(a) Show that the sine-Gordon equation = sin u %3D is invariant under the transformations =...

 

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(a) Show that the sine-Gordon equation = sin u %3D is invariant under the transformations = m(x- Ut) and i=m(x- Ut) where m = 3(1-U²) and -1<U<1. %3D 498 Nonlinear Klein-Gordon and Sine-Gordon Equations (b) Hence or otherwise, derive that (1-6*) sin{aß{t -Utz-4)} sin{aß{1-Ux-t)} -1 u(x,t)=4 tan 1 cosh a(1-B*) (x-U - is a solution of the sine-Gordon equation, where B 0, x,,, and U are real constants, -1<B<1 and -1<U<1. hou that four solutions (a) Show that the sine-Gordon equation = sin u %3D is invariant under the transformations = m(x- Ut) and i=m(x- Ut) where m = 3(1-U²) and -1<U<1. %3D 498 Nonlinear Klein-Gordon and Sine-Gordon Equations (b) Hence or otherwise, derive that (1-6*) sin{aß{t -Utz-4)} sin{aß{1-Ux-t)} -1 u(x,t)=4 tan 1 cosh a(1-B*) (x-U - is a solution of the sine-Gordon equation, where B 0, x,,, and U are real constants, -1<B<1 and -1<U<1. hou that four solutions