clear all
Nx=81;Nt=501;
X=linspace(-3,3,Nx);
T=linspace(0,2,Nt)';
XX=zeros(Nt,Nx);
a=0;
a=1*sqrt(exp(1)/2)-0.5;

for ix=1:length(X);
    xi=X(ix);
    if a==0;
    G=exp(-xi^2)*T+xi;
    else;
    G=exp(-xi^2)*(1-exp(-a*T))/a+xi;
    end;
    
    figure(10)
    plot(G,T,'k')
    drawnow
    hold on
    
end;
figure(10)
plot(X,exp(-X.^2))
xi=1/sqrt(2);
Gi=exp(-xi^2)*T+xi;
xi=0;
G0=exp(-xi^2)*T+xi;
plot(Gi,T,'r',G0,T,'k')
xlabel('$x$','Interpreter','Latex','Fontsize',20)
ylabel('$t$','Interpreter','Latex','Fontsize',20)

%figure(10);hold off


xp=linspace(sqrt(2),4,401);
xn=-xp;
xi1p=1/2*(xp+sqrt(xp.^2-2));
xi1n=1/2*(xp-sqrt(xp.^2-2));
xi2p=1/2*(xn+sqrt(xn.^2-2));
xi2n=1/2*(xn-sqrt(xn.^2-2));


t1p=exp(xi1p.^2)./(2*xi1p);
t1n=exp(xi1n.^2)./(2*xi1n);


figure(10)
plot(xp,t1p,'g','Linewidth',2)
plot(xp,t1n,'g','Linewidth',2)
axis([0 2 0 2])
title('Characteristics and theie Envelop for $u_t + u u_x=0$ with $u(x,0)=e^{-x^2}$','Interpreter','Latex','Fontsize',18) 

hold off