Dispersionless 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 λ 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 Davey–Stewartson equation

Dispersionless Novikov–Veselov equation

The dispersionless Novikov-Veselov equation is most commonly written as the following equation on function v = v(x₁,x₂,t):

$\begin{align} & \partial_{ t } v = \partial_{ z }( v w ) + \partial_{ \bar z }( v \bar w ), \\ & \partial_{ \bar z } w = - 3 \partial_{ z } v, \end{align}$

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.

Dispersionless Hirota equation

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