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Ce sunt solitonii ?

Studiul unor simulări pe calculator ale propagării undelor solitare (solitonilor) Korteweg-de Vries. Ce sunt solitonii ?. De ce apar ?. Ecuatia Korteweg - de Vries. Aproximari prin diferente finite :. Metoda Runge-Kutta de aproximare iterativa a solutiilor unei ecuatii diferentiale.

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Ce sunt solitonii ?

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  1. Studiul unor simulări pe calculator ale propagării undelor solitare (solitonilor) Korteweg-de Vries

  2. Ce sunt solitonii ? • De ce apar ?

  3. Ecuatia Korteweg - de Vries

  4. Aproximari prin diferente finite :

  5. Metoda Runge-Kutta de aproximare iterativa a solutiilor unei ecuatii diferentiale - I=[x0, x0+a]  R, -f:IxRR,(x,y)f(x,y), y’ = f(x,y). y(x0) = y0. • y(x)f(x,y(x))

  6. Pasii iteratiei asociate metodei RK

  7. Implementare in Matlab • dx=0.1; • x=(-18+dx:dx:18)'; • nx=length(x); • k=dx^3; • nsteps=8.0/k; • function u=onesoliton(x,v,x0) u=-v/2./cosh(.5*sqrt(v)*(x-x0)).^2; • function dudt=kdvequ(u,dx) Ecuatia KdV(simplificata si particulara pentru n=-6 si β=1): du/dt - 6*u*du/dx + d^3u/dx^3 = 0 dudt = 6*(u(3:end-2)).*(u(4:end-1)-u(2:end-3))/2/dx - (u(5:end)-2*u(4:end-1)+2*u(2:end-3)-u(1:end-4))/2/dx^3;

  8. Calculul prin metodaRunge-Kutta • k1=k*kdvequ(u,dx); • k2=k*kdvequ(u+k1/2,dx); • k3=k*kdvequ(u+k2/2,dx); • k4=k*kdvequ(u+k3,dx); u=u+k1/6+k2/3+k3/3+k4/6; Particularizez constantele din metoda RK : h=1 si =dx

  9. u(x,t)=0.5/(cosh(.5*(x-0.1*t)))^2

  10. u(x,t)=-2/(cosh((x-4*t)))^2

  11. u(x,t)=2*.01/(cosh(.1*(x-.04*t)))^2

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