Uniqueness of the Kadomtsev-Petviashvili and Boussinesq Equations

Uniqueness of the Kadomtsev-Petviashvili and Boussinesq Equations

Wen-Xiu Ma^aand Aslı Pekcan^b

aDepartment of Mathematics and Statistics, University of South Florida, Tampa, FL 33620- 5700, USA

bDepartment of Mathematics, Istanbul University, 34134, Vezneciler, Istanbul, Turkey

Reprint requests to W.-X. M.; Tel.: (813)974-9563, Fax: (813)974-2700, E-mail:mawx@cas.usf.edu

Z. Naturforsch.66a,377 – 382 (2011); received December 2, 2010

The Kadomtsev-Petviashvili and Boussinesq equations(uxxx−6uux)x−utx±uyy=0, (uxxx− 6uux)_x+uxx±utt =0,are completely integrable, and in particular, they possess the three-soliton solution. This article aims to expose a uniqueness property of the Kadomtsev-Petviashvili (KP) and Boussinesq equations in the integrability theory. It is shown that the Kadomtsev-Petviashvili and Boussinesq equations and their dimensional reductions are the only integrable equations among a class of generalized Kadomtsev-Petviashvili and Boussinesq equations (ux₁x₁x₁−6uux₁)x₁+

∑^M_i,j=1ai juxixj=0,where theai j’s are arbitrary constants andMis an arbitrary natural number, if the existence of the three-soliton solution is required.

Key words:Integrable Equations; Hirota’s Bilinear Form; Three-Soliton Condition.

PACS numbers:02.30.Ik; 02.30.Xx; 05.45.Yv

1. Introduction

It is interesting to search for nonlinear integrable equations and study their integrable characteristics in mathematical physics. The task is remarkably difficult due to the nonlinearity involved. No general theory is available for dealing with nonlinear differential equa- tions, indeed. Each method focuses on a specific aspect or is based on a specific mathematical subject.

Hirota’s bilinear method [1], however, proposes a direct algebraic approach to nonlinear integrable equations [1–3], and it is pretty powerful in presenting multi-soliton solutions, particularly three-soliton solu- tions [2,4]. It is a common sense that the existence of the three-soliton solution usually implies the integra- bility [5] of the considered equations.

In this article, we will consider a class of gener- alized Kadomtsev-Petviashvili (KP) and Boussinesq equations:

(u_x1x1x1−6uu_x1)_x1+

M i,j=1

∑

a_{i j}u_xixj=0, (1) whereM is a natural number and we assume that the constantsa_{i j}’s satisfy the symmetric propertya_{i j}=a_ji, 1 ≤i,j≤M, without loss of generality. This is the most general class of generalizations of the station- ary Korteweg-de Vries (KdV) equation by adding the

second-order partial derivatives. Using the Hirota bi- linear technique, we would like to show a kind of uniqueness property for the KP and Boussinesq equa- tions

(u_xxx−6uu_x)_x−u_tx±u_yy=0,

(u_xxx−6uu_x)_x+u_xx±u_tt=0, (2) in mathematical physics. That is, we will show that among the above class of nonlinear differential equa- tions, the KP and Boussinesq equations and their di- mensional reductions are the only integrable equations, if the existence of the three-soliton solution is required.

We also mention that Hirota’s bilinear method is used to determine nonlinear superposition formulas for the KP and Boussinesq equations [6,7].

2. The Three-Soliton Condition

A general Hirota bilinear equation reads

P(D_x,D_t,· · ·)f·f =0, (3) wherePis a polynomial in the indicated variables just to satisfy

P(0,0,· · ·) =0, (4)

0932–0784 / 11 / 0600–0377 $ 06.00 c2011 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen·http://znaturforsch.com

and D_x,D_t,· · · are Hirota’s differential operators de- fined by

D_y^pf(y)·g(y) = (∂_y−∂_y⁰)^pf(y)g(y⁰)|_y⁰_=y

=∂_y^p0f(y+y⁰)g(y−y⁰)|_y⁰₌₀, p≥1.

Let us introduce new variables

ηm=kmx+ω_mt+· · ·+ηm,0,m≥1, (5) and define a set of constants

A_mn=−P(p¯_m−p¯_n)

P(p¯_m+p¯_n), m,n≥1, (6) where the involved parameters

¯

p_m= (k_m,ω_m,· · ·), m≥1, (7) satisfy the dispersion relations

P(p¯_m) =0, m≥1, (8)

and ηm,0, m≥1, are arbitrary constant shifts. Obvi- ously, we have the one-soliton and two-soliton solu- tions to the bilinear equation (3):

f=1+εe^η1, f =1+ε(e^η1+e^η2)

+ε²A₁₂e^η1+η2, (9) whereεis an arbitrary perturbation parameter. Noting that

P(D_x,D_t,· · ·)e^η1·e^η2=P(p¯1−p¯2)e^η1+η2, the existence of the two-soliton solution requires that the constantA₁₂ must be determined by (6). The one- periodic and two-periodic wave solutions have the same situation of existence of solutions [8].

However, in general, we do not have the three- soliton solution automatically. Let us fix

f=1+ε e^η1+e^η2+e^η3

+ε² A₁₂e^η1+η2 (10) +A₁₃e^η1+η3+A₂₃e^η2+η3

+ε³A₁₂₃e^η1+η2+η3, whereA₁₂₃=A₁₂A₁₃A₂₃andεis an arbitrary perturba- tion parameter. Then generally we have a three-soliton condition

∑

σ₁,σ₂,σ₃=±1

P(σ₁p¯1+σ2p¯2+σ3p¯3)P(σ₁p¯1−σ2p¯2)

·P(σ₂p¯₂−σ₃p¯₃)P(σ₁p¯₁−σ₃p¯₃) =0,(11) to guarantee the existence of the three-soliton solution (10). If this condition is automatically satisfied, then the considered equation (3) is called integrable in the sense of existence of the three-soliton solution.

Let us now turn back to the class of nonlinear equa- tions defined by (1). It is direct to see that under the

dependent variable transformation

u=−2(lnf)_x1x1 (12) every nonlinear equation defined by (1) can be written as

P(D_x1,D_x2,· · ·,D_xM)f·f=

D⁴_x

1+

M

∑

i,j=1

a_{i j}D_xiD_xj

f·f=0, (13) which exactly gives

f_x1x1x1x1f−4f_x1x1x1f_x1+3f_x²

1x1

+

M i,

∑

a_{i j}(f f_xixj−f_xif_xj) =0.

We assume that the three-soliton solution f to (13) is given by (10) with

ηm=k_mx₁+

M

∑

l_m,jx_j+ηm,0,

A_mn=−R_mn

S_mn, 1≤m,n≤3,

(14)

where

R_mn= (k_m−k_n)⁴+a₁₁(k_m−k_n)² +

M

∑

j=2

2a_1j(k_m−k_n)(l_m,j−l_n,j)

+

M i,j=2

∑

a_{i j}(l_m,i−l_n,i)(l_m,j−l_n,j),1≤m,n≤3, S_mn= (k_m+k_n)⁴+a11(k_m+k_n)²

+

M

∑

j=2

2a_1j(k_m+k_n)(l_m,j+l_n,j)

+

M i,j=2

∑

a_{i j}(l_m,i+l_n,i)(l_m,j+l_n,j),1≤m,n≤3.

Taking advantage of the dispersion relations of (1), P(p¯_m) =0, p¯_m= (k_m,lm,2,· · ·,lm,M), 1≤m≤3,(15) which leads to

k_m⁴ =−a₁₁k²_m−

M

∑

2a1jkmlm,j−

M i,j=2

∑

ai jlm,ilm,j, 1≤m≤3,

(16)

we can expand the three-soliton condition (11) for (1) and show that the three-soliton condition (11) is equiv-

alent to a determinant relation k²₁k²₂k₃²

M i,j,p,q=2

∑

a_{i j}a_pqdet(K,L_i,L_p)det(K,L_j,L_q)

=0,

(17)

where

K= (k₁,k₂,k₃)^T,L_i= (l_1,i,l_2,i,l_3,i)^T,2≤i≤M. (18) The proof is given in the appendix. Obviously, as an example, the three-soliton condition (17) gives rise to

k₁²k₂²k²₃ a22a33−a²₂₃

det(K,L2,L3)²=0, (19) whenM=3 [9].

The condition (17) is an integrability condition for the bilinear equation (13). Not every equation in (1) has this property, and two counterexamples are the (2+1)- dimensional Boussinesq equation [10]

(u_xxx−6uu_x)_x+uxx−utt+uyy=0, (20) and the (3+1)-dimensional KP equation [11]

(u_xxx−6uu_x)_x−u_tx+u_yy+u_zz=0. (21) 3. Uniqueness Property

Based on the above three-soliton condition (17), we would like to prove that for whatever value M, any nonlinear equation defined by (1) can be transformed into one of the KP and Boussinesq equations (2) and their dimensional reductions. This exposes a unique- ness property of the KP and Boussinesq equations in the integrability theory. The result includes all cases of the value ofM, generalizing the caseM≤3 discussed in [9].

In what follows, let us present our proof in five steps.

Step 1: Take an invertible linear transform of inde- pendent variables

X₂=QY₂,X₂= (x₂,· · ·,x_M)^T,

Y₂= (y₂,· · ·,y_M)^T, (22) whereQis an orthogonal matrix transforming the sym- metric matrix

A₂= (a_{i j})2≤i,j≤M (23)

into a diagonal matrix:

Q^TA₂Q=diag(b₂,· · ·,b_M). (24) Therefore, under the transform (22), we have

M i,

∑

a_{i j}u_xixj =

M j=2

∑

b_ju_yjyj, (25) and further, an original equation defined by (1) be- comes

(u_x1x1x1−6uu_x1)_x1+a₁₁u_x1x1+

M

∑

j=2

c_1ju_x1yj (26) +

M

∑

bjuy_jy_j =0

for some constantsc_1j, 2≤j≤M.

Step 2: Now, apply the three-soliton condition (17) to the transformed equation (26), and then we see from the arbitrariness character of the parameters li,j that there is at most one non-zero constant, let us sayb₂, among the coefficientsb_i,2≤i≤M.

Step 3: Assume that there is at least one non-zero constant, sayc₁₃6=0, among the coefficientsc_1j, 3≤ j≤M. Then making another invertible linear trans- form of independent variables

r=x₁, s=y₂,

(t,z₄,· · ·,z_M)^T=R(y₃,y4,· · ·,y_M)^T, (27) where the invertible constant matrixRsatisfies

R(c₁₃,c₁₄,· · ·,c_1M)^T= (c₁₃,0· · ·,0)^T, (28) the transformed equation (26) becomes

(u_rrr−6uu_r)_r+a₁₁u_rr+c₁₂u_rs

+c₁₃u_rt+b₂u_ss=0. (29) This equation withc₁₃=0 corresponds to the trans- formed equation (26) with all c_1j =0, 3 ≤ j ≤M.

Therefore, we only need to consider (29) with arbitrary constant coefficients.

Step 4: Let b2=0. If c12=c13=0, then (29) becomes the stationary Boussinesq equation when a₁₁ 6=0 and the stationary derivative KdV equation

when a11=0, both of which are the dimensional re- ductions of the KP and Boussinesq equations. Other- wise, let us assumec₁₂6=0 without loss of generality, and choose two constantsα andβ satisfying

a₁₁+αc₁₂+βc₁₃=0. (30) Then the invertible linear transform ofr,s, andt,

r⁰=r+αs+βt,t⁰=c12t−c₁₃s,s⁰=s, (31) can transform (29) into

(u_r0r⁰r⁰−6uu_r0)_r0+c12u_r⁰_s⁰=0. (32) This presents the derivative KdV equation – the dimen- sional reduction of the KP equation.

Step 5: Letb₂6=0. Then an invertible linear trans- form of independent variables,

r⁰=r−c₁₂

2b₂s, t⁰=t,s⁰=s, (33) removes the mixed partial-derivative termu_rs, and (29) becomes

(u_r0r⁰r⁰−6uu_r0)_r0+

a11−c²₁₂ 4b₂

u_r⁰_r⁰ +c13u_r⁰_t⁰+b2u_s⁰_s⁰=0.

(34)

Now ifc13=0, then this presents the Boussinesq equa- tion, and it can be further transformed into the standard Boussinesq equation for whatever values of a11, c12, andb₂6=0 [12]. Ifc₁₃6=0, then under a further invert- ible linear transform of independent variables

r⁰⁰=r⁰− a11

c₁₃− c²₁₂ 4c₁₃b₂

t⁰,t⁰⁰=t⁰,s⁰⁰=s⁰, (35) (34) becomes

(u_r00r⁰⁰r⁰⁰−6uu_r00)_r00+c13u_r⁰⁰_t⁰⁰+b2u_s⁰⁰_s⁰⁰=0, (36) which presents the KP equation.

4. Concluding Remarks

To conclude, we discussed a class of generalized KP and Boussinesq equations (1), and proved that among the considered class of equations, the only integrable equations are the KP and Boussinesq equations (2) and their dimensional reductions. This shows that the KP and Boussinesq equations possess a uniqueness prop-

erty in the integrability theory, presenting a kind of par- ticular integrable equations. In particular, the (2+1)- dimensional Boussinesq equation (20) and the (3+1)- dimensional KP equation (21) do not have the three- soliton solution (see also [13,14] for exact solutions to the (3+1)-dimensional KP equation).

In analyzing the existence of the three soliton solu- tion for the generalized KP and Boussinesq equations (1), the difficulty is to compute the three-soliton condi- tion (11), and our success is to rewrite the three-soliton condition (11) as a determinant relation (17), which is put in the appendix. An approach of Darboux transfor- mations [15] could be used to generate multi-soliton solutions directly from the three-soliton solution.

There are various discussions about the (2+1)-di- mensional Boussinesq equation and the (3+1)-dimen- sional KP equation as well as another class of higher- dimensional generalizations of the Boussinesq equa- tion [16]. Those equations are shown to be connected with Ricatti-type integrable ordinary differential equa- tions, and correspondingly, abundant exact solutions can be worked out [16–20].

Acknowledgements

The work was supported in part by the Established Researcher Grant, the CAS faculty development grant, and the CAS Dean research grant of the University of South Florida, Chunhui Plan of the Ministry of Edu- cation of China, the State Administration of Foreign Experts Affairs of China, and the Scientific and Tech- nological Research Council of Turkey.

Appendix: A Proof of the Three-Soliton Determinant Condition

We verify that the three-soliton condition (11) can be written as a determinant relation (17). Noting the even property of the polynomial

P(x₁,x₂,· · ·,x_M) =x⁴₁+

M

∑

i,j=1

a_{i j}x_ix_j

for the generalized KP and Boussinesq bilinear equa- tion (13), the three-soliton condition (11) can be com- puted as follows:

Sum :=1

2

∑

σ₁,σ₂,σ₃=±1

P(σ₁p¯1+σ2p¯2+σ3p¯3)

·P(σ₁p¯1−σ2p¯2)P(σ₂p¯2−σ3p¯3)P(σ₁p¯1−σ3p¯3)

=

∑

(σ₁,σ₂,σ₃)∈S

P(σ₁p¯1+σ2p¯2+σ3p¯3)

·P(σ₁p¯₁−σ₂p¯₂)P(σ₂p¯₂−σ₃p¯₃)P(σ₁p¯₁−σ₃p¯₃),

where S ={(1,1,1),(1,1,−1),(1,−1,1),(−1,1,1)}

and ¯p_m, 1≤m≤3, are defined as in (15). Then we expand it to obtain

Sum =576 k⁶₁k⁶₃k⁴₂+k⁶₂k⁶₃k⁴₁+k⁶₁k₂⁶k₃⁴

+1152a₁₁ k₁⁶k₂⁴k⁴₃+k⁴₁k⁶₂k⁴₃+k⁴₁k⁴₂k⁶₃

+1728a²₁₁k⁴₁k⁴₂k₃⁴ +1152

M j=2

∑

a_1j k³₁k⁶₂k⁴₃l_1,j+k⁶₁k³₂k⁴₃l_2,j+k₁⁴k₂³k⁶₃l_2,j+k⁴₁k⁶₂k₃³l_3,j+k⁶₁k⁴₂k³₃l_3,j+k³₁k⁴₂k⁶₃l_1,j

+2304

M i,

∑

a1ia1j k⁴₁k³₂k⁴₃l2,il3,j+k⁴₃k³₁k₂³l2,jl1,i+k₂⁴k³₁k³₃l1,jl3,i

+2304a₁₁

M

∑

j=2

a_1j k⁴₁k³₂k₃⁴l2,j+k⁴₁k⁴₂k³₃l3,j+k³₁k⁴₂k⁴₃l1,j

+1152a₁₁

M i,

∑

a_{i j} k³₁k³₂k⁴₃l_1,il_2,j+k⁴₁k³₂k₃³l_2,il_3,j+k₁³k⁴₂k³₃l_1,jl_3,i

+2304k³₁k³₂k³₃

M i,p,q=2

∑

a1iapq l1,il2,pl3,q+l1,pl2,ql3,i+l_1,ql2,il3,p

+1152

M

∑

i,j=2

a_{i j} k₁³k³₂k⁶₃l1,il2,j+k⁶₁k³₂k³₃l2,il3,j+k³₁k₂⁶k₃³l1,jl3,i

+576

M

∑

i,j,p,q=2

a_{i j}a_pq 2k³₁k²₃k³₂l_1,il_3,jl_2,pl_3,q+2k³₁k₃³k²₂l_1,il_2,jl_2,pl_3,q+2k²₁k³₃k³₂l_1,il_2,jl_1,pl_3,q

−k⁴₁k²₃k²₂l_2,il_3,jl_2,pl_3,q−k⁴₂k²₃k₁²l_1,il_3,jl_1,pl_3,q−k²₂k₃⁴k²₁l_1,il_2,jl_1,pl_2,q .

Plugging a consequence of the dispersion relations (16), k_m⁶=−a₁₁k⁴_m−

M

∑

j=2

2a_1jk³_mlm,j−

M

∑

i,j=2

a_{i j}k_m²lm,ilm,j, 1≤m≤3, into the above expression and carrying out cancelations, we have Sum =576k²₁k²₂k²₃

k²₂

M

i,

∑

a_{i j}l_1,il_1,j

M

i,

∑

a_{i j}l_3,il_3,j

+k²₁

M

i,

∑

a_{i j}l_2,il_2,j

M

i,

∑

a_{i j}l_3,il_3,j

+k²₃

M

i,

∑

a_{i j}l1,il1,j

M

i,

∑

a_{i j}l2,il2,j

−1152k₁²k²₂k²₃

k1k2

M

i,j=2

∑

a_{i j}l1,il2,j

M

i,j=2

∑

a_{i j}l3,il3,j

+k₁k₃

_M

∑

i,j=2

a_{i j}l_3,il_1,j

_M

∑

i,j=2

a_{i j}l_2,il_2,j

+k₂k₃

_M

∑

i,j=2

a_{i j}l_2,il_3,j

_M

∑

i,j=2

a_{i j}l_1,il_1,j

+576k²₁k₂²k²₃

2k₁k₂

M i,j,p,q=2

∑

a_{i j}a_pql_1,il_3,jl_2,pl_3,q+2k₁k₃

M i,j,p,q=2

∑

a_{i j}a_pql_1,pl_2,jl_2,ql_3,i

+2k2k3 M i,j,p,q=2

∑

a_{i j}a_pql1,ql2,il1,jl3,p−k²₁

M i,j,p,q=2

∑

a_{i j}a_pql2,il3,jl2,pl3,q

−k²₂

M i,j,p,q=2

∑

a_{i j}a_pql1,jl3,il1,ql3,p−k²₃

M i,j,p,q=2

∑

a_{i j}a_pql1,il2,jl1,pl2,q

=576k²₁k₂²k²₃

M

∑

i,j,p,q=2

a_{i j}a_pqdet(K,L_i,L_p)det(K,L_j,L_q).

This implies that the three-soliton determinant condition (17) holds for the generalized KP and Boussinesq bilin- ear equation (13).

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