolson scheme (2.53) was developed and implemented in MATLAB [

In

In the numerical computation, The Crank-Nicolson implicit scheme was used to compute multi-soliton solutions in KdV equation. In Figures 14-18 we implemented the Crank-Nicolson algorithm in MATLAB, and studied the interactions of three solitons with soliton initial profile 6 k n 2 α sech 2 k ( x − x 0 n ) , n = 1 , 2 , 3 . Computed at τ = 0.01 , α = 0.134 , β = 1 , k 1 = 1.5 , k 2 = 2 , k 3 = 2.5 and

initial points x 01 = 3 L 4 , x 02 = L 4 , x 03 = L 2 of propagation at an interval, L, at

four different times. We note here however that τ = 0.01 is chosen sufficiently small because of stability issues (to avoid blowup solutions which we have seen for larger time step τ ) in the approximation of the quantity w ^ in the nonlinear term, estimated by predictor corrector technique.

Solitons in the numerical simulations move either left to right or right to left depending on the sign (positive or negative) in nonlinear term in KdV equation

with the tallest one moving faster. It is also observed that as two solitons collide with each other they momentarily form a single soliton pulse as represented in

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.

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 simple computer codes to compute the three solutions. To study interactions or collision of solitons we preformed simulations using computer symbolic programming language—Wolfram Mathematica and MATLAB.

It was observed how two or more of solitons interact or collide with one another. It was fascinating to note in the study of analytical solution and numerical results obtained by Crank-Nicolson scheme that solitons pass through each other and come out unchanged. Thus we affirm that solitons indeed collide with one another and come without change of properties or identities. The analytical and numerical comparison of solutions agreement is very good.

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Orapine, H.O., Ayankop-Andi, E. and Ibeh, G.J. (2020) Analytical and Numerical Computations of Multi-Solitons in the Korteweg-de Vries (KdV) Equation. Applied Mathematics, 11, 511-531. https://doi.org/10.4236/am.2020.117037