salim-barzani

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These are answers submitted by salim-barzani


 

restart

with(PDEtools)

interface(showassumed = 0)

Setup(mathematicalnotation = true)

assume(x::real); assume(t::real); assume(lambda1::complex); assume(lambda2::complex); assume(A1::real); assume(A2::real); assume(B1::real); assume(B2::real)

declare(q(x, t), qr(x, t))

q(x, t)*`will now be displayed as`*q

 

qr(x, t)*`will now be displayed as`*qr

(1)

pde := diff(diff(q(x, t), t), x)+q(x, t)-(2*I)*q(x, t)*qr(x, t)*(diff(q(x, t), x))

diff(diff(q(x, t), t), x)+q(x, t)-(2*I)*q(x, t)*qr(x, t)*(diff(q(x, t), x))

(2)

Q := phi(-c*t+x)*exp(I*(k*x-Omega*t+theta(-c*t+x))); rel := {q(x, t) = Q, qr(x, t) = subs({t = -t, x = -x}, Q)}; pde1 := simplify(eval(subs(subs({}, rel), pde))) = 0

phi(-c*t+x)*exp(I*(k*x-Omega*t+theta(-c*t+x)))

 

{q(x, t) = phi(-c*t+x)*exp(I*(k*x-Omega*t+theta(-c*t+x))), qr(x, t) = phi(c*t-x)*exp(I*(-k*x+Omega*t+theta(c*t-x)))}

 

-2*phi(c*t-x)*phi(-c*t+x)*((-(D(theta))(-c*t+x)-k)*phi(-c*t+x)+I*(D(phi))(-c*t+x))*exp(I*(-Omega*t+k*x+theta(c*t-x)+2*theta(-c*t+x)))-2*exp(I*(k*x-Omega*t+theta(-c*t+x)))*((1/2)*(-1-(D(theta))(-c*t+x)^2*c+(-c*k-Omega)*(D(theta))(-c*t+x)+I*c*((D@@2)(theta))(-c*t+x)-Omega*k)*phi(-c*t+x)+I*(D(phi))(-c*t+x)*(D(theta))(-c*t+x)*c+((1/2)*I)*(c*k+Omega)*(D(phi))(-c*t+x)+(1/2)*((D@@2)(phi))(-c*t+x)*c) = 0

(3)

pde2 := simplify(subs({phi(c*t-x) = phi(c*t-x), theta(c*t-x) = -theta(-c*t+x)}, pde1))

-2*exp(I*(k*x-Omega*t+theta(-c*t+x)))*(((-(D(theta))(-c*t+x)-k)*phi(-c*t+x)+I*(D(phi))(-c*t+x))*phi(-c*t+x)*phi(c*t-x)+(1/2)*(-1-(D(theta))(-c*t+x)^2*c+(-c*k-Omega)*(D(theta))(-c*t+x)+I*c*((D@@2)(theta))(-c*t+x)-Omega*k)*phi(-c*t+x)+I*(D(phi))(-c*t+x)*(D(theta))(-c*t+x)*c+((1/2)*I)*(c*k+Omega)*(D(phi))(-c*t+x)+(1/2)*((D@@2)(phi))(-c*t+x)*c) = 0

(4)

pde3 := simplify(exp(-I*(k*x-Omega*t+theta(-c*t+x)))*pde2)

-2*((-(D(theta))(-c*t+x)-k)*phi(-c*t+x)+I*(D(phi))(-c*t+x))*phi(-c*t+x)*phi(c*t-x)+((D(theta))(-c*t+x)^2*c+(c*k+Omega)*(D(theta))(-c*t+x)-I*c*((D@@2)(theta))(-c*t+x)+Omega*k+1)*phi(-c*t+x)-(2*I)*(D(theta))(-c*t+x)*c*(D(phi))(-c*t+x)+I*(-c*k-Omega)*(D(phi))(-c*t+x)-((D@@2)(phi))(-c*t+x)*c = 0

(5)

Seperat := Re(pde3)

Re(-2*((-(D(theta))(-c*t+x)-k)*phi(-c*t+x)+I*(D(phi))(-c*t+x))*phi(-c*t+x)*phi(c*t-x)+((D(theta))(-c*t+x)^2*c+(c*k+Omega)*(D(theta))(-c*t+x)-I*c*((D@@2)(theta))(-c*t+x)+Omega*k+1)*phi(-c*t+x)-((D@@2)(phi))(-c*t+x)*c)+Im(2*(D(theta))(-c*t+x)*c*(D(phi))(-c*t+x)-(-c*k-Omega)*(D(phi))(-c*t+x)) = 0

(6)

Real := -(2*((-(D(theta))(-c*t+x)-k)*phi(-c*t+x)+I*(D(phi))(-c*t+x)))*phi(-c*t+x)*phi(c*t-x)+((D(theta))(-c*t+x)^2*c+(c*k+Omega)*(D(theta))(-c*t+x)-I*c*((D@@2)(theta))(-c*t+x)+Omega*k+1)*phi(-c*t+x)-((D@@2)(phi))(-c*t+x)*c = 0

-2*((-(D(theta))(-c*t+x)-k)*phi(-c*t+x)+I*(D(phi))(-c*t+x))*phi(-c*t+x)*phi(c*t-x)+((D(theta))(-c*t+x)^2*c+(c*k+Omega)*(D(theta))(-c*t+x)-I*c*((D@@2)(theta))(-c*t+x)+Omega*k+1)*phi(-c*t+x)-((D@@2)(phi))(-c*t+x)*c = 0

(7)

im := 2*(D(theta))(-c*t+x)*c*(D(phi))(-c*t+x)-(-c*k-Omega)*(D(phi))(-c*t+x) = 0

2*(D(theta))(-c*t+x)*c*(D(phi))(-c*t+x)-(-c*k-Omega)*(D(phi))(-c*t+x) = 0

(8)

NULL

NULL


 

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