1 Introduction

## Abstract

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 dimensions and with quadratic nonlinearity, this equation is integrable through a novel IST, 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 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 . At last, the analytic aspects of such a 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 discontinuous shock. These results, contained in the 2012 master thesis of one of the authors (FS) [1], generalize those obtained in [2] for the dKP equation in dimensions with quadratic nonlinearity, and are obtained following the same strategy.

On the dispersionless Kadomtsev-Petviashvili equation with arbitrary nonlinearity and dimensionality:

exact solutions, longtime asymptotics of the Cauchy problem, wave breaking and discontinuous shocks

F. Santucci and P. M. Santini

Istituto Superiore IIS G.Marconi, via Reno snc, 04100 Latina, Italy

Dipartimento di Fisica, Università di Roma ”La Sapienza”, and

Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1

Piazz.le Aldo Moro 2, I-00185 Roma, Italy

e-mail: paolo.santini@roma1.infn.it

March 6, 2018

## 1 Introduction

In this paper we investigate the following generalized dispersionless Kadomtsev - Petviashvili equation

 (ut+umux)x+△⊥u=0,  u=u(x,→y,t)∈R,,△⊥:=∑n−1i=1∂2yi,   →y=(y1,⋯,yn−1)∈Rn−1,  x,t∈R, (1.1)

in dimensions, where is the degree of the nonlinearity, and . Hereafter, we refer to (1.1) as the equation. Equations (1.1) contain, as particular cases, the integrable (through the method of characteristics) Riemann equation [3]

 ut+umux=0 (1.2)

for , the integrable dispersionless Kadomtsev - Petviashvili (dKP) equation [4, 5, 6, 7]

 (ut+uux)x+uyy=0 (1.3)

for , the nonintegrable Khokhlov - Zobolotskaya (KZ) equation [6]

 (ut+uux)x+uy1y1+uy2y2=0 (1.4)

for , and the nonintegrable modified dKP (mdKP) equation [8]

 (ut+u2ux)x+uyy=0 (1.5)

for .

Equation (1.1) describes weakly nonlinear and quasi one-dimensional waves, with negligeable dissipation and dispersion, if the linear approximation of the original theory is given by the - dimensional wave equation, at least in some limiting cases like the long wave approximation [2]. In equation (1.1) there are two competing terms: the nonlinear term , responsible for the steepening of the profile, and the term, describing diffraction in the transversal dimensional hyperplane (reminiscence, through the multiscale expansion leading to (1.1), of the diffraction described by the wave operator); therefore diffraction increases as increases [2]. The degree of the nonlinearity originates from expanding the nonlinear terms of the original PDEs in power series, when the first powers of the expansion are absent due, usually, to some symmetry of the problem. In this sense, the degree of universality of (1.1) decreases as the degree of nonlinearity grows (see §2). are relevant in acoustics [4, 5, 6, 2], and also in plasma physics [7, 2, 1] in the long wave approximation; in hydrodynamics, again in the long wave approximation [9, 10]; and in nonlinear optics [1] and in the study of sound waves in antiferromagnets [11].

We remark that equation (1.1) arises as the -dispersionless limit of the natural generalization in dimensions and with nonlinearity of degree :

 (ut+uxxx+umux)x+△⊥u=0,  u=u(x,→y,t)∈R. (1.6)

of the dimensional Kadomtsev-Petviashvili (KP) equation [7, 12, 9, 10].

Apart from the Riemann equation (1.2), integrable by the method of characteristics, in the family of equations (1.1) only the dKP equation (1.3) is integrable through a novel Inverse Scattering Transform (IST) for integrable dispersionless PDEs [13, 14, 15], recently made rigorous in [16] on the example of the Pavlov equation [17]. This IST allows one to show, in particular, that solutions of dKP depend on through the combination [18]; i.e., these solutions can be written in the characteristic form [18, 19]

 u=F(ζ,y,t),  ζ=x−2F(ζ,y,t)t, (1.7)

in analogy with the case of the Riemann equation (1.2), for which the dependence of the solution on is through the combination . For this reason, the IST for dKP can be viewed as a significant generalization of the method of characteristics. The formulation (1.7) has allowed one to study in an analytically explicit way the interesting features of the gradient catastrophe of two dimensional waves at finite time [18, 19] and in the longtime regime [18] in terms of the initial data.

The other examples of equations are not integrable; therefore the possibility to investigate a generic wave breaking through equations like (1.7) and the precise form that these equations should take are, in our opinion, challenging open problems; in addition, blow up of the solutions is expected for sufficiently large [8] to complicate the picture. In the recent paper [20], f.i., the formal dependence , motivated by the independent limit, was used to study the generic breaking features of and its dispersive shock formation.

In our paper we give some light on the problem of finding a convenient characteristic form of the type (1.7) for the study of wave breaking of solutions, i) from the construction of a family of exact solutions of exhibiting wave breaking, and ii) from the construction of the longtime behavior of solutions of the Cauchy problem of , for small initial data.

Indeed, after showing in §2 the universality of (1.5) starting from a family of nonlinear wave equations, in §3 we use the invariance of the equations under motions on the paraboloid, to construct a family of exact solutions involving an arbitrary function of one variable, and describing waves constant on their paraboloidal wave front, breaking simultaneously in all points of it, and developing, after breaking, either multivalued overturning profiles or single valued discontinuous shocks. Then we use in §4 such solutions to build a uniform approximation of the solution of the Cauchy problem, in the longtime regime and for small and localized initial data, showing that such a small and localized data evolving according to the equation break, in the longtime regime, iff . In §5 we study the analytic aspects of such a wave breaking, given explicitely in terms of the initial data, providing, in particular, a description of the overturning profile and of the development of a discontinuous shock immediately after breaking; we concentrate, in particular, on the mdKP (1.5) case and on its comparison with the already known dKP (1.3) case. The results of this paper, contained in the 2012 master thesis of one of the authors (FS) [1], generalize analogous ones for the equation in [2], and are obtained following the same strategy.

We end this introduction considering the following consequence of these results: for the family of exact solutions of and for the longtime asymptotics of the solutions of the small data Cauchy problem, the dependence of the solutions on is, f.i., through the combinations , if , and , if , where

 cm,n=1−m(n−1)2. (1.8)

Therefore this dependence involves in a simple way the degree of the nonlinearity and the dimensionality of the problem through the coefficient , and it is consistent with the case of the Riemann equation (1.2), for which , and with the dKP equation (1.3), for which (confirming the apparently mysterious factor in (1.7)); it is not consistent, instead, with the formal dependence used in [20].

In analogy with the dKP case, the results of our paper seem to suggest the following conjecture (admittedly a weak conjecture, due to the absence, in the generic case, of an IST generalizing the method of characteristics).
Conjecture. The solutions of the equation around breaking, for , are described by the following characteristic formulae

 u=G(ζ,y,t),ζ={x−c−1m,nGm(ζ,→y,t)t,%ifcm,n>0,x−Gm(ζ,→y,t)tlogt,if cm,n=0. (1.9)

This paper is dedicated to the memory of S. V. Manakov.

## 2 Universality and applicability of dKP(m,n)

The universality (and therefore the applicability) of (1.1), through a multiscale expansion, is well described by the following simple model: the family of nonlinear wave equations

 (f(w))TT=△w,△=n∑i=1∂2Xi,w=w(→X,T). (2.1)

For small amplitudes: , , we substitute in (2.1) the nonlinear term by its Taylor expansion

 f(ϵw)=f(0)+f′(0)ϵw+12f′′(0)ϵ2w2+⋯. (2.2)

obtaining, at , the wave equation

 wTT=c2△w,c=1/√f′(0), (2.3)

where is assumed to be positive.

If the waves are quasi one-dimensional and we choose as the direction of propagation, the wave lengths in the trasversal directions are small: , where is the transversal wave vector and has to be fixed. Then the dispersion relation becomes [2]

 ω=c√k21+→k2⊥=ck1 ⎷1+ϵ2α→κ⊥2k21≃ck1(1+ϵ2α→κ2⊥2k21), (2.4)

and the phase of a monochromatic wave reads

 →k⋅→X−ωT=k1(X1−cT)+ϵα→κ⊥⋅→X⊥−cϵ2α2→κ⊥2k21T, (2.5)

motivating the introduction of the new variables

 ⎧⎪ ⎪⎨⎪ ⎪⎩x =X1−cT,→y =ϵα→X⊥,  yi=Xi+1, i=1,…,n−1,t =ϵ2αc2T. (2.6)

Rewriting (2.1) in the new variables and imposing to get the maximal balance, one obtains, at , the equation

 (ut+uux)x+△⊥u=0, (2.7)

where , and is the tranversal Laplacian given in (1.1b).

If the term vanishes, the maximal balance must involve the cubic term; if also , the maximal balance must involve the quartic term, and so on. In the very special case in which , , and , then the maximal balance must involve the term of order , and is achieved for at the first nontrivial order , obtaining the equation (1.1), with and

 C:=−c2f(m+1)(0)m!, (2.8)

if is odd, and

 (ut−sgn(f(m+1)(0)) umux)x+△⊥u=0, (2.9)

with , if is even.

The above considerations explain well why equations (1.1) are less and less universal and applicable, as increases.

## 3 Exact solutions of dKP(m,n)

It was observed in [2] that is invariant under the following Lie symmetry group of transformations

 ⎧⎪ ⎪⎨⎪ ⎪⎩~x =x+∑n−1i=1(δiyi−δ2it),~yj =yj−2δjt,j=1,⋯,n−1~t =t, (3.1)

where are the arbitrary parameters of the group, leaving invariant the paraboloid

 ξ=x+14tn−1∑i=1y2i. (3.2)

Such a symmetry was used to construct a family of exact and implicit solutions of exhibiting wave breaking and playing a relevant role in the longtime regime of the small data Cauchy problem for [2].

Since is also invariant under the transformation (3.1), following the same strategy as in [2], such a symmetry will be used in this section to construct a family of exact and implicit solutions of . In this paper we add, to the exact solutions exhibiting a gradient catastrophe and multivaluedness after breaking, also the exact weak solutions of developing, after breaking, single valued discontinuous shocks of dissipative nature.

Therefore we look for solutions of (1.1) in the form

 u=v(ξ,t),   ξ=x+14tn−1∑i=1y2i, (3.3)

reducing (1.1) to the dimensional PDE

 vt+vmvξ+n−12tv=0. (3.4)

The following change of variables

 v(ξ,t)=t−n−12q(ξ,τ), (3.5)

where

 τ(t)={1cm,n tcm,n+α,%ifcm,n≠0lnt+β,if cm,n=0, (3.6)

are real constants and the coefficient is defined in (1.8), transforms (3.4) into the Riemann equation (1.2) in the variables

 qτ+qmqξ=0. (3.7)

We recall [3] that (3.7) has the general implicit solution

 q=A(ζ),  ζ=ξ−Am(ζ)τ   ⇔   q=A(ξ−qmτ), (3.8)

where is an arbitrary differentiable function of one argument. If, in particular, describes a localized positive hump, the solution breaks first at , on the characteristic , where

 τb=min ζ(−1mAm−1(ζ)A′(ζ))=−1mAm−1(ζb)A′(ζb)>0. (3.9)

For , the solution becomes multivalued and not acceptable in many physical contexts. Alternatively, for , the regular multivalued solution can be replaced by a weak solution, a single valued discontinuous shock of dissipative nature, whose wave front discontinuity is described by equations [3]

 dsdτ=1m+1Am+1(ζ1)−Am+1(ζ2)A(ζ1)−A(ζ2),s=ζ1+Am(ζ1)τ=ζ2+Am(ζ2)τ,\par (3.10)

with the initial conditions

 s(τb)=ξb,  ζ1(τb)=ζ2(τb)=ζb. (3.11)

Behind and ahead the shock, the solution is given by

 q={q2=A(ζ2),if ξs(τ), (3.12)

where and are respectively the maximum and the minimum among the (three, in the case of a single hump) branches of the implicit equations (3.8).

We remark that in (3.6) is a monotonically increasing function of ; to have it positive, we choose the constants and the -intervals as follows:

 τ(t)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩1cm,n tcm,n, t>0,% if cm,n>0,lnt, t>1,if cm,n=0,1|cm,n|(t−|cm,n|0−t−|cm,n|), t>t0,if cm,n<0, (3.13)

Recalling (3.3), (3.5), and (3.13), the exact solutions of in the original variables corresponding to (3.8) read

 u=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩t−n−12A(x+14t∑n−1i=1y2i−1cm,numt), t>0,if cm,n>0t−n−12A(x+14t∑n−1i=1y2i−umtlnt), t>1,if cm,n=0t−n−12A(x+14t∑n−1i=1y2i−1|cm,n|umt((tt0)|cm,n|−1)), t>t0,if cm,n<0, (3.14)

where is defined in (1.8).

If, again, is a localized hump, (3.14) describes a wave constant on the paraboloidal wave front (3.2) and simultaneously breaking on it. More precisely, the solution breaks at time defined by

 tb={(cm,nτb)1/cm,n,if cm,n>0,eτb,if cm,n=0, (3.15)

on the paraboloid

 x+14tn−1∑i=1y2i=ζb+Am(ζb)τb, (3.16)

where are defined in (3.9) (see Figures 1).

Figures 1. Plotting of the exact solution of the mdKP equation (1.5) (), before, at and after breaking. Here we have chosen ; consequently: . If , as , and the wave freezes asymptotically. If , then , and the wave breaks before freezing at

 tb=(t−|cm,n|0−τb|cm,n|)−1/|cm,n|, (3.17)

on the paraboloid (3.16). If, instead, , then and no breaking takes place before the wave freezes.

If , (3.14) reduces to the general solution of the Riemann equation (1.2); if , (3.14) reduces to the class of particular solutions of the dKP equation with quadratic nonlinearity and arbitrary dimensions () constructed in [2]. It is interesting to remark that this type of exact solutions was first derived for the integrable dKP equation (1.3) using its nonlinear Riemnann - Hilbert inverse problem [21].

After breaking: , the regular multivalued solutions (3.14) can be replaced by the discontinuous single valued solutions of obtained using (3.3), (3.5), and (3.13) in (3.10), (3.11), (3.12). More precisely, for , where is defined in (3.15) and (3.17), the solution described by (3.14) for , develops a discontinuous shock on the parabola

 x+y24t=S(t),   S(t)≡s(τ(t)), (3.18)

where is characterized by equations (3.10), (3.11), and is defined in (3.13). Behind and ahead the shock, the solution is given by

 u=⎧⎪⎨⎪⎩u2(x,→y,t),if x+y44tS(t), (3.19)

where and are respectively the maximum and the minimum among the (three, in the case of a localized bump) branches of the implicit equations (3.14).

We end this section remarking that, as already mentioned in §1, in the above class of exact solutions of (1.1) the dependences of on through the combinations (if ), (if ), and , if , are not consistent with the formal dependence recently postulated to study the breaking features of in [20].

## 4 The Cauchy problem for small and localized initial data and longtime wave breaking

In analogy with [2], now we use the exact solutions of the previous section to construct the longtime behaviour of solutions of the Cauchy problem of (1.1) for small and localized initial data, showing in particular that small and localized initial data evolving according to (1.1) break, in the longtime regime, only if ().

For small and localized initial data of the type

 u(x,→y,0)=ϵu0(x,→y),  0<ϵ≪1 (4.1)

evolving according to equation (1.1), the behavior of the solution may be approximated by the solution of the linearized equation until nonlinearity becomes relevant.

For initial data, equations (3.7),(3.9) imply that the breaking time is

 τb=O(ϵ−m)≫1; (4.2)

so, the nonlinear regime for is characterized by the condition

 t=O(τ−1(ϵ−m))=⎧⎪⎨⎪⎩O(ϵ−mcm,n),cm,n>0O(eϵ−m),cm,n=0 (4.3)

where is the inverse of (3.6), while, for , large values of are not compatible with (4.2), and a nonlinear regime never occurs.

So, for , the solution of the Cauchy problem is approximated by the solution of the linearized equation, while for , a matching will be made between the linearized solution and the exact solution (3.14).

In the linear regime, when , the solution is well described by that of the linearized , i.e.:

 u(x,→y,t)∼ϵ(2π)n∫Rn^u0(k1,→k⊥)exp[ı(k1x+→k⊥⋅→y−k2⊥k1t)]dk1d→k⊥ (4.4)

where

 ^u0(k1,→k⊥)=∫Rnu0(x,→y)exp[−ı(k1x+→k⊥⋅→y)]dxd→y.

In the longtime regime

 1≪t≪O(τ−1(ϵ−m)), (4.5)

the multiple integral can be evaluated by the stationary phase method, which gives [2]

 u(x,→y,t)∼t−n−12ϵG(x+14tn−1∑i=1y2i,→y2t), (4.6)

where

 G(ξ,→η):=12nπn+12∫Rdk1|k1|n−12^u0(k1,→ηk1)exp[ık1ξ−ı(n−1)π4sgn(k1)]. (4.7)

in the space-time region

 (x−ξ)/t, yi/t=O(1),i=1,…,n−1, (4.8)

on the paraboloid (3.2), and decays faster outside it. Therefore, small and localized initial data set, in the longtime regime (4.5) governed by the linear theory, on the paraboloid (3.2).

When nonlinearity becomes relevant, i.e., when , the asymptotic solution of is built matching the longtime solution (4.6),(4.7) of the linearized Cauchy problem with the exact solution (3.14). If, f.i., , the first argument of function defined in (4.7) is replaced by , or, equivalently, the arbitrary function in (3.8) acquires the dependence on the second argument , and is identified with . As a result of this matching, in the nonlinear regime (4.2), (4.3) the solution reads

 u(x,→y,t)≃uasm,n(x,→y,t),uasm,n(x,→y,t)≡⎧⎪ ⎪⎨⎪ ⎪⎩t−n−12ϵG(x+14t∑n−1i=1y2i−1cm,numt,→y2t),cm,n>0,t−n−12ϵG(x+14t∑n−1i=1y2i−umtlnt,→y2t),cm,n=0, (4.9)

with for , and for , becoming infinitesimal in the linear regime, as it has to be.

For , the solution is described by its linear form (4.6); therefore wave breaking takes place for the particular values of described by the condition

 cm,n≥0   ⇔   m(n−1)≤2; (4.10)

i.e., for

 n=1 n=2 n=3 ∀m m=1,2 m=1
(4.11)

For , reduces to the general solution of the Riemann equation (1.2) [3]. The cases and , the dKP and the KZ equations, have been already investigated in [18] and [2] respectively; therefore, in the following section we mainly focus on the case , corresponding to the nonintegrable modified dKP equation (1.5) and to its asymptotic solution

 u≃uas2,2(x,y,t)=1√tϵG(x+y24t−u2tlnt,y2t), (4.12)

comparing the results with the dKP case investigated in [18], with its asymptotic solution

 u≃uas1,2(x,y,t)=1√tϵG(x+y24t−2ut,y2t) (4.13)

Remark 1. The estimate of the first correction to the asymptotics (4.9) reads

 u=uasn,m(x,→y,t)(1+O(t−1)). (4.14)

Indeed, if , one has

 u∼uasn,m(x,→y,t)+ϵ1+mcm,ntn−12H(x+14tn−1∑i=1y2i,→y2t,ˇτ), (4.15)

where and is expressed in terms of ( is defined in the second of equations (5.1)), through the PDE

 (Hˇτ+GmHξ1+mGm−1Gζˇτ)ξ+14(cm,nˇτ)1+c−1m,nn−1∑j=1(Gηj1+mGm−1Gζˇτ)ηj=0. (4.16)

Since, from (4.3), , equation (4.15) yields (4.14). Similarly, if ,

 u∼uasn,m(x,→y,t)+ϵtn+12H(x+14tn−1∑i=1y2i,→y2t,ˇτ), (4.17)

where now solves the PDE

 (H1+mGm−1Gζˇτ)ξ=14n−1∑j=1(Gηj1+mGm−1Gζˇτ)ηj. (4.18)

Then equation (4.17) yields equation (4.14) too. Remark 2. As is was observed in [2], for a gaussian initial condition

 u0(x,y)=d exp[−x2+→y24], (4.19)

where is constant, the asymptotic solution can be written in terms of special functions, obtaining for :

 G(ξ,→η)=d√π1(1+→η2)n+14[cos(π4(n−1))Γ(n+14) 1F1(n+14,12,−Y24)+sin(π4(n−1))Γ(n+34)Y 1F1(n+34,32,−Y24)] (4.20)

where

 Y=ξ√1+→η2, (4.21)

is the Euler gamma function and is the Kummer confluent hypergeometric function [22, 23]. For and , it yields, respectively:

 G(ξ,η)=1(1+η2)34[Γ(34) 1F1(34,12,−Y24)+YΓ(54) 1F1(54,32,−Y24)]. (4.22)

Remark 3. The fact that small data break only when the inequality (4.10) is satisfied was expected from physical considerations. Indeed, as already mentioned in the introduction, in equation (1.1) there are two competing terms: the nonlinear term , responsible for the steepening of the profile, and the term, describing diffraction in the transversal dimensional hyperplane (reminiscence, through the multiscale expansion leading to (1.1), of the wave operator); therefore diffraction increases as increases [2]. For small initial data, the term is initially smaller and smaller increasing , and the solution evolves in a linear way for a longer and longer time, diffracting transversally through the diffraction channels. So, diffraction, increasing with , acts on a very long time, increasing with , before the nonlinear regime could become relevant, and if and/or are sufficiently large, one expects that the solution would be diffracted away almost completely, before reaching the nonlinear regime, and will not break.
We have, in particular, that breaking takes place in the following longtime regimes (see (4.3)):
(the Riemann equation with quadratic nonlinearity): ;
(the dKP equation): ;
(the KZ equation): ;
(the Riemann equation with cubic nonlinearity): ;
(the mdKP equation): .

It is interesting to remark that, as it was observed in [2], if (the physically more relevant case), wave breaking takes place only for ; i.e., in physical space!

We end this section remarking that also in the longtime behaviour of the solutions of the small data Cauchy problem for equation (1.1), the dependence of the solution on , as in the case of the exact solutions (3.14), is not consistent with the formal ansatz recently made in [20].

## 5 Analytic aspects of the wave breaking

In this section we use the longtime behavior of solutions of the small data Cauchy problem for to show explicitly the analytic aspects of the solution in the neighbourhood of the breaking time in terms of the initial data, represented by function G defined in (4.7), as it was already done in [18, 2].

Choosing , we rewrite equation (4.9) in the characteristic form

 q∼ϵG(ζ,→η),ξ=ϵmF(ζ,→η;m)τ+ζ,  F(ζ,→η;m)≡Gm(ζ,→η), (5.1)

where

 q=utn−12,ξ=x+14tn−1∑j=1y2j,→η=→y2t (5.2)

and

 τ(t)={1cm,n tcm,n,if cm,n>0,lnt,if cm,n=0. (5.3)

One solves the second of equations (5.1) with respect to the parameter , obtaining and replaces it into the first, to obtain the solution . The inversion of the second of equations (5.1) is possible iff its -derivative is different from zero. Therefore the singularity manifold (SM) of the solution is the - dimensional manifold characterized by the equation

 S(ζ,→η,τ)≡1+ϵmFζ(ζ,→η,m)τ=0   ⇒   τ=−1ϵmFζ(ζ,→η,m). (5.4)

Since

 ∇(ξ,→η)q=ϵ∇(ζ,→η)G(ζ,→η)1+ϵ