Rosenau–Hyman equation

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^[1]

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)=\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)=\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)=\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., 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, American Physical Society, 70 (5): 564–567, Bibcode:1993PhRvL..70..564R, doi:10.1103/PhysRevLett.70.564

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