Perturbation in PDE
Can anyone please hep me solve this problem?
I think it's ment to be solved by Lindstedt method:
Consider the PDE u_tt-u_xx+k^2u+eps*u^3=0.
a) Assume that k>0. Look for a solution u(t,x)=U(w(eps)t,x), where U=U0+eps*U1+eps^2U2+..., and w=1+eps*w1+eps^2*w2+.... We also assume u(0,x)=A*cos(x)+eps*B*cos(3x)+O(eps^2), u_t(0,x)=0, A~=0, and U_j is bounded for all j.
We need to find a positive value k0 of k for which not all resonant terms can be eliminated from the equation for U2. Show that for k~=k0 all resonant terms have been eliminated from the equations for U1 and U2.
b) Again assume that k>0. Let u(t,x,eps) be a solution to the PDE that is periodic in x with period 2*pi, periodic in t with minimal period P(eps), and is even in x and in t. Assume that as eps->0 the solution u(t,x,eps) converges to a nonzero function u0(t,x) and its period P(eps) converges to a nonzero P0. Show that there is a countable set E of positive real numbers that if k is not in E there exist a nonzero A and an integer n such that u0(t,x)=A*cos(sqrt(n^2+k^2)t)*cos(nx).
c) Is the k0 from (a) in E from (b)?
It would mean the world to me if someone could help me!
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