Abstract
The Kadomstev–Petviashvili (KP) equation is a well-known modulation equation normally derived by starting with the trivial state and an appropriate dispersion relation. In this paper, it is shown that the KP equation is also the relevant modulation equation for bifurcation from periodic travelling waves when the wave action flux has a critical point. Moreover, the emergent KP equation arises in a universal form, with the coefficients determined by the components of the conservation of wave action. The theory is derived for a general class of partial differential equations generated by a Lagrangian using phase modulation. The theory extends to any space dimension and time, but the emphasis in the paper is on the case of 3+1. Motivated by light bullets and quantum vortex dynamics, the theory is illustrated by showing how defocusing NLS in 3+1 bifurcates to KP in 3+1 at criticality. The generalization to N >3 is also discussed.
Highlights
The Kadomstev–Petviashvili (KP) equation in 3 + 1 can be scaled so that it takes the formx = ±uyy ± uzz, (1.1)and in N + 1 with N > 3 one just adds additional second derivative terms for each new space dimension on the right-hand side
We find that the governing equations are satisfied exactly up to fifth order in ε if and only if q satisfies (1.10)
The theory came full circle in the work of Bridges [10,11] where modulating parameters and new scaling was included in the Lagrangian setting giving a new approach to modulation in the conservative setting. This theory led to a new universal form for the codimension one emergence of the KdV equation
Summary
Relating the wavenumber and frequency modulation to derivatives of the phase using the new slow variables, the conservation of waves gives. The theory came full circle in the work of Bridges [10,11] where modulating parameters and new scaling was included in the Lagrangian setting giving a new approach to modulation in the conservative setting This theory led to a new universal form for the codimension one (only one assumption needed) emergence of the KdV equation. The strategy of this paper—introduce an ansatz, substitute into the Euler–Lagrange equation, derive exact equations up to fifth order and show that the coefficients are determined by a conservation law—is similar to [10] and so we will be brief, highlighting those features that are new and different.
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