The software provides numerical simulation of the real solutions of the Cauchy problem for the nonlinear wave equation

(1)     [ d_tt - d_xx ] u( t,x ) = f( u( t,x ) ),     0 <= t <  +infinity

   	u( 0, x ) = u0( x );    d_t u( 0, x ) = v0( x );

where f( u ) = - d_u P( u ) and P( u ) = 0.25 * ( u^2 - 1 )^2. We consider finite energy solutions: the total energy is given by

	E( t ) = Integral from -infinity to +infinity of r( t,x ) * dx,    E( t ) < +infinity

where the density of energy r( t,x ) is defined by

	r( t, x ) = 0.5 * [ d_t ( u( t, x ) ) ]^2 + 0.5 * [ d_x ( u( t, x ) ) ]^2 + P( u( t, x ) )

Let us consider the soliton solutions, i.e. u( t, x ) = U( x - v * t ). Substituting into (1) we get the stationary equation

	( 1 - v^2 ) * U'' = P'( U )

It can be integrated using the "energy conservation"

	( 1 - v^2 ) * U' / 2 - P( U ) = const

The solitons correspond to the separatrices connecting two stationary points with U = -1 and U = +1.

The separatrices with U' > 0 describes the kinks, while U' < 0 antikinks.

Thus, we have two twoparametrical families of the solitons U( x - v * t + c ).

The program allows us to study the collisions of the solitons and their perturbations.

METHODS OF SOLUTION

Explicit difference second order scheme with standing mesh. The solution is observed inside a light cone.

DEMONSTRATION EXAMPLE

Choose 2-nd tab "Initial conditions", choose in the left upper window the file 3kinks_1.we1 and click the button  "Get Data". You observe the superposition of three kinks: the initial functions u( 0, x ) and d_t u( 0, x ).

Go to 3-rd tab "Solution" and click on the left upper button "Let's go". - You see below the graphs u( t, . ) and r( t, . ), and in the middle the red line on the yellow background: the graph of oscillations of the total energy, which has to be constant, i.e. reflecting the quality of the difference scheme.

Go out to any another tab - this will accelerate the rate of computation (avoiding the graphs construction). At moment calculating ended ( t = 140.0 ) the head of window will changed by our equation.

Return to 3-rd tab "Solution" and click the button "Refresh" below on the right. You observe the colour field with 3 kinks.

Pressing on any point of the colour field, you get the value of the solution at this point.

Close the program and go to its further possible extensions.

MODIFICATION OF NONLINEAR TERM

You can change the nonlinear term. Change the "version of potential" in the file "WE_ini.txt" (=1 by default).

Call "WaveEq.exe": the rose colour marks choosen nonlinear term.

Go to the 1-st tab "Kinks-antikinks" and click the button "One plot": you see the graph of the corresponding potential energy, the phase space of the stationary equation and the kink.

If you would modify the length of the support of initial functions (=10 by default): introduce new value in the window "Initial interval's coordinate" on the 1-st tab "Kinks-antikinks" (upper right window). Click again "One plot".

MODIFICATION OF INITIAL FUNCTIONS

I. Compositions of kinks.

The program allows to compose kinks and antikinks in the alternating order. You can create initial functions u0 and v0 as the "superposition" of  the kinks and antikinks of type U( x - v * t - c ) where U( 0 ) = 0.

Go to the 1-st tab "Kinks-antikinks" and click the button ">" - you have choosen the initial functions corresponding to the standing kink corresponding to the v = 0.0 and c = 0.0.

To add new kink: click "One plot", introduce the needed "v" and displacement  "c", and click ">".

On the right you see the list of choosen kinks.

If you would drop some of them: select it by mouse and delete clicking on the button  "<".

You can see the initial function u0 composed from  the choosen kinks, clicking on the button "All plots" in the middle below.

II. Perturbation of kinks.

Go to 2-nd tab "Initial conditions". You can work either with pure kinks choosing the upper left button (by default), or  with their perturbations clicking  on the second button.

Click the middle button "Plots" to see the initial conditions and to put them into special massive for further computations.

Edit an appropriate perturbation with 4 parameters.

You can save the intial data you like, editing the title of the file "MyNewFile" and clicking the button "Put Data".

NUMERICAL SIMULATION

Go to the 3-rd tab "Solution" and click left upper button "Let's go". - You see below the graphs u( t, x ) and r( t, x ) (logarithmig scale), and the red line in the middle on the yellow background.

Go out to any another tab - this will accelerate the rate of computation. At moment calculating ended the head of window will changed by our equation.

Return to the 3-rd tab "Solution" and click the button "Refresh" below on the right.

Pressing on any point of the colour field, you get the value of the solution at this point.

If you want to continue the computation, click the upper right button "The END?". Then the button becomes "Continue", click it once more. - The computation continuous.

Novelty: the screen reflects only a central part of the solution: the "fast part" of the solution (and the fast kinks) leaves the screen though its computation continuous.

INTERVALS OF MONITORING

To observe the fast kinks, click the button  "+1" below in the middle, and then draw, by left finger,  a horizontal interval around the kink position on the blue graph of the solution. After freeing the mouse, a rectangle of the monitoring around the kink apprears. It has the center at the maximal point of the energy density.

Click the button "Refresh" below on the right - you see the parameters of the kink in the left small window:

	u - the value of u( t, x ) in the middle of the interval,

	w - the width of the energy support (r at the ends =0.1 of the maximal value at the center)

	e - total energy over the interval of the width w or over all interval of monitoring (if w = 0)

	t - current time

	E - total energy

You can edit the intervals of the monitoring by the buttons "+1", "-1", and "-All" below.

Any interval of the monitoring is centered automatically at the point of the local maximum. Just this allows to observe the kinks leaving the center field.

Moreover, all the parameters t, x, u, w, e and E are registered in the text file "PrintFile" (by default)  if the "iteration number" in the file "WE_ini.txt" does not wanish (it is 20 by default). (The name "PrintFile" can be changed at file "WE_ini.txt").

This file is convenient for a consequent analysis in Excel.

It is possible change the resolution of the colour scale: editing N <= 200 and push Enter.

RESEARCH OF THE BASIC PICTURE

On the 3-rd tab "Solution" in the middle above make a choice " With stops ". - Then the account will stop at filling a field of a picture.

On the right below there are buttons of a choice: 
	 
	"Solution" - on "Refresh" to show u (t, x), 

	"Dens. energy " - on "Refresh" to show r (t, x). 

It is possible to replace a degree of a detail of a color scale - choose in the middle on the right suitable "N".

At the left below there is a field of a choice: 

	 "Hint", 

	 "Micro-1" (1-dimensional microscope), 

	 "Micro-2" (2-dimensional microscope).

Let's choose "Micro-1". The left key of the mouse we shall lead a line on the basic picture and let's press "Refresh". Fields below will show graphs u(t, x) and r(t, x) along this line on a blue background.

Let's choose "Micro-2". The left key of the mouse we shall lead a diagonal on the basic picture and let's press "Refresh". On the right in the bottom of the basic picture there will be an increased copy the chosen rectangular.

Let's choose "Hint". Clicking on the basic picture and its copy we shall be convinced, that the hint operates everywhere.


NEW EXPERIMENT

I. New calculation with the same initial data:

Go to 3-rd tab "Solution" and click on the small left upper window (with either "1024" or "512" or "256").

Then the upper right button "Continue" becomes "Let's go". Click on it - the calculation starts from t = 0.

Novelty: if you change the number ("1024" or "512" or "256"), you will change step of scheme.

II. New initial data: 

Go to 1-st tab "Kinks-antikinks" and repeate all steps: choosing initial data, etc.

END OF SIMULATION

Click the button "memory" and then "close".

Another way: press right upper "X".

FORMULATION OF PROBLEM

Prof. Alexander KOMECH

COMPOSITION
Dr. Arkadii VINNICHENKO, e-mail: arkadi_vinn@mtu-net.ru, http://www.apv.narod.ru/