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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![2. Given the nonlinear sine-Gordon eqn.
Utz = Sin u, u(t, r), (uz = u2), k: constant (1)
Also, given the (Bäcklund transform) equations
(,"), - ASin (""), (2)
("놀), -ISm (""), (3)
%3D
=D
a) Find viz and vzt using eqn. (2) – (3).
b) Subtract v, and vVt side by side, you have found above, (use ea = ve) and show that
u satisfies cqn. (1): utz = Sin u.
c) Add v, and va side by side, you have found above, (use v = vz) and show that v
also satisfies the same eqn.: v = Sin v.
This means you can construct a new solution v 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.)]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6eb9ee4-e0dd-4b45-8eea-6745fb15f56c%2F7a3e774c-867a-418b-a1d8-16def96c2a57%2F5hewftj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Given the nonlinear sine-Gordon eqn.
Utz = Sin u, u(t, r), (uz = u2), k: constant (1)
Also, given the (Bäcklund transform) equations
(,"), - ASin (""), (2)
("놀), -ISm (""), (3)
%3D
=D
a) Find viz and vzt using eqn. (2) – (3).
b) Subtract v, and vVt side by side, you have found above, (use ea = ve) and show that
u satisfies cqn. (1): utz = Sin u.
c) Add v, and va side by side, you have found above, (use v = vz) and show that v
also satisfies the same eqn.: v = Sin v.
This means you can construct a new solution v 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.)]
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