GideonP.DaspanandMichaelM.Tom ComparisonofKPandBBM-KPModels ResearchArticle

Volume 2007, Article ID 37274,20pages doi:10.1155/2007/37274

Research Article

Comparison of KP and BBM-KP Models

Received 26 January 2007; Accepted 17 May 2007 Recommended by Nils Ackermann

It is shown that the solutions of the pure initial-value problem for the KP and regularized KP equations are the same, within the order of accuracy attributable to either, on the time scale 0≤t≤^−3/2, during which nonlinear and dispersive effects may accumulate to make an order-one relative difference to the wave profiles.

Copyright © 2007 G. P. Daspan and M. M. Tom. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Bona et al. [1] compared the solutions of the Cauchy problem of the Korteweg-de Vries (KdV)

Pt+Px+PPx+Pxxx=0 (1.1)

and the Benjamin-Bona-Mahony (BBM) equation

Qt+Qx+QQx−Qxxt=0. (1.2)

In (1.1) and (1.2),PandQare functions of two real variablesxandt.

Such equations have been derived as models for nonlinear dispersive waves in many different physical contexts, and in most cases where they arise,xis proportional to dis- tance measured in the direction of wave propagation, whiletis proportional to elapsed time. Interest is often focused on the pure initial-value problem for (1.1) and (1.2) in whichPandQare specified for all realxat some beginning value oft, sayt=0, and then the evolution equation is solved fort≥0 subject to the restriction that the solution re- spects the given initial condition. The thrust of their theory was that for suitably restricted

initial conditions, the solutionsPandQemanating therefrom are nearly identical at least for values oftin an interval [0,T], whereTis quite large.

This paper is concerned with mathematical models representing the unidirectional propagation of weakly nonlinear dispersive long waves with weak transverse effects. In- terest will be directed toward two particular models. One is the Kadomtsev-Petviashvili (KP) equations

η_t+η_x+ηη_x+η_xxxx+αη_{y y}=0 (1.3) and the other is a regularized version, the Benjamin-Bona-Mahony (BBM-KP) equation ξ_t+ξ_x+ξξ_x−ξ_xxtx+αξ_{y y}=0, (1.4) whereα= ±1. Ifα=+1 in (1.3), the equation is known as the KP II equation, while for α= −1, it is the KP I equation. The Kadomtsev-Petviashvili equations are two-dimen- sional extensions of the Korteweg-de-Vries equation. They occur naturally in many phys- ical contexts as “universal” models for the propagation of weakly nonlinear dispersive long waves which are essentially one directional, with weak transverse effects. Observe that the linearized dispersion relation for (1.3) is

w(k,l)=k

1−k²+αl² k²

, (1.5)

while that of (1.4) is

w(k,l)= k²+αl²

k1 +k², (1.6)

within the scaling assumption that k²=O(δ), andl=O(δ). As δ→0,w(k,l) may be approximated byw(k,l) to the same order of δ(sincek²l²is of higher order).

The Cauchy problem for these equations have been studied by a number of authors.

Bourgain [2], using Picard iteration, has proved that the pure initial-value problem for the KP II equation is locally well-posed, and hence in light of the conservation laws for the equation, globally well-posed for data inL²(R). The same method has been used to extend local well-posedness to some Sobolev spaces of negative indices.

A compactness method that uses only the divergence form of the nonlinearity and the skew-adjointness of the linear dispersion operator was employed by I ´orio and Nunes in [3] to establish local well-posedness for data inH^s(R²), fors >2, for the KP I equation.

The I ´orio-Nunes approach applies equally well to KP II-type equation. Molinet et al. [4]

using the local well-posedness of I ´orio and Nunes obtained a version of the classical en- ergy method coupled with some of the known conserved quantities and delicate estimates of Strichartz type for the KP I equation to show global well-posedness for KP I equation in the space

Z=

ϕ∈L²R²

:ϕL²+ϕxxx

L²+ϕy

L²

+ϕxy

L²+∂⁻_x¹ϕy

L²+∂⁻_x²ϕy y

L²<∞ .

(1.7)

Kenig [5] improved on this local well-posedness result given by the classical energy esti- mate by showing local well-posedness in the space

Y_s=

ϕ∈L²R²

:ϕL²+J^sϕ_L2+∂⁻_x¹ϕ_yL2<∞ (1.8) fors >3/2, whereJ^sf(k,l)=(1 +k²)^s/2f(k,l).

In [6], global well-posedness is established in Z0=

ϕ∈L²R²

:ϕL²+∂⁻_x¹ϕy

L²+ϕxx

L²+∂⁻_x²ϕy y

L²<∞ (1.9) for the KP I equation. It is worth mentioning the result of Colliander et al. [7], dealing with the well-posedness of KP I equation when the initial data has low regularity. Bona et al. [8] have shown that (1.4) can be solved by Picard iteration yielding to local and global well-posedness results for the associated Cauchy problem. In particular, it is shown that the pure initial-value problem for (1.4), regardless of the sign ofα, is globally well-posed in

W1=

ϕ∈L²R²

:ϕL²+ϕ_xL2+ϕ_xxL2+∂⁻_x¹ϕ_yL2+ϕ_yL2<∞ . (1.10) Saut and Tzvetkov [9] improved this global well-posedness to the space

Y=

ϕ∈L²R²

:ϕ_x∈L²R²

. (1.11)

We remark that providedg satisfies an appropriate constraint, (1.3) and (1.4) are equiv- alent to the integrated forms

ηt+ηx+ηηx+ηxxx+α∂⁻_x¹ηy y=0, ξt+ξx+ξξx−ξxxt+α∂⁻_x¹ξy y=0.

(1.12)

To illustrate the kind of results we have in mind, we briefly outline below what Saut and Tzvetkov generally discussed concerning the relationship between the two models in [9].

Since both KP and BBM-KP equations model weakly nonlinear dispersive long waves which are valid to order², they can be considered as orderperturbations of the linear transport equation, and hence can be written as

η_t+η_x+

ηη_x+η_xxx±∂⁻_x¹η_{y y}=0, (1.13) ξ_t+ξ_x+

ξξ_x−ξ_xxt±∂⁻_x¹ξ_{y y}=0, (1.14)

with initial data η0,ξ0, respectively, which are of order one. The neglected terms in the right-hand sides of (1.13) and (1.14) are of order². After performing the change of variables

M=η, N=ξ, x1=^−1/2x, y1=⁻¹y, τ=^−1/2t, (1.15) one can rewrite (1.12) and (1.13) as

M_τ+M_x1+MM_x1+M_x1x1x1±∂⁻_x1¹M_y1y1=0, N_τ+N_x1+NN_x1−N_x1x1τ±∂⁻_x1¹N_y1y1=0,

(1.16)

with initial data, respectively,

Mx1,y1, 0=η0

^1/2x1,y1

, Nx1,y1, 0=ξ0

^1/2x1,y1

.

(1.17)

The dispersive and nonlinear terms in (1.13) and (1.14) may have a significant influence on the structure of the waves on the time scalet1=⁻¹ (corresponding toτ1=^−3/2), while the neglected order²terms might affect the solution at order one on time scale t2=⁻² (τ2=^−5/2). It is therefore of interest to compare the solutions of (1.13) and (1.14) on time scales betweent1, andt2. Such analysis was performed for the KdV and BBM models in [1].

To give an idea of the results one can expect that we consider the Cauchy problem for the linear versions of (1.13) and (1.14)

ηt+ηx+

+ηxxx±∂⁻_x¹ηy y

=0, ξt+ξx+

−ξxxt±∂⁻_x¹ξy y

=0

(1.18)

together with initial condition

η(x,y, 0)=ξ(x,y, 0)=f(x,y), (1.19) where f is of order one. Taking the Fourier transforms in bothxandyvariables, we have

η_t+ikη+

(ik)³η±−l² ik η

=0, ξ_t+ikξ+

k²ξ_t±il² k ξ

=0,

(1.20)

withη(k,l, 0)=ξ(k,l, 0)=f(k,l). Solving forηandξ, there obtains

η=e^{−it[k−k3±(l2/k)]}f(k,l,t), ξ₌e^{(−it/(1+k2))(k±(l2/k))}f(k,l,t).

(1.21)

It is readily inferred that providedk⁵f(k,l),l²kf(k,l)∈L¹(R²), then

η(·,·,t)−ξ(·,·,t)_∞≤η(k,l,t)−ξ(k,l,t)_L1(_R²)≤²tCf ≤C (1.22) sinceη andξ are of order one, the estimate (1.22) proves that up to time scalet1= ^−(1+δ), 0< δ <1,ηandξare^1−δclose to each other.

2. Notations

We will employ the following notations. We will let| · |p, · s denote the norms in L^p(R²) and the classical Sobolev spacesH^s(R²), respectively, where

f²_s=

R²

1 +k²+l^2s f(k,l)²dk dl (2.1)

andconnotes Fourier transformation. Thus, the norm inL²(R²) will simply be denoted by · 0. ByH^∞, we denote_s≥0H^s. The elements ofH^∞ are infinitely differentiable functions, all of whose derivatives lie inL². IfXis an arbitrary Banach space andT >0, the spaceC(0,T,X) is the collection of continuous functionsu: [0,T]→X. This collection is a Banach space with the norm sup_0≤t≤Tu(t)X, where · Xdenotes the norm inX.

Define the spaceH₋^s₁(R²)= {η∈S(R²) :ηH₋^s1(_R²)<∞}equipped with the norm ηH−^s1(R²)=

R²

1 +|k|⁻¹2

1 +k²+l^2sη(k,l)²dk dl 1/2

. (2.2)

3. Summary of existence theory

As earlier mentioned, the pure initial-value problems for these model evolution equa- tions have been studied. For our analysis, we will use the result of [6] for the Cauchy problem for KP I equation, and the result for the well-posedness of BBM-KP I equation is contained in [8].

We will first consider the initial-value problems for the KP I and BBM-KP I equations η_t+η_x+ηη_x+η_xxxx−η_{y y}=0, (3.1) ξ_t+ξ_x+ξξ_x−ξ_xxtx−ξ_{y y}=0, (3.2)

with initial condition

η(x,y, 0)=ξ(x,y, 0)=g(x,y). (3.3) The theoretical results relating to the initial-value problems (3.1) and (3.2) are presented in the following theorems without proof.

Theorem 3.1. Lets≥2. Then for anyg∈H₋^s₁(R²), there exist a positiveT0=T0(| g|L^∞) (limρ→0T0(ρ)= ∞) and a uniqueηof the integrated KP I equation (3.1) with initial data gon the time interval [0,T0] satisfyingη_∈C([0,T0];H₋^s₁(R²)),η_t∈C([0,T0];H^s−3(R²)).

Furthermore, the Mapg→ηis continuous fromH₋^s₁(R²) toC([0,T0];H₋^s₁(R²)).

Theorem 3.2. Letg∈H₋^s1withs >3/2. Then, there existT0>0 such that the BBM-KP I equation (3.2) has a unique solutionξ∈C([0,T0];H₋^s1(R²)),∂⁻_x¹ξy∈C([0,T0];H₋^s−1¹(R²)), withξ_t∈C([0,T0];H^s−2(R²)). Moreover, the map g→ξ is continuous fromH₋^s₁(R²) to C([0,T0] :H₋^s1(R²)).

4. Main result

In this section, we compare the solutions of the initial-value problems η_t+η_x+ηη_x+η_xxx−∂⁻_x¹η_{y y}=0,

ξ_t+ξ_x+ξξ_x−ξ_xxt−∂⁻_x¹ξ_{y y}=0

(4.1)

both with initial condition

η(x,y, 0)=ξ(x,y, 0)=g^1/2x,y. (4.2) Our main result in this paper is the following.

Theorem 4.1. Let g∈H₋^k+5₁ (R²), wherek≥0 and letη andξ be the unique solutions guaranteed by Theorems3.1and3.2for the initial-value problems (4.1). Then there exist positive constantsCandTwhich depend only onkandgsuch that the solutionsηandξof the initial-value problems

ηt+ηx+ηηx+ηxxx−∂⁻_x¹ηy y=0, ξt+ξx+ξξx−ξxxt−∂⁻_x¹ξy y=0, η(x,y, 0)=ξ(x,y, 0)=g^1/2x,y,

(4.3)

satisfy the inequalities

∂x^jη(·,·,t)−∂x^jξ(·,·,t)₀≤C^j/2+5/4, ∂^jyη(·,·,t)−∂^jyξ(·,·,t)₀≤C^j+5/4

(4.4)

for 0<≤1, and 0≤t≤^−3/2min{T,T0}, where 0≤j≤k.

Before we proveTheorem 4.1, we introduce two new dependent variables u(x,y,t)=⁻¹η^−1/2x+^−3/2t,⁻¹y,^−3/2t,

v(x,y,t)=⁻¹ξ^−1/2x+^−3/2t,⁻¹y,^−3/2t.

(4.5)

A brief calculation shows thatuandvsatisfy, respectively, the initial-value problems u_t+uu_x+u_xxx−∂⁻_x¹u_{y y}=0,

v_t+vv_x+v_xxx−v_xxt−∂⁻_x¹v_{y y}=0, u(x,y, 0)=v(x,y, 0)=g(x,y).

(4.6)

By virtue of Theorems3.1and3.2, the existence and uniqueness ofuandvare assured.

Theorem 4.2. Letg∈H₋^s₁(R²), wheres≥2, then the initial-value problem for both equa- tions (4.6) have solutions inC([0,T0];H₋^s1(R²)) for someT0>0. Moreover, ifg∈H₋^k+51 (R²), then there exist positive constantsCandTdepending only onkandgsuch that the difference u−vsatisfies

∂x^ju−∂x^jv₀≤Ct, ∂y^ju−∂y^jv₀≤Ct

(4.7)

for allandtfor which 0≤≤tand 0≤t≤min[T;T0] where 0≤j≤k.

We will make use of the following anisotropic Sobolev inequalities

|f|∞≤2f^1/40 fy^1/2

0 fxx^1/4

0 , fx

∞≤fxx^1/4

0 fy^1/2

0 fxxx^1/4

0 ,

(4.8)

the proofs of which can be found in [10].

Proof ofTheorem 4.2. Letw=u−v. Then,wis seen to satisfy

wt+wwx+wxxx−wxxt−∂⁻_x¹wy y= −uxxt−(wu)x, (4.9)

w(x,y, 0)=0. (4.10)

We now venture into the task of estimating∂x^jw0+∂^jyw0, for j=0, 1, 2, 3, 4,. . . . In light ofTheorem 3.1, we note that

uk≤cg_H^k_−1(R²₎=A_k. (4.11) This fact justifies the following computations.

We first estimate∂x^jw0. Apply the operator∂x^j to both sides of the differential equa- tion (4.10), multiply the result by∂x^jw, and integrate the result overR²and over [0,t].

After a few integrations by parts, and taking into account the fact thatw(x,y, 0)≡0, it follows that

R²

∂x^jw²+

∂x^j+1w²dx d y=2 _t

0

R²∂x^jw

∂⁽x^j+2)uτ−∂^(j+1)x

wu+1

2w²

dx d y dτ, (4.12) which holds for j=0, 1, 2,. . . . Similarly, apply the operator∂y^j on (4.10), multiply the result by∂y^jw, and integrate the result overR²and over [0,t]. After a few integrations by parts, and taking into account the fact thatw(x,y, 0)≡0, it follows also that

R²

∂^jyw²+

∂^jywx2

dx d y=2 _t

0

R²∂y^jw

∂y^juxxτ−∂^jy

wu+1

2w²

x

dx d y dτ (4.13) forj=0, 1, 2, 3, 4,. . . .The relations (4.12) and (4.13) will be used repeatedly.

First, for j=0, (4.12) or (4.13) may be used, since they will both given the same esti- mate. Making use of (4.12), there appears after two integrations by parts that

R²

w²+w²_xdx d y= _t

0

R²

2 wuxxτ

−

uxw²dx d y dτ, (4.14) from this, the following inequality is derived

w²0≤ _t

0

2w0u_xxτ0+u_x∞w²0

dτ. (4.15)

By a variant of Gronwall’s lemma, it follows that w0≤

C2

C1

e^C1t−1≤tC2e^C1=M0t, (4.16)

whereC1andC2are bounds for (1/2)ux∞anduxxt0, respectively, andtis restricted to the range [0,1].

Using the equation satisfied byu, the following estimates can be derived:

sup

t≥0

u_xxt0=sup

t≥0

u_{xy y}−3u_xu_xx−uu_xxx−u_xxxxx0

≤sup

t≥0

uxy y

0+ 2u^1/40 uy^1/2

0 uxx^1/4

0 uxxx

0

+ 6u_xx^5/4₀ u_y^1/2₀ u_xxxx^1/4₀ +u_xxxxx0.

(4.17)

Hence,C2may be defined by sup_t≥0uxxt0≤C2. ForC1, we estimate it from the aniso- tropic Sobolev inequality

ux

∞≤2uxx^1/4

0 uy^1/2

0 uxxxx^1/4

0 ≤C1. (4.18)

We therefore infer by an application of Gronwall’s lemma that w0≤

C2

C1

e^C1t−1≤tC2e^C1=M0t≤B (4.19)

for 0≤t≤1.

Forj=1, integrate (4.12) by parts to get the following relation:

R²

w_x²+w²_xxdx d y= _t

0

R²

2w_xu_xxxτ−w³_x−3w_x²u_x−ww_xu_xxdx d y dτ. (4.20)

Similarly, forj=1, integrate (4.13) by parts to get

R²

w²_y+w²_xydx d y= _t

0

R²

2w_yu_xxyτ−2w_xw_yu_y+w²_yu_x+w_ywu_xy−w²_yw_xdx d y dτ.

(4.21) Adding these two equations above, we obtain

R²

w²_x+w²_y+w_xx² +w²_xydx d y

= _t

0

R²

2w_xu_xxxτ−w³_x−3w_x²u_x−w_xwu_xx+ 2w_yu_xxyτ

−2wxwyuy+w²_yux+wywuxy−w²_ywx

dx d y dτ.

(4.22)

The integrand on the right-hand side of (4.22) may be bounded above by 2w_x0u_xxxτ0+ 2w_y0u_xxyτ0+4u_x∞+v_x∞w_x²₀

+2u_x∞+v_x∞w_y²₀+w_x0w0u_xx∞+w_y0w0u_xy∞ + 2wx

0wy

0uy

∞.

(4.23)

Also from the anisotropic Sobolev inequalities, we infer that u_y∞≤2u_y^1/4₀ u_{y y}^1/2₀ u_xxy^1/4₀ ≤C, u_x∞≤2u_xx^1/4₀ u_y^1/2₀ u_xxx^1/4₀ ≤C, u_xx∞≤2u_xxx^1/4₀ u_xy^1/2₀ u_xxxx^1/4₀ ≤C, u_xy∞≤2u_xxy^1/4₀ u_{y y}^1/2₀ u_xxxy^1/4₀ ≤C.

(4.24)

Now using the equation satisfied byu, we may derive the following estimates, valid for 0≤τ≤1:

sup

t≥0

uxxxτ

0

≤sup

t≥0

u_{xxy y0}+ 2u^1/40 u_y^1/2₀ u_xx^1/4₀ u_xxxx0

+ 8u_xx^1/4₀ u_y^1/2₀ u_xxxx^1/4₀ u_xxx0+ 3u_xx²₀+u_xxxxxx0≤C, sup

t≥0

uxxyτ

0

≤sup

t≥0

u_xxxy0+ 4u_xy^3/2₀ u_xxx^1/4₀ u_xxxx^1/4₀

+ 6u_xx^1/4₀ u_y^1/2₀ u_xxx^1/4₀ u_xxy0+ 2u_y^1/4₀ u_{y y}^1/2₀ u_yxx^1/4₀ u_xxx0 + 2u_x^1/4₀ u_y^1/2₀ u_xx^1/4₀ u_xxxy0+u_xxxxxy0+u_{y y y0}≤C.

(4.25) Putting all these estimates together, the right-hand side of (4.20) may be bounded above by

2C4w_x0+w_y0+C3w_x²₀+w_y²₀, (4.26)

whereC3andC4are order-one quantities. The following is then seen to hold from (4.19) wx²

0+wy²

0≤ _t

0

2C4wx

0+wy

0

+C3wx²

0+wy²

0

dτ. (4.27)

LetA1(t)=[wx²0+wy²0]^1/2. Observe that

w_x0+w_y0²=w_x²₀+w_y²₀+ 2w_x0w_y0≤2w_x²₀+w_y²₀, (4.28) so that

wx

0+wy

0≤√

2A1(t). (4.29)

Therefore, d

dtw_x²₀+w_y²₀≤2C4w_x0+w_y0+C3w_x²₀+w_y²₀, (4.30) which is equivalent to

d

dtA²1(t)≤2C4

√2A1(t) +C3A²1(t) (4.31) or

d

dtA1(t)≤C4

√2 +1

2C3A1(t), (4.32)

and hence by a variant of Gronwall’s inequality, we obtain A1(t)≤2^√2

C4

C3

e^(1/2)C3t−1≤2^√2tC4e^(1/2)C3=tM1. (4.33)

We thus infer that

wx

0≤tM1, w_y0≤tM1

(4.34) for 0≤t≤1. SinceC3andC4are order-one quantities,M1is also an order-one quantity.

For j=2, integrate (4.12) and (4.13) by parts, combine the two operations to get the following:

R²

w²_xx+w_xxx² +w²_{y y}+w_{xy y}² dx d y

= _t

0

R²

2w_xxu_xxxxτ+ 2w_{y y}u_{y yxxτ}+ 6w_xw_xxu_xx−4w_yw_{y y}u_xy + 2wwxxuxxx−wwy yuxy y−4wxw²_xx−2w²_{y y}ux

dx d y dτ.

(4.35)