Whitham modulation theory for (2  +  1)-dimensional equations of Kadomtsev–Petviashvili type

Abstract

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 twodimensional Benjamin-Ono (2DBO) equation and a modified KP (m2KP) equation. A unified derivation is also provided. In the case of the m2KP equation, the corresponding Whitham modulation system exhibits features different from the other two. The approach presented here does not require integrability of the original evolution equation. Indeed, while the KP equation is known to be a completely integrable equation, the 2DBO equation and the m2KP equation are not known to be integrable. In each of the cases considered, the Whitham modulation system obtained consists of five first-order quasilinear partial differential equations. The Riemann problem (i.e. the analogue of the Gurevich-Pitaevskii problem) for the one-dimensional reduction of the m2KP equation is studied. For the m2KP equation, the system of modulation equations is used to analyze the linear stability of traveling wave solutions.Recently, however, several studies have been devoted to Whitham theory for (2+1)-dimensional systems. In particular, [3] demonstrated the derivation and physical relevance of the Whitham systems for certain reductions of the KP and two-dimensional Benjamin-Ono (2DBO) equations. These reductions lead to the cylindrical KdV and cylindrical BO equations, hence they are quite different from the standard (1+1)dimensional KdV and BO reductions of the KP and 2DBO equations, respectively. From the point of view of Whitham theory, the reductions of these (2+1)-dimensional PDEs can be considered on the same footing as the (1+1)-dimensional ones. Subsequently Whitham theory has been considered for the (2+1) dimensional KP [1] and 2DBO [2] without using any one-dimensional reductions. We also mention that the underlying structure of solutions that will develop dispersive shocks in the small dispersion limit of the generalized KP equations has been studied in [11]. We refer the reader to these papers for additional background and references.In this work we present an approach that applies equally well to integrable and non-integrable PDEs, since at the heart of it lies Whitham's WKB type expansion and separation of fast and slow scales. We consider (2+1)-dimensional PDEs of the form

On the Whitham system for the radial nonlinear Schrödinger equation

“…As mentioned earlier, the Whitham theory for cylindrical KdV considered as an exact reduction of the KP equation 15 set the stage for the subsequent development of the full (2 + 1)-dimensional theory resulting in the "hydrodynamic" form of the Whitham system (ie, the analog of the -equations) for the KP equation itself 16 and more generally for (2 + 1)-dimensional PDEs of KP type. 18 Our present rNLS Whitham theory can be considered as a similar predecessor to the derivation and study of the Whitham system for the 2d NLS equation and other (2 + 1)-dimensional PDEs related to it and, more generally, as a further step in development of nonlinear modulation theory for NLS-type systems in more than one spatial dimension. It is important to emphasize that we use a unified approach to derive Whitham systems that is equally applicable to both integrable and nonintegrable PDEs.…”

“…15 This spawned further development of Whitham theory for the KP equation, the (2 + 1)-dimensional Benjamin-Ono equation, and more generally for equations of KP type. [16][17][18] See also earlier work 19,20 about plane dark solitons and oblique DSWs of (2 + 1)dimensional NLS equation in supersonic fluid (BEC) flow around obstacles. The present study provides another example of Whitham theory applied to (2 + 1)-dimensional PDEs and their reductions; as indicated below, our rNLS results are quite different from those for the corresponding 1d NLS system.…”

“…The general nonlinear WKB approach and singular perturbation theory which we use will be an indispensable tool in this research. It has already proven to be extremely useful for (2 + 1)-dimensional PDEs of KP-type, [16][17][18] which are important mathematical models in many applications.…”

“…means terms order 1 in of the expression in the brackets. Integrating over the period and using the leading-order equation(18) gives Equation C.9 with the identification equation (C.10).The-derivative of Equation 13 reads Φ ′ + ( 2 + ) ′ = 2 ( With it, Equation C.26 can also be derived in the approach of Section 3. We now combine Equation C.35 with Equation 12 and, after some rearrangement of terms and cancellations, get of Equation C.36 is another total -derivative that integrates to the integration "constant" 1 is fixed by comparison with Equation 32 as 2 1 = 4 0 + 1 = 2 0 (2 − ) − 2 1 (also Equation C.5 is used).…”

Dispersive shock waves in three dimensional Benjamin–Ono equation

“…We note that the method used in [22] works only under the special choice of parabolic front. Later, a generalization of Whitham theory for DSWs in the KP-type equations with any initial conditions was developed [23,24]. The main result of these works, the 5x5 system of (2+1)-dimensional hydrodynamic-type equations [25] was derived which describes the slow modulations of the periodic solutions of the corresponding KP-type equation.…”