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2. Given the nonlinear sine-Gordon eqn. ua - Sin u, u(t, z), (4 = tig), k: constant (1) Also, given the (Bücklund transform) equations (),- Si -(), (2) -kSin Sin a) Find t, and ty tasing eqn. (2) – (3).

2. Given the nonlinear sine-Gordon eqn. ua - Sin u, u(t, z), (4 = tig), k: constant (1) Also, given the (Bücklund transform) equations (),- Si -(), (2) -kSin Sin a) Find t, and ty tasing eqn. (2) – (3).

Advanced Engineering Mathematics
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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this lesson is PDE partial differential eqution

2. Given the nonlinear sine-Gordon eqn. Uz = Sin u, u(t, r), (u = u), k: constant (1) Also, given the (Bäcklund transform) equations (S).- (). - kSin (""). (2) Sin a) Find vz and Vt 1using eqn. (2) – (3). b) Subtract vg and ve side by side, you have found sbove, (use v = Vz) and show thst u satisfies eqn. (1): uz = Sin u. c) Add vz and v, side by side, you have found above, (use v = Vz) and show that v also satisfies the same eqn.: Vz = Sin v. This means you can construct a new solution v to the sine-Gordon eqn. Vg = Sin v starting from a simple, known solution u to the same eqn. ug = 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(z) = In() for %3D a clear result.
Transcribed Image Text:2. Given the nonlinear sine-Gordon eqn. Uz = Sin u, u(t, r), (u = u), k: constant (1) Also, given the (Bäcklund transform) equations (S).- (). - kSin (""). (2) Sin a) Find vz and Vt 1using eqn. (2) – (3). b) Subtract vg and ve side by side, you have found sbove, (use v = Vz) and show thst u satisfies eqn. (1): uz = Sin u. c) Add vz and v, side by side, you have found above, (use v = Vz) and show that v also satisfies the same eqn.: Vz = Sin v. This means you can construct a new solution v to the sine-Gordon eqn. Vg = Sin v starting from a simple, known solution u to the same eqn. ug = 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(z) = In() for %3D a clear result.
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