Camassa–Holm equation

Dispersionless (or quasi-classical) limits of integrable partial differential equations (PDE) arise in various problems of mathematics and physics and are intensively studied in the recent literature (see, f.i., [1]-[5]).

Examples

Dispersionless KP equation

The dispersionless Kadomtsev–Petviashvili equation (dKPE) has the form

$(u_{t}+uu_{x})_{x}+u_{yy}=0,\qquad (1)$

It arises from the commutation

$[L_{1},L_{2}]=0.\qquad (2)$

of the following pair of 1-parameter families of vector fields

$L_{1}=\partial _{y}+\lambda \partial _{x}-u_{x}\partial _{\lambda },\qquad (3a)$

$L_{2}=\partial _{t}+(\lambda ^{2}+u)\partial _{x}+(-\lambda u_{x}+u_{y})\partial _{\lambda },\qquad (3b)$

where $\lambda$ is a spectral parameter. The dKPE is the $x$-dispersionless limit of the celebrated Kadomtsev–Petviashvili equation.

Dispersionless Korteweg–de Vries equation

The dispersionless Korteweg–de Vries equation (dKdVE) reads as

$u_{t}={\frac {3}{2}}uu_{x}.\qquad (4)$

It is the dispersionless or quasiclassical limit of the Korteweg–de Vries equation.

Dispersionless Novikov–Veselov equation

The dispersionless Novikov-Veselov equation is most commonly written as the following equation on function $v=v(x_{1},x_{2},t)$:

${\begin{aligned}&\partial _{t}v=\partial _{z}(vw)+\partial _{\bar {z}}(v{\bar {w}}),\\&\partial _{\bar {z}}w=-3\partial _{z}v,\end{aligned}}$

where the following standard notation of complex analysis is used: $\partial _{z}={\frac {1}{2}}(\partial _{x_{1}}-i\partial _{x_{2}})$, $\partial _{\bar {z}}={\frac {1}{2}}(\partial _{x_{1}}+i\partial _{x_{2}})$. The function $w$ here is an auxiliary function defined via $v$ up to a holomorphic summand. The function $v$ is generally assumed to be a real-valued function.

See also

• Integrable systems
• Nonlinear Schrödinger equation
• Nonlinear systems
• Davey–Stewartson equation
• Dispersive partial differential equation
• Kadomtsev–Petviashvili equation
• Korteweg–de Vries equation

References

• Kodam Y., Gibbons J. "Integrability of the dispersionless KP hierarchy"
• Zakharov V.E. "Dispersionless limit of integrable systems in 2+1 dimensions"
• Takasaki K., Takebe T. Rev. Math. Phys., 7, 743 (1995)
• Konopelchenko B.G. "Quasiclassical generalized Weierstrass representation and dispersionless DS equation", ArXiv: 0709.4148
• Konopelchenko B.G., Moro A. "Integrable Equations in Nonlinear Geometrical Optics", Studies in Applied Mathematics, 113(4), pp. 325–352 (2004)
• Dunajski M. "Interpolating integrable system". ArXiv: 0804.1234

External links

• Ishimori_system at the dispersive equations wiki