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Arnold's Trivium
1991 Arnold's Trivium
82
Problem 82
Problem 82
Source:
July 1, 2010
Problem Statement
For what values of the velocity
c
c
c
does the equation
u
t
=
u
−
u
2
+
u
x
x
u_t = u -u^2 + u_{xx}
u
t
=
u
−
u
2
+
u
xx
have a solution in the form of a traveling wave
u
=
φ
(
x
−
c
t
)
u = \varphi(x-ct)
u
=
φ
(
x
−
c
t
)
,
φ
(
−
∞
)
=
1
\varphi(-\infty) = 1
φ
(
−
∞
)
=
1
,
φ
(
∞
)
=
0
\varphi(\infty) = 0
φ
(
∞
)
=
0
,
0
≤
u
≤
1
0 \le u \le 1
0
≤
u
≤
1
?
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