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Rosenau–Hyman equation

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The Rosenau–Hyman equation or K(n,n) equation is a KdV-like equation having compacton solutions. This nonlinear partial differential equation is of the form

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.

The K(n,n) equation has the following traveling wave solutions:

  • when a > 0
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)},}
  • when a < 0
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

  1. Polyanin, Andrei D.; Zaitsev, Valentin F. (28 October 2002), Handbook of Nonlinear Partial Differential Equations (Second ed.), CRC Press, p. 891, ISBN 1584882972
  2. 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
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