Integrability, exact reductions and special solutions of the KP–Whitham equations

  title={Integrability, exact reductions and special solutions of the KP–Whitham equations},
  author={Gino Biondini and Mark A. Hoefer and Antonio Moro},
  pages={4114 - 4132}
Reductions of the KP–Whitham system, namely the (2+1)-dimensional hydrodynamic system of five equations that describes the slow modulations of periodic solutions of the Kadomtsev–Petviashvili (KP) equation, are studied. Specifically, the soliton and harmonic wave limits of the KP–Whitham system are considered, which give rise in each case to a four-component (2+1)-dimensional hydrodynamic system. It is shown that a suitable change of dependent variables splits the resulting four-component… 
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We obtain the necessary and sufficient conditions for a two-component (2 + 1)-dimensional system of hydrodynamic type to possess infinitely many hydrodynamic reductions. These conditions are in
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Numerical study of the KP equation for non-periodic waves
Whitham modulation theory for the Kadomtsev– Petviashvili equation
The basic properties of the KP-Whitham system, including symmetries, exact reductions and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg–de Vries equation are discussed.
Whitham modulation theory for (2  +  1)-dimensional equations of Kadomtsev–Petviashvili type
Whitham modulation theory for certain two-dimensional evolution equations of Kadomtsev–Petviashvili (KP) type is presented. Three specific examples are considered in detail: the KP equation, the
KP solitons, total positivity, and cluster algebras
This paper explains how the theory of total positivity and cluster algebras provides a framework for understanding these soliton solutions to the KP equation and uses this framework to give an explicit construction of certain soliton contour graphs.
Solitons and the Inverse Scattering Transform
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On a family of solutions of the Kadomtsev–Petviashvili equation which also satisfy the Toda lattice hierarchy
We describe the interaction pattern in the x–y plane for a family of soliton solutions of the Kadomtsev–Petviashvili (KP) equation, The solutions considered also satisfy the finite Toda lattice
On a family of solutions of the KP equation which also satisfy the Toda lattice hierarchy
We describe the interaction pattern in the x-y plane for a family of soliton solutions of the Kadomtsev-Petviashvili (KP) equation, (−4ut +uxxx+6uux)x +3uyy = 0. Those solutions also satisfy the
Whitham modulation theory for the two-dimensional Benjamin-Ono equation.
A system of five quasilinear first-order partial differential equations is derived and this system describes modulations of the traveling wave solutions of the 2DBO equation, which is transformed to a singularity-free hydrodynamic-like system referred to here as the2DBO-Whitham system.