Dear friends,
I am try to find a period or quasiperiod solution for Aut+Bux-Cuux=0 equation, where u=u(x,t) is variable with two independant variables t and x. A,B and C are positive constants. The solution of u is periodic or quasiperiodic on t, with u(0,t)=-u(1,t)=sin(2*pi*f*t) as boundary conditions. Does anyone have idea how to solve it, please? Thank you very much.
Best,
Wei
If I understand correctly your notation, the PDE you are talking about can be seen as the inviscid Burgers equation with a forcing term
du/dt +d/dx(u2/2)=f(u)
The classical Burgers equation admits a strong solution for smooth initial data until a time where the regular solution breaks. On the other hand, if you rewrite the equation as
du/dt + (1+u)*du/dx =0
considering that du=dt*du/dt+dx*du/dx, you see that du=0 on the curves in the x,t plane, dx/dt=1+u. That means the u is constant along such curves that are, consequently, nothing but straight lines and the value is computed along the interesection at t=0.
The equation can be indeed recast into the inviscid Burgers equation after a shift and multiplication by -1. However, one must be careful when drawing conclusions about this problem by using the image of what happens in the whole real line. In that case, the solution follows from the method of characteristics and indeed there's a finite time at which a shock appears. However, with this constraint at the extremes of the interval [0,1], things may be different.
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