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I wish to solve the following system of PDEs for h1, h2 using periodic boundary conditions and specified initial conditions. When I execute Mathematica complains that the boundary conditions are not consistent

    System1 = {D[h1[x, t], t] == -D[Q1[x, t], x], 
  D[h2[x, t], t] == -D[Q1[x, t], x] - D[Q2[x, t], x], 
  h1[x, 0] == 
   0.5 + A0*h1L*Exp[Sqrt[-1]*alpL*x] + B0*h1S*Exp[Sqrt[-1]*alpS*x], 
  h2[x, 0] == 
   1 + A0*h2L*Exp[Sqrt[-1]*alpL*x] + B0*h2S*Exp[Sqrt[-1]*alpS*x], 
  h1[-40, t] == h1[40, t], h2[-40, t] == h2[40, t]}

I also tried it with PeriodicBoundaryCondition function but it gives "Equation or list of equations expected instead of h1[x,t] in the first argument {h1[x,t],h2[x,t]}".

System = {D[h1[x, t], t] == -D[Q1[x, t], x], 
  D[h2[x, t], t] == -D[Q1[x, t], x] - D[Q2[x, t], x], 
  h1[x, 0] == 
   0.5 + A0*h1L*Exp[Sqrt[-1]*alpL*x] + B0*h1S*Exp[Sqrt[-1]*alpS*x], 
  h2[x, 0] == 
   1 + A0*h2L*Exp[Sqrt[-1]*alpL*x] + B0*h2S*Exp[Sqrt[-1]*alpS*x], 
  PeriodicBoundaryCondition[{h1[x, t], h2[x, t]}, x == 2*Pi/alpL, 
   Function[x, x - 2*Pi/alpL]]}

The constants and the expressions are provided in the attached .nb files. Kindly help!

click here for the first file click here for the second file

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