The Rosenau–Hyman equation or K (n ,n ) equationKdV-like equation having compacton solutions. This nonlinear partial differential equation is of the form[ 1]
u
t
+
a
(
u
n
)
x
+
(
u
n
)
x
x
x
=
0.
{\displaystyle u_{t}+a(u^{n})_{x}+(u^{n})_{xxx}=0.\,}
The equation is named after Philip Rosenau and James M. Hyman , who used in their 1993 study of compactons.[ 2]
The K (n ,n ) equation has the following traveling wave solutions:
u
(
x
,
t
)
=
(
2
c
n
a
(
n
+
1
)
sin
2
(
n
−
1
2
n
a
(
x
−
c
t
+
b
)
)
)
1
/
(
n
−
1
)
,
{\displaystyle u(x,t)=\left({\frac {2cn}{a(n+1)}}\sin ^{2}\left({\frac {n-1}{2n}}{\sqrt {a}}(x-ct+b)\right)\right)^{1/(n-1)},}
u
(
x
,
t
)
=
(
2
c
n
a
(
n
+
1
)
sinh
2
(
n
−
1
2
n
−
a
(
x
−
c
t
+
b
)
)
)
1
/
(
n
−
1
)
,
{\displaystyle u(x,t)=\left({\frac {2cn}{a(n+1)}}\sinh ^{2}\left({\frac {n-1}{2n}}{\sqrt {-a}}(x-ct+b)\right)\right)^{1/(n-1)},}
u
(
x
,
t
)
=
(
2
c
n
a
(
n
+
1
)
cosh
2
(
n
−
1
2
n
−
a
(
x
−
c
t
+
b
)
)
)
1
/
(
n
−
1
)
.
{\displaystyle u(x,t)=\left({\frac {2cn}{a(n+1)}}\cosh ^{2}\left({\frac {n-1}{2n}}{\sqrt {-a}}(x-ct+b)\right)\right)^{1/(n-1)}.}
References
^ Polyanin, Andrei D.; Zaitsev, Valentin F. (28 October 2002), Handbook of Nonlinear Partial Differential Equations (Second ed.), CRC Press, p. 891, ISBN 1584882972 ^ Rosenau, Philip; Hyman, James M. (1993), "Compactons: Solitons with finite wavelength", Physical Review Letters , 70 (5), American Physical Society: 564– 567, Bibcode :1993PhRvL..70..564R , doi :10.1103/PhysRevLett.70.564 , PMID 10054146 
