Generation and Interaction of Solitons
G.P. Chuiko amd S.I.Shyan
Petro Mohyla BlackSea State University, Mykolayiv,Ukraine
gp47@mail.ru

Modelling of the origin and interaction of solitons



Classic computer experiments


Write down the nonlinear differential equation of Kortevegde Vries (KdV) in such a form:
>

KdV:= diff(u(x,t),t) + u(x,t)*diff(u(x,t),x) + delta^2*diff(u(x,t),x$3) = 0;

 (2.1.1) 
here  is the wave function within a nonlinear environment with the dispersion. Authors [1 and 2] were accepting , hence
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KdV:=eval(KdV,delta=0.022);

 (2.1.2) 
Let us rewrite now the periodic boundary conditions of [1,2]as a set:
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bc:={u(2,t)=u(0,t), D[1](u)(2,t)=D[1](u)(0,t), D[1$2](u)(2,t)=D[1$2](u)(0,t)};

 (2.1.3) 
Thus, the wave lenght is equal to 2, hence the wave function as well as its derivatives are periodic along the coordinate axis. The initial conditions were chosen by the authors [1,2] in the form of harmonic perturbation with unit amplitude:
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ic:={u(x,0)=cos(Pi*x)};

 (2.1.4) 
We can to obtain the numerical solution of above mentioned Cauchy problem as a module, following [2]:
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sol:=pdsolve(KdV,bc union ic,numeric, spacestep=0.01, timestep=0.01);

 (2.1.5) 
Now we can to present the spacetime evolution of solution in two different ways
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sol:plot3d(t=0..1.2,x=0..2,axes=framed,orientation=[60,45,0],labelfont=[HELVETICA,14],color=gold,lightmodel=light4,title=`Fig.1. Evolution of the initial perturbation`,font=[HELVETICA,14]);

>

sol:animate(u(x,t), t=0..Pi/3, frames=120,color=blue,thickness=3,gridlines=true,title=`Рис. 2 Increasing of the wave front steepness \n and the emergence of solitons`,font=[HELVETICA,14],axes=boxed);

The animation of Fig. 2 illustrates such effects:
i

The increasing of the steepness of the wave front of the initial perturbation: its top is faster of own bottom:

ii

The appearence of solitons  the decay of the primary cosine wave on the system of individual solìtons of varying amplitude and speed;

iii

Stabilizing the front lines after the appearance of solitons – "tipping" of the wave stops



Other form of initial perturbations


It easy to chek that the other forms of perturbations leads to same results as these mentioned above. Let us exchange the cosine perturbation with the Gauss impulse having the mean value and the dispersion equal to
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ic_new:={u(x,0)=exp(9*(x1/2)^2)};

 (2.2.1) 
The new solution is:
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sol_new:=pdsolve(KdV,bc union ic_new,numeric, spacestep=0.01, timestep=0.01);

 (2.2.2) 
sol_new:animate(u(x,t), t=0..2, frames=100,color=blue,thickness=3,gridlines=true,title=`Fig.3 Increasing of the wave front steepness \n and the emergence of solitons2`,font=[HELVETICA,14],axes=boxed);
This animation shows all the same effects as on the Fig.2. Therefore, the conclusions, mentioned above, are independent of the form of initial perturbations.


Multisoliton solutions and interaction of solitons within doublets


A more direct method of studying the interaction between solitons is the modeling of the solitons doublets, which is one of the cases of the multisoliton solutions of the Kortevegde Vries equation [3]. All multisoliton solution, the simplest of which is a soliton doublet, that is linked to a pair solitons, are wellknown and exact solutions of the equation Korpteveg  de Vrìes. However these multisolitons are not a simple linear combinations of the individual solitons, though, because the equation of KdS is nonlinear, therefore, does not provide a linear combinations of individual solutions as standalone solutions. Solìtons in multiplets are bonded in nonlinear way. A soliton doublet, or a pair of solitons, which satisfies the equation KdV as its exact analytical solution is [3]:
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sol2:=unapply(2*(v2v1)*(v1*sech(sqrt(v1/2)*(x2*v1*t))^2+v2*csch(sqrt(v2/2)*(x2*v2*t))^2)/(sqrt(2*v1)*tanh(sqrt(v1/2)*(x2*v1*t))sqrt(2*v2)*coth(sqrt(v2/2)*(x2*v2*t)))^2,v1,v2,x,t);

 (2.3.1) 
here  are velosities of solitons wich are proportional to its amplitudes, so the faster KdV soliton is also higher and narrower.
Let us show the evolution of doublet into time and space by different ways:
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plot3d(sol2(1/3,2/3,x,t),x=20..20,t=20..20,axes=framed,grid=[200,200],orientation=[45,45,0],font=[HELVETICA,14],labelfont=[helvetica,14],lightmodel=`light4`,color=gold,title=`Fig.4 A soliton doublet: \n evolution in time and space`);

>

plots[densityplot](sol2(1/3,2/3,x,t),x=20..20,t=20..20,axes=framed,title="Fig.5 A soliton doublet:\n evolution and collision at (0,0) point",style=patchnogrid,font=[HELVETICA,14],labelfont=[helvetica,14]);

The figure 5 shows that the up to time the coordinates of soliton with smaller amplitude (darker and wider line) are above the coordinates of solìton with higher amplitude (lighter and narrower line), so the "small" solìton ahead of the "big". After a collision at the point solìtons exchanged places, now the "big" is ahead of the "small". We can notice that the line of evolution of solìtons after the collision is not direct sequels of their lines to clash (there has been some visible shifting).
Animation is probably most popular of these ways:
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plots[animate](sol2(1/3,2/3,x,t),x=55..55,t=30..30,frames=120,color=blue,thickness=3,gridlines=true,numpoints=280,axes=boxed, title=`Fig.6 Interaction of two solitons of a doublet`,font=[HELVETICA,14],labelfont=[helvetica,14]);

The animation Fig.6 clearly evident the interaction of solìtons: faster solìton catching up slower. Then they "exchange amplitudes", and front solìton gets the higher speed and amplitude, and the back  lower speed and amplitude. The situation is reminiscent of the collision two elastic balls with various pulses. Note: both solìtons do not change own shapes after the interaction. While directly at the time of the collision (), both solitons form the wider and lower united complex. It is specially evident at "frametoframe" animation.



References


[1] N. J. Zabusky and M. D. Kruskal, Interaction of "Solitons" in a Collissionless Plasma and the Recurrence of Initial States //Physical Review Letters 15, p/240243 (1965).
[2] F. Wang. Demonstrating Soliton Interactions using 'pdsolve' (available by adresse http://www.maplesoft.com/applications/view.aspx?SID=1733 (12.04.2012))
[3] K. Brauer. Two Soliton Solution of KdV Equation (available by adresse http://www.usf.uos.de/~kbrauer/solitons/soli2.html) (09.23.2012))

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