2. Given the nonlinear sine-Gordon eqn. Me - Sin u, u(t, r), (ut = ), k : constant (1) Also, given the (Bäcklund transform) equations (,"), - kSin (""). (2) ()- Sin (","). (3) a) Find v and vt using eqn. (2) – (3). b) Subtract v, and vt side by side, you have found above, (use v = Vez) and show that u satisfies cqn. (1): Uta = Sin u. c) Add t, and side by side, you have found above, (use v = Va) and show that v also satisfies the same eqn.: , = Sin v. This means you can construct a new solution e to the sine-Gordon eqn. V = Sinv starting from a simple, known solution u to the same eqn. u = Sin u, using eqn. (2) – (3). For this purpose start with the trivial solution u(t, r) = 0 of eqn (1). d) Write eqn. (2) and (3) in case u = 0. e) Integrate these two equations consistently. You can use Arctanh(2) = in() for a clear result. ] f) Write v(t, r) in terms of t and r clearly. [Homework: Study/Learn sine-Gordon eqn. and its relation with Pseudospherical sur- faces and also Bäcklund transforms. (Though this will not help you either solving the problem above or gaining any points, this is part of Mathematics and Physics culture.)]

2. Given the nonlinear sine-Gordon eqn. Me - Sin u, u(t, r), (ut = ), k : constant (1) Also, given the (Bäcklund transform) equations (,"), - kSin (""). (2) ()- Sin (","). (3) a) Find v and vt using eqn. (2) – (3). b) Subtract v, and vt side by side, you have found above, (use v = Vez) and show that u satisfies cqn. (1): Uta = Sin u. c) Add t, and side by side, you have found above, (use v = Va) and show that v also satisfies the same eqn.: , = Sin v. This means you can construct a new solution e to the sine-Gordon eqn. V = Sinv starting from a simple, known solution u to the same eqn. u = Sin u, using eqn. (2) – (3). For this purpose start with the trivial solution u(t, r) = 0 of eqn (1). d) Write eqn. (2) and (3) in case u = 0. e) Integrate these two equations consistently. You can use Arctanh(2) = in() for a clear result. ] f) Write v(t, r) in terms of t and r clearly. [Homework: Study/Learn sine-Gordon eqn. and its relation with Pseudospherical sur- faces and also Bäcklund transforms. (Though this will not help you either solving the problem above or gaining any points, this is part of Mathematics and Physics culture.)]

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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