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Math Help - Help me for nonlinear hyperbolic first order system of pde

  1. #1
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    Help me for nonlinear hyperbolic first order system of pde

    Hi,

    i have a problem with the solving of nonlinear hyperbolic first oder system of pde. This look like following



    Can anyone help me to solve it?

    Thank you in advance.
    Best regards
    Minh
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  2. #2
    Super Member Rebesques's Avatar
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    Well... that's just great


    This system is not trivial to solve -- at least in general. Start by scaling and then define \underbar{u}=(p, q), to obtain (P): \  \frac{\partial}{\partial t}\underbar{u}=A\frac{\partial}{\partial x}\underbar{u}, where A=\left[ \begin{matrix} -a_2/a_1 & -a_3/a_1  \\ -b_2/b_1 & -b_3/b_1\end{matrix} \right]. By saying this is hyperbolic, I presume you mean the (constant!) matrix has no real eigenvalues. Then, prescribing differentiable Cauchy data (C): \ u(0,x)=u_0(x) might not mean there is a solution at all! In fact, u_0 has to be at least analytic, and I have no idea on obtaining an actual solution.


    For the sake of argument, suppose...


    1. The matrix of the system has real and distinct eigenvalues. Then the Cauchy problem (P)&(C) has always a solution of the form

      \underbar{u}(t,x)=(p(t,x),q(t,x))=\left( \int\exp(c_1ts+c_2xs)u_0(s)ds,\int\exp(d_1ts+d_2xs  )u_0(s)ds\right)

      where we have to pick the constants involved so that the integrals converge and \underbar{u} is truly a solution, the latter part being algebraic manipulations.

    2. The matrix of the system is self adjoint. Then, for the Cauchy data (C) being a tempered distribution, there is a unique solution for (P)&(C). In fact, a stronger result on regularity is true: This solution must necessarily be C^{\infty}!



    Hope this helps.
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