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Perturbation in PDE

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Perturbation in PDE


I really need help. Can anyone explain me the way to solve this:

I think this is ment to be solved using Lindstedt method.

Consider the PDE u_tt-u_xx+k^2*u+eps*u^3=0.

a) We assume k>0, and look for a solution u(t,x)=U(w(eps)t,x), where U=U0+eps*U1+eps^2*U2+O(eps^2), and w(eps)=1+eps*w1+eps^2*w2+O(eps^3).
We assume u(0,x)=A*cos(x)+eps*B*cos(3x)+O(eps^2), and u_t(0,x)=0. A~=0, and U_j is bounded for all j.

We look for 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 of 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 such that if k is not in E then there exist a nonzero A and an integer n such that u0(t,x)=A*cos(sqrt(n^2+k^2)*t)*cos(n*x).

c) Is k0 from (a) is in E from (b)?

This would mean the world if anyone would help me!
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I’m think k0 come from the cutoff frequency designing , the media . Hence, k0 from (b) process !.

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