92 REFERENCES
(i) Introducing a potential function where
D
x
C h.x/; u D
t
x
; (4.149)
shows that the shallow water equations become
2
x
tt
2
t
x
tx
C
2
t
g
3
x
xx
gh
0
.x/u
3
x
D 0: (4.150)
(ii) Show that under a modified Hodograph transformation, this equation for h.x/ D
˛x can be transformed to
U
3
X
U
T T
gU
XX
D 0: (4.151)
Further show that under a Legendre transformation, this equation can be linearized.
Compose the two transformations.
4.5 REFERENCES
[63] S. Lie, Göttinger Nachrichten, p. 480, 1872. 80
[64] D. J. Arrigo and J. M. Hill, Transformations and equation reductions in finite elastic-
ity. I. Plane strain deformations, Math. Mech. Solids, 1(2), pp. 155–175, 1996. DOI:
10.1177/108128659600100201. 79, 80
[65] J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51(6),
pp. 1498–1521, 1991 DOI: 10.1137/0151075. 88
[66] O. L. Morozov, Contact equivalence of the generalized hunter—Saxton Equation and the
Euler-Poisson Equation, ArXiv:math-ph/0406016v2. 88
[67] W. F. Ames, Nonlinear Partial Differential Equations in Engineering, vol. I, 1968. DOI:
10.1063/1.3034120. 89
[68] E. W. Weisstein, Minimal Surfaces, From MathWorld–A Wolfram Web Resource. http:
//mathworld.wolfram.com 86, 87
[69] D. J. Arrigo, L. Le, and J. W. Torrence, Exact solutions for a class of ratholes in highly
frictional granular materials, DCDIS Series B: Appl. Alg., 19, pp. 497–509, 2010. 91
[70] J. J. Stoker, Water Waves—e Mathematical eory with Applications, Wiley-Interscience,
New York, Interscience, 1958. DOI: 10.1002/9781118033159. 91
[71] C. Rogers and W. F. Shadwick, Backlünd Transformations and their Applications, Academic
Press, 1982. 90
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