# Kadomtsev-Petrashvili equation

The Kadomtsev-Petrashvili equation

(u_t + uu_x + \epsilon^2u_{xxx})_x + \lambda u_{yy} = 0, \quad \lambda = \pm1

essentially describes one dimensional wave phenomena in the long wave-length limit with weak transversal effects, for instance waves in shallow water.

### Solitons and almost periodic solutions

From ph/0601025 and nlin/0512066.

#### KP line 2-soliton

The KP equation has solitonic solutions which are exponentially localized in one spatial direction and infinitely extended in the other, so-called line solitons. They generalize the KdV solitons (no y-dependence) to y-dependent solutions of KP. The video shows a 2-soliton solution.

#### KP line 4-soliton

The KP equation has solitonic solutions which are exponentially localized in one spatial direction and infinitely extended in the other, so-called line solitons. They generalize the KdV solitons (no y-dependence) to y-dependent solutions of KP. The video above shows a 4-soliton solution.

#### KP, perturbed line soliton

For 𝜀 = 1, the KP equation has solitonic solutions which are localized in both spatial directions with an algebraic fall off. These solutions are called lump solitons. In this case the line solitons are unstable and will form lump solitons if perturbed. The video above shows a perturbed KdV soliton where the perturbations grow up to the formation of lump solitons.

#### Hyperelliptic KP solution, genus 2

Almost periodic solutions to the KP equation can be obtained in terms of multi-dimensional theta functions on arbitrary compact Riemann surfaces. The video above shows a periodic solution on a hyperelliptic surface of genus 2.

#### Hyperelliptic KP solution, genus 4

Almost periodic solutions to the KP equation can be obtained in terms of multi-dimensional theta functions on arbitrary compact Riemann surfaces. The video shows an almost periodic solution on a hyperelliptic surface of genus 4.

See ph/0601025 and nlin/0512066 for references and further details. For Hyperelliptic KP solution, genus 2, see also 1202.0962.

### Small dispersion limit

KP solution for initial data with compact support in the small dispersion limit, from ph/0601025.

#### Small dispersion KP

In the small dispersion limit (𝜀 ≪ 1), KP solutions develop a zone of rapid modulated oscillations near shocks of the dispersionless KP equation. For small initial data of order 𝜀 with rapid oscillations in x-direction of order 1/𝜀, a multi-scales expansion of the KP equation shows that the amplitude of the solution is asymptotically described  by a solution of the Davey-Stewartson equation. The video shows the solution for 𝜀 = 0.1 for initial data with compact support. The corresponding Davey-Stewartson solution can be seen in DS solution.

#### Small dispersion KP solution (0.1)

In the small dispersion limit (𝜀 ≪ 1), KP solutions develop a zone of rapid modulated oscillations near shocks of the dispersionless KP equation. The video shows the solution for 𝜀 = 0.1 for the initial data with compact support. There are two oscillatory regions for the shown initial data.

#### Small dispersion KP solution (0.01)

In the small dispersion limit (𝜀 ≪ 1), KP solutions develop a zone of rapid modulated oscillations near shocks of the dispersionless KP equation. In Small dispersion KP solution (𝜀 = 0.1) (above), we show the solution for 𝜀 = 0.1 and for initial data with compact support which develop two oscillatory zones. Here we present these two oscillatory zones for the same initial data for 𝜀 = 0.01.

See ph/0601025 for references and further details.