# Difference between revisions of "Kadomtsev-Petviashvili equation"

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The '''Kadomtsev-Petviashvili equations''' (KP) are given by | The '''Kadomtsev-Petviashvili equations''' (KP) are given by | ||

− | :<math>(u_t + u_{xxx} + uu_x)_x + \lambda | + | :<math>(u_t + u_{xxx}^{} + uu_x)_x + \lambda u_{yy} = 0</math> |

− | where u=u(x,y,t) is a real-valued function of two space variables and one time variable, and <math>\lambda</math> is a constant scalar. When <math>\lambda = 0</math> the evolution is trivial in the y variable and the equation collapses to the KdV equation. When <math>\lambda < 0</math> the equation is known as the [[KP-I equation]], and when <math>\lambda > 0</math> the equation is the [[KP-II equation]]. These equations are a model for shallow long waves in the x direction, with some mild dispersion in the y direction [KdPv1970], [PvYn1989]. Generally speaking, KP-I is a good model when surface tension is strong, and KP-II is a good model when surface tension is weak. | + | where <math>u=u(x,y,t)</math> is a real-valued function of two space variables and one time variable, and <math>\lambda</math> is a constant scalar. When <math>\lambda = 0</math> the evolution is trivial in the y variable and the equation collapses to the [[KdV]] equation. When <math>\lambda < 0</math> the equation is known as the [[KP-I equation]], and when <math>\lambda > 0</math> the equation is the [[KP-II equation]]. These equations are a model for shallow long waves in the x direction, with some mild dispersion in the y direction ([[KdPv1970]], [[PvYn1989]]). Generally speaking, KP-I is a good model when surface tension is strong, and KP-II is a good model when surface tension is weak. |

− | KP (or the 2D Boussinesq equations) can also arise as a model for the (somewhat unphysical) 3D water wave equation in the shallow case, similar to how KdV arises from the 2D water wave equation. Under further limiting assumptions (slowly varying amplitude, weak non-linearity) one can obtain the Davey-Stewatson equation (formally, at least). KP-I also arises as a model for sound waves in ferromagnetic media [TrFl1985]. | + | KP (or the 2D Boussinesq equations) can also arise as a model for the (somewhat unphysical) 3D water wave equation in the shallow case, similar to how KdV arises from the 2D water wave equation. Under further limiting assumptions (slowly varying amplitude, weak non-linearity) one can obtain the Davey-Stewatson equation (formally, at least). KP-I also arises as a model for sound waves in ferromagnetic media ([[TrFl1985]]). |

− | The equations are completely integrable, and thus have an infinite number of conserved quantities. However for the LWP and GWP theory the most important conserved quantities are the | + | The equations are completely integrable, and thus have an infinite number of conserved quantities. However for the LWP and GWP theory the most important conserved quantities are the mass |

:<math>\int u^2 dx dy</math> | :<math>\int u^2 dx dy</math> | ||

− | and the Hamiltonian | + | and the Hamiltonian |

:<math>\int u_x^2 - u^3/3 - \lambda (\nabla_x^{-1} u_y)^2 dx dy.</math> | :<math>\int u_x^2 - u^3/3 - \lambda (\nabla_x^{-1} u_y)^2 dx dy.</math> | ||

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The next conserved quantity contains terms which resemble the L^2 norm of u_xx or of <math>\nabla_x^{-2} u_{yy}</math>. (Note that one y derivative has the same scaling as two x derivatives.) | The next conserved quantity contains terms which resemble the L^2 norm of u_xx or of <math>\nabla_x^{-2} u_{yy}</math>. (Note that one y derivative has the same scaling as two x derivatives.) | ||

− | Explicit solutions can be obtained by inverse scattering [AxPgSac1997], [FsSng1992], [Zx1990], although this does not appear to be directly helpful to the low-regularity LWP and GWP theory. | + | Explicit solutions can be obtained by inverse scattering ([[AxPgSac1997]], [[FsSng1992]], [[Zx1990]]), although this does not appear to be directly helpful to the low-regularity LWP and GWP theory. |

− | The Cauchy problem is usually studied in two-parameter Sobolev spaces <math>H_x^{s_1} H_y^{s_2}</math>. However, the presence of inverse derivatives such as <math>nabla_x^{-1}</math> necessitates some technical modifications to these spaces at low frequencies which we will not detail here. | + | The Cauchy problem is usually studied in two-parameter Sobolev spaces <math>H_x^{s_1} H_y^{s_2}</math>. However, the presence of inverse derivatives such as <math>\nabla_x^{-1}</math> necessitates some technical modifications to these spaces at low frequencies which we will not detail here. |

Despite the apparent similarity of KP-I and KP-II, the two equations behave very differently, both qualitatively and quantitatively. KP-I has the advantage of having a Hamiltonian which is positive definite in the leading order terms, however it contains resonances which make an X^{s,b}-based analysis somewhat delicate. The KP-II equation does not have any non-trivial resonances, but its Hamiltonian does not have a definite sign. | Despite the apparent similarity of KP-I and KP-II, the two equations behave very differently, both qualitatively and quantitatively. KP-I has the advantage of having a Hamiltonian which is positive definite in the leading order terms, however it contains resonances which make an X^{s,b}-based analysis somewhat delicate. The KP-II equation does not have any non-trivial resonances, but its Hamiltonian does not have a definite sign. | ||

− | Just as KdV can be generalized to gKDV-k, KP can be generalized to gKP-k, by replacing uu_x with u^k u_x (and u^3/3 with u^{k+1}/(k+1) in the Hamiltonian). The value k=4/3 is critical in the sense that the potential energy term in the Hamiltonian can be controlled by the other two terms, and thus one expects blow up in general for long times when k > 4/3. This is supported by numerics [BnDgKar1986], [ | + | Just as KdV can be generalized to [[gKdV|gKDV-k]], KP can be generalized to gKP-k, by replacing <math>uu_x </math> with <math>u^k u_x</math> (and <math>u^3/3</math> with <math>u^{k+1}/(k+1)</math> in the Hamiltonian). The value <math>k=4/3</math> is critical in the sense that the potential energy term in the Hamiltonian can be controlled by the other two terms, and thus one expects blow up in general for long times when <math>k > 4/3</math>. This is supported by numerics ([[BnDgKar1986]], [[WgxAbSe1994]], [[Wc1987]]). |

+ | |||

+ | [[Category:Integrability]] | ||

+ | [[Category:Equations]] |

## Latest revision as of 19:46, 3 February 2011

The **Kadomtsev-Petviashvili equations** (KP) are given by

where is a real-valued function of two space variables and one time variable, and is a constant scalar. When the evolution is trivial in the y variable and the equation collapses to the KdV equation. When the equation is known as the KP-I equation, and when the equation is the KP-II equation. These equations are a model for shallow long waves in the x direction, with some mild dispersion in the y direction (KdPv1970, PvYn1989). Generally speaking, KP-I is a good model when surface tension is strong, and KP-II is a good model when surface tension is weak.

KP (or the 2D Boussinesq equations) can also arise as a model for the (somewhat unphysical) 3D water wave equation in the shallow case, similar to how KdV arises from the 2D water wave equation. Under further limiting assumptions (slowly varying amplitude, weak non-linearity) one can obtain the Davey-Stewatson equation (formally, at least). KP-I also arises as a model for sound waves in ferromagnetic media (TrFl1985).

The equations are completely integrable, and thus have an infinite number of conserved quantities. However for the LWP and GWP theory the most important conserved quantities are the mass

and the Hamiltonian

The next conserved quantity contains terms which resemble the L^2 norm of u_xx or of . (Note that one y derivative has the same scaling as two x derivatives.)

Explicit solutions can be obtained by inverse scattering (AxPgSac1997, FsSng1992, Zx1990), although this does not appear to be directly helpful to the low-regularity LWP and GWP theory.

The Cauchy problem is usually studied in two-parameter Sobolev spaces . However, the presence of inverse derivatives such as necessitates some technical modifications to these spaces at low frequencies which we will not detail here.

Despite the apparent similarity of KP-I and KP-II, the two equations behave very differently, both qualitatively and quantitatively. KP-I has the advantage of having a Hamiltonian which is positive definite in the leading order terms, however it contains resonances which make an X^{s,b}-based analysis somewhat delicate. The KP-II equation does not have any non-trivial resonances, but its Hamiltonian does not have a definite sign.

Just as KdV can be generalized to gKDV-k, KP can be generalized to gKP-k, by replacing with (and with in the Hamiltonian). The value is critical in the sense that the potential energy term in the Hamiltonian can be controlled by the other two terms, and thus one expects blow up in general for long times when . This is supported by numerics (BnDgKar1986, WgxAbSe1994, Wc1987).