Dispersionless Kadomtsev-Petviashvili Equation Research Articles

Numerical study of break-up in solutions to the dispersionless Kadomtsev–Petviashvili equation

Numerical study of break-up in solutions to the dispersionless Kadomtsev–Petviashvili equation

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

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 systems into two parts: (i) a decoupled, independent two-component system comprised of the dispersionless KP equation, (ii) an auxiliary, two-component system coupled to the mean flow equations, which describes either the evolution of a linear wave or a soliton propagating on top of the mean flow. The integrability of both four-component systems is then demonstrated by applying the Haantjes tensor test as well as the method of hydrodynamic reductions. Various exact reductions of these systems are then presented that correspond to concrete physical scenarios.

Analysis of the symmetry group and exact solutions of the dispersionless KP equation in n + 1 dimensions

The Lie algebra of the symmetry group of the (n + 1)-dimensional generalization of the dispersionless Kadomtsev–Petviashvili equation is obtained and identified as a semi-direct sum of a finite dimensional simple Lie algebra and an infinite dimensional nilpotent subalgebra. Group transformation properties of solutions under the subalgebra sl(2,R) are presented. Known explicit analytic solutions in the literature are shown to be actually group-invariant solutions corresponding to certain specific infinitesimal generators of the symmetry group.

On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3,5)

Let Gr(d; n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V n. A submanifold X Gr(d; n) gives rise to a differential system ⊂(X) that governs d-dimensional submanifolds of V n whose Gaussian image is contained in X. Systems of the form Σ(X) appear in numerous applications in continuum mechanics, theory of integrable systems, general relativity and differential geometry. They include such wellknown examples as the dispersionless Kadomtsev-Petviashvili equation, the Boyer-Finley equation, Plebansky's heavenly equations, and so on. In this paper we concentrate on the particularly interesting case of this construction where X is a fourfold in Gr(3; 5). Our main goal is to investigate differential-geometric and integrability aspects of the corresponding systems Σ(X). We demonstrate the equivalence of several approaches to dispersionless integrability such as • the method of hydrodynamic reductions, • the method of dispersionless Lax pairs, • integrability on solutions, based on the requirement that the characteristic variety of system Σ(X) defines an Einstein-Weyl geometry on every solution, • integrability on equation, meaning integrability (in twistor-theoretic sense) of the canonical GL(2;R) structure induced on a fourfold X ⊂ Gr(3; 5). All these seemingly different approaches lead to one and the same class of integrable systems Σ(X). We prove that the moduli space of such systems is 6-dimensional. We give a complete description of linearisable systems (the corresponding fourfold X is a linear section of Gr(3; 5)) and linearly degenerate systems (the corresponding fourfold X is the image of a quadratic map P4 99K Gr(3; 5)). The fourfolds corresponding to `generic' integrable systems are not algebraic, and can be parametrised by generalised hypergeometric functions.

On the dispersionless Kadomtsev–Petviashvili equation with arbitrary nonlinearity and dimensionality: exact solutions, longtime asymptotics of the Cauchy problem, wave breaking and shocks

We study the generalization of the dispersionless Kadomtsev–Petviashvili (dKP) equation in dimensions and with nonlinearity of degree , a model equation describing the propagation of weakly nonlinear, quasi one-dimensional waves in the absence of dispersion and dissipation, and arising in several physical contexts, like acoustics, plasma physics, hydrodynamics and nonlinear optics. In 2 + 1 dimensions and with quadratic nonlinearity, this equation is integrable through a novel inverse scattering transform, and it has been recently shown to be a prototype model equation in the description of the two-dimensional wave breaking of localized initial data. In higher dimensions and with higher nonlinearity, the generalized dKP equations are not integrable, but their invariance under motions on the paraboloid allows one to construct in this paper a family of exact solutions describing waves constant on their paraboloidal wave front and breaking simultaneously in all points of it, developing after breaking either multivaluedness or single-valued discontinuous profiles (shocks). Then such exact solutions are used to build the longtime behavior of the solutions of the Cauchy problem, for small and localized initial data, showing that wave breaking of small initial data takes place in the longtime regime if and only if . Lastly, the analytic aspects of such wave breaking are investigated in detail in terms of the small initial data, in both cases in which the solution becomes multivalued after breaking or it develops a shock. These results, contained in the 2012 master’s thesis of one of the authors (FS) [], generalize those obtained in [] for the dKP equation in dimensions with quadratic nonlinearity, and are obtained following the same strategy.

Shock formation in the dispersionless Kadomtsev–Petviashvili equation

The dispersionless Kadomtsev–Petviashvili (dKP) equation is one of the simplest nonlinear wave equations describing two-dimensional shocks. To solve the dKP equation numerically we use a coordinate transformation inspired by the method of characteristics for the one-dimensional Hopf equation . We show numerically that the solutions to the transformed equation stays regular for longer times than the solution of the dKP equation. This permits us to extend the dKP solution as the graph of a multivalued function beyond the critical time when the gradients blow up. This overturned solution is multivalued in a lip shape region in the (x, y) plane, where the solution of the dKP equation exists in a weak sense only, and a shock front develops. A local expansion reveals the universal scaling structure of the shock, which after a suitable change of coordinates corresponds to a generic cusp catastrophe. We provide a heuristic derivation of the shock front position near the critical point for the solution of the dKP equation, and study the solution of the dKP equation when a small amount of dissipation is added. Using multiple-scale analysis, we show that in the limit of small dissipation and near the critical point of the dKP solution, the solution of the dissipative dKP equation converges to a Pearcey integral. We test and illustrate our results by detailed comparisons with numerical simulations of both the regularized equation, the dKP equation, and the asymptotic description given in terms of the Pearcey integral.

Numerical study of shock formation in the dispersionless Kadomtsev–Petviashvili equation and dispersive regularizations

The formation of singularities in solutions to the dispersionless Kadomtsev–Petviashvili (dKP) equation is studied numerically for different classes of initial data. The asymptotic behavior of the Fourier coefficients is used to quantitatively identify the critical time and location and the type of the singularity. The approach is first tested in detail in 1+1 dimensions for the known case of the Hopf equation, where it is shown that the break-up of the solution can be identified with prescribed accuracy. For dissipative regularizations of this shock formation as the Burgers equation and for dispersive regularizations as the Korteweg–de Vries equation, the Fourier coefficients indicate as expected global regularity of the solutions. The Kadomtsev–Petviashvili (KP) equation can be seen as a dispersive regularization of the dKP equation. The behavior of KP solutions for small dispersion parameter ϵ≪1 near a break-up of corresponding dKP solutions is studied. It is found that the difference between KP and dKP solutions for the same initial data at the critical point scales roughly as ϵ2/7 as for the Korteweg–de Vries equation.

On a new generalization of dispersionless Kadomtsev–Petviashvili hierarchy and its reductions

The dispersionless Kadomtsev–Petviashvili hierarchy is generalized by introducing two new time series γn and σk with two parameters ηn and λk. By this hierarchy, we obtain the first type, the second type as well as mixed type of dispersionless Kadomtsev–Petviashvili equation with self‐consistent sources and their related conservation equations. In addition, the reduction and constrained flow of this new hierarchy are studied. The first type, the second type and the mixed type of dispersionless Korteweg–de Vries equation with self‐consistent sources and of dispersionless Boussinesq equation with self‐consistent sources are obtained. Copyright © 2013 John Wiley & Sons, Ltd.

Holomorphic eigenfunctions of the vector field associated with the dispersionless Kadomtsev–Petviashvili equation

Vector fields naturally arise in many branches of mathematics and physics. Recently it was discovered that Lax pairs for many important multidimensional integrable partial differential equations (PDEs) of hydrodynamic type (also known as dispersionless PDEs) consist of vector field equations. These vector fields have complex coefficients and their analytic, in the spectral parameter, eigenfunctions play an important role in the formulations of the direct and inverse spectral transforms.In this paper we prove existence of eigenfunctions of the basic vector field associated with the celebrated dispersionless Kadomtsev–Petviashvili equation, which are holomorphic in the spectral parameter λ in the strips |Imλ|>C0.

Grassmannians Gr(N − 1, N + 1), closed differential N − 1-forms and N-dimensional integrable systems**In memory of S V Manakov.

Integrable flows on the Grassmannians Gr(N − 1, N + 1) are defined by the requirement of closedness of the differential N − 1-forms ΩN − 1 of rank N − 1 naturally associated with Gr(N − 1, N + 1). Gauge-invariant parts of these flows, given by the systems of the N − 1 quasi-linear differential equations, describe coisotropic deformations of (N − 1)-dimensional linear subspaces. For the class of solutions which are Laurent polynomials in one variable these systems coincide with N-dimensional integrable systems such as the Liouville equation (N = 2), dispersionless Kadomtsev–Petviashvili equation (N = 3), dispersionless Toda equation (N = 3), Plebanski second heavenly equation (N = 4) and others. Gauge-invariant part of the forms ΩN − 1 provides us with the compact form of the corresponding hierarchies. Dual quasi-linear systems associated with the projectively dual Grassmannians Gr(2, N + 1) are defined via the requirement of the closedness of the dual forms Ω⋆N − 1. It is shown that at N = 3 the self-dual quasi-linear system, which is associated with the harmonic (closed and co-closed) form Ω2, coincides with the Maxwell equations for orthogonal electric and magnetic fields.

Wave breaking in solutions of the dispersionless kadomtsev-petviashvili equation at a finite time

We discuss some interesting aspects of the wave breaking in localized solutions of the dispersionless Kadomtsev-Petviashvili equation, an integrable partial differential equation describing the propagation of weakly nonlinear, quasi-one-dimensional waves in 2+1 dimensions, which arise in several physical contexts such as acoustics, plasma physics, and hydrodynamics. For this, we use an inverse spectral transform for multidimensional vector fields that we recently developed and, in particular, the associated inverse problem, a nonlinear Riemann-Hilbert problem on the real axis. In particular, we discuss how the derivative of the solution blows up at the first breaking point in any direction of the plane (x, y) except in the transverse breaking direction and how the solution becomes three-valued in a compact region of the plane (x, y) after the wave breaking.

Frobenius Manifold for the Dispersionless Kadomtsev-Petviashvili Equation

We consider a Frobenius structure associated with the dispersionless Kadomtsev – Petviashvili equation. This is done, essentially, by applying a continuous analogue of the finite dimensional theory in the space of Schwartz functions on the line. The potential of the Frobenius manifold is found to be a logarithmic energy with quadratic external field. Following the construction of the principal hierarchy, we construct a set of infinitely many commuting flows, which extends the classical dKP hierarchy.

On the dispersionless Kadomtsev–Petviashvili equation in n+1 dimensions: exact solutions, the Cauchy problem for small initial data and wave breaking

We study the (n + 1)-dimensional generalization of the dispersionless Kadomtsev–Petviashvili (dKP) equation, a universal equation describing the propagation of weakly nonlinear, quasi-one-dimensional waves in n + 1 dimensions, and arising in several physical contexts, such as acoustics, plasma physics and hydrodynamics. For n = 2, this equation is integrable, and has been recently shown to be a prototype model equation in the description of the two-dimensional wave breaking of localized initial data. We construct an exact solution of the (n + 1)-dimensional model containing an arbitrary function of one variable, corresponding to its parabolic invariance, describing waves, constant on their paraboloidal wave front, breaking simultaneously in all points of it. Then, we use such a solution to build a uniform approximation of the solution of the Cauchy problem, for small and localized initial data, showing that such a small and localized initial data evolving according to the (n + 1)-dimensional dKP equation break, in the long time regime, if and only if 1 ⩽ n ⩽ 3, i.e., in physical space. Such a wave breaking takes place, generically, in a point of the paraboloidal wave front, and the analytic aspects of it are given explicitly in terms of the small initial data.

Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian

We investigate integrable second order equations of the form F(u_{xx}, u_{xy}, u_{yy}, u_{xt}, u_{yt}, u_{tt})=0. Familiar examples include the Boyer-Finley equation, the potential form of the dispersionless Kadomtsev-Petviashvili equation, the dispersionless Hirota equation, etc. The integrability is understood as the existence of infinitely many hydrodynamic reductions. We demonstrate that the natural equivalence group of the problem is isomorphic to Sp(6), revealing a remarkable correspondence between differential equations of the above type and hypersurfaces of the Lagrangian Grassmannian. We prove that the moduli space of integrable equations of the dispersionless Hirota type is 21-dimensional, and the action of the equivalence group Sp(6) on the moduli space has an open orbit.

Cartan’s Structure of Symmetry Pseudo-Group and Coverings for the r-th Modified Dispersionless Kadomtsev–Petviashvili Equation

We derive two non-equivalent coverings for the r-th dKP equation from Maurer–Cartan forms of its symmetry pseudo-group. Also we find Bäcklund transformations between the obtained covering equations.

An interpolating dispersionless integrable system

We introduce a dispersionless integrable system which interpolates between the dispersionless Kadomtsev–Petviashvili equation and the hyper-CR equation. The interpolating system arises as a symmetry reduction of the anti-self-dual Einstein equations in (2, 2) signature by a conformal Killing vector whose self-dual derivative is null. It also arises as a special case of the Manakov–Santini integrable system. We discuss the corresponding Einstein–Weyl structures.

Cauchy problem on the plane for the dispersionless Kadomtsev-Petviashvili equation

We construct the formal solution of the Cauchy problem for the dispersionless Kadomtsev - Petviashvili equation as application of the Inverse Scattering Transform for the vector field corresponding to a Newtonian particle in a time-dependent potential. This is in full analogy with the Cauchy problem for the Kadomtsev - Petviashvili equation, associated with the Inverse Scattering Transform of the time dependent Schroedinger operator for a quantum particle in a time-dependent potential.

Integrable Equations in Nonlinear Geometrical Optics

Geometrical optics limit of the Maxwell equations for nonlinear media with the Cole–Cole dependence of dielectric function and magnetic permeability on the frequency is considered. It is shown that for media with slow variation along one axis such a limit gives rise to the dispersionless Veselov–Novikov equation for the refractive index. It is demonstrated that the Veselov–Novikov hierarchy is amenable to the quasiclassical ‐dressing method. Under more specific requirements for the media, one gets the dispersionless Kadomtsev–Petviashvili equation. Geometrical optics interpretation of some solutions of the above equations is discussed.

Einstein–Weyl spaces and dispersionless Kadomtsev–Petviashvili equation from Painlevé I and II

We present two constructions of new solutions to the dispersionless KP (dKP) equation arising from the first two Painlev\'e transcendents. The first construction is a hodograph transformation based on Einstein--Weyl geometry, the generalised Nahm's equation and the isomonodromy problem. The second construction, motivated by the first, is a direct characterisation of solutions to dKP which are constant on a central quadric. We show how the solutions to the dKP equations can be used to construct some three-dimensional Einstein--Weyl structures, and four--dimensional anti-self-dual null-K\"ahler metrics.

Bi-Hamiltonian structures of d-Boussinesq and Benney-Lax equations

The dispersionless-Boussinesq and Benney-Lax equations are equations of hydrodynamic type which can be obtained as reductions of the dispersionless Kadomtsev-Petviashvili equation. We find that for the three-component reduction, the dispersionless Boussinesq and Benney-Lax equations are the same up to a diffeomorphism. This equivalence becomes manifest when the equations of motion are cast into the form of a triplet of conservation laws. Furthermore, in this form we are able to recognize a non-trivial scaling symmetry of these equations which plays an important role in the construction of their bi-Hamiltonian structure. We exhibit a pair of compatible Hamiltonian operators which belong to a restricted class of Dubrovin and Novikov operators appropriate to a system of conservation laws. The recursion operator for this system generates three infinite sequences of conserved Hamiltonians.