Question

Consider the sine-Gordon equation (SGE)
θxt =sinθ, (1)

which governs a function θ(x,t). For any given λ denote the following B ̈acklund trans- formation by BTλ:

1
θ −θx=2λsin 2(θ +θ) , (2a)

2 1
θ +θt=λsin 2(θ −θ) , (2b)

1. (a) Given a solution θ(x, t) of the SGE, show that θ(x, t) also satisfies the SGE. Hint: Try calculating the t derivative of Equation (2a) and the x-derivative of Equation (2b) and then taking a sum or difference.

Answer 1

Sol.

Given the sine-Gordon equation (SGE)

-----(1)

Also for any denotethe following Backlund trasformation by BT :

-------(2a)

--------(2b)

To Prove:-   also satisfies SGE. i.e.

Now differentiating eq.(2a) w.r.t. t we get

[By using eq.(2b)]

---------(3)

Now differentiating eq.(2b) w.r.t. x we get

[by using eq.(2a)]

----(4)  

Now on adding eq.(3)and eq.(4) we get

[by using trignometric identity

1.

2. ]

This implies that also satisfies SGE.

Hence Proved.