Rosenau–Hyman equation

Last updated February 19, 2019

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]

In mathematics, the Korteweg–de Vries (KdV) equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified. KdV can be solved by means of the inverse scattering transform. The mathematical theory behind the KdV equation is a topic of active research. The KdV equation was first introduced by Boussinesq and rediscovered by Diederik Korteweg and Gustav de Vries (1895).

In the theory of integrable systems, a compacton, introduced in, is a soliton with compact support.

Partial differential equation differential equation that contains unknown multivariable functions and their partial derivatives

In mathematics, a partial differential equation (PDE) is a differential equation that contains beforehand unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.

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]

Philip Rosenau, is an Israeli mathematician and a poet. He is a Professor at the Department of Applied Mathematics at Tel Aviv University. He introduced compactons, along with James M. Hyman.

James Macklin "Mac" Hyman is an applied mathematician formerly at Los Alamos National Laboratory and currently at Tulane University in the United States. He received his undergraduate degree from Tulane University and his PhD in 1976 from NYU's Courant Institute of Mathematical Sciences under Peter Lax with thesis The method of lines solution of partial differential equations.

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

when a > 0

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)=\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)}.

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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, PMID 10054146

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[1] Polyanin, Andrei D.; Zaitsev, Valentin F., 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, American Physical Society, 70 (5): 564–567, Bibcode:1993PhRvL..70..564R, doi:10.1103/PhysRevLett.70.564, PMID 10054146