Xiao-Bo Wang(王曉波), Man Jia(賈曼), and Sen-Yue Lou(樓森岳)

School of Physical Science and Technology,Ningbo University,Ningbo 315211,China

Keywords: Painlev′e property, residual symmetry, Schwartz form, B¨acklund transforms, D’Alembert waves,symmetry reductions,Kac–Moody–Virasoro algebra,(2+1)-dimensional KdV equation

1. Introduction

Recently, a novel (2+1)-dimensional Korteweg–de Vries (KdV) extension, the combined KP3 (Kadomtsev–Petviashvilli)and KP4(cKP3-4)equation

In Ref.[1],the multiple solitons of the model(1)are obtained by using Hirota’s bilinear approach. Applying the velocity resonant mechanism[11–13]to the multiple soliton solutions, the soliton molecules with arbitrary number of solitons are also found in Ref[1]. It is further discovered that the model permits the existence of the arbitrary D’Alembert-type waves which implies that there are one special type of solitons and soliton molecules with arbitrary shapes but fixed modeldependent velocity.

In this paper, we investigate other significant properties such as the Painlev′e property (PP), Schwartz form, B¨aclund transformations, infinitely many local and nonlocal symmetries, Kac–Moody–Virasoro symmetry algebras, groupinvariant solutions, and symmetry reductions for the cKP3-4 equation (1). To study the PP of a nonlinear partial differential equation system, there are some equivalent ways such as the Weiss–Tabor–Carnevale(WTC)approach,[14]Kruskal’s simplification,Conte’s invariant form,[15]and Lou’s extended method.[16]

The rest of the present article is arranged as follows. In Section 2, the PP of Eq. (1) is tested by using the Kruskal’s simplification. Using the truncated Panlev′e expansion, one can find many interesting results for integrable systems including the B¨acklund/Levi transformation, Schwarz form, bilinearization,and Lax pair. In Ref.[17],it is found that the nonlocal symmetries,the residual symmetries can also be directly obtained from the truncated Painlev′e expansion. The residual symmetries can be used to find Dabourx transformations[18,19]and the interaction solutions between a soliton and another nonlinear wave such as a cnoidal wave and/or a Painlev′e wave.[20,21]In Section 3, the nonlocal symmetry (the residual symmetry)is localized by introducing a prolonged system.Whence a nonlocal symmetry is localized, it is straightforward to find its finite transformation which is equivalent to the B¨acklund/Levi transformation. In Section 4, it is found that similar to the usual KP equation,the general Lie point symmetries of the cKP3-4 equation possess also three arbitrary functions of the time t and constitute a centerless Kac–Moody–Virasoro symmetry algebra. Using the general Lie point symmetries, two special types of symmetry reductions are found.The first type of(1+1)-dimensional reduction equation is Laxintegrable with the fourth-order spectral problem. The second type of the symmetry reduction equation is just the usual KdV equation. In Section 5,we study the finite transformation theorem of the general Lie point symmetries via a simple direct method instead of the traditional complicated method by solving an initial value problem. The last section includes a short summary and discussion.

2. Painlev′e property, B¨acklund transformation,and Schwartz form of cKP3-4 equation

According to the standard WTC approach, if the model (1) is Painlev′e-integrable, all the possible solutions of the model can be written as

with four arbitrary functions among ujand vjin addition to the fifth arbitrary function, the arbitrary singular manifold φ,where α and β should be positive integers. In other words,all the solutions of the model are single-valued about the arbitrary movable singular manifold φ.

To fix the constants α and β, one may use the standard leading order analysis.Substituting u ～u0φ-αand v ～v0φ-βinto Eq.(1),and comparing the leading terms for φ ～0,we get the only possible branch with Substituting Eq.(6)with Eq.(7)into Eq.(1)yields the recursion relation on the functions{uj, vj}

where

After introducing the M¨obious transformation (φ →(c0+c1φ)/(b0+b1φ)with c0b1／=c1b0)invariants,

and substituting Eq. (12) with u2v2／=0 into Eq. (1), one can directly obtain the auto and/or non-auto B¨acklund transformation(BT)theorem and the residual symmetry theorem.

Theorem 1 B¨acklund transformation theorem

If φ is a solution of the Schwartz cKP3-4 equation

then both

are solutions of the cKP3-4 equation(1).

Theorem 2 Residual symmetry theorem

If φ is a solution of the Schwartz cKP3-4 equation (17),and the fields {u, v}={ua, va} are related to the singular manifold φ by Eq.(18),then

is a nonlocal symmetry (residual symmetry) of the cKP3-4 equation(1). In other words,the solution(20)solves the symmetry equations,the linearized equations of Eq.(1)

From Eq. (17), one can find that when b=0, the Schwartz cKP3-4 is reduced back to the following usual Schwartz KP equation

The B¨acklund transformation (18) is a non-auto-BT because it changes a solution of the Schwartz cKP3-4 equation(17)to that of the usual cKP3-4 equation(1). The B¨acklund transformation(19)may be considered as a non-auto-BT if uaand vaare replaced by Eq. (18). The B¨acklund transformation (19)may also be considered as an auto-BT which changes one solution{ua, va}to another{ub, vb}for the same equation(1).

From the auto-B¨acklund transformation(19)and the trivial seed solution {ua=0, va=0}, one can obtain some interesting exact solutions. Substituting {ua=0, va=0} into Eq.(18),we have

After solving the over-determined system Eqs.(17),(22),and(23),one can find various exact solutions from the BT(19)with{ua=0, va=0}. Here, we discuss only for the travelling wave solutions of the system Eqs. (17), (22), and (23). For the travelling wave,φ =Φ(kx+py+ωt),the Schwartz equation(17)becomes an identity while equations(22)and(23)become as the following equations

Here we list three special solution examples of the cKP3-4 equation(1)related to Eqs.(24)and(25).

Example 1 D’Alembert-type arbitrary travelling waves moving in one direction with a fixed model-dependent velocity

where Φ is an arbitrary function of ξ =b2x-2a3t-aby.

Because of the arbitrariness of Φ, the localized excitations with special fixed model-dependent velocity{-2a3/b2, -2a2/b} possess rich structures including kink shapes,plateau shapes,molecule forms,few cycle forms,periodic solitons,etc.in addition to the usual sech2form.[1]

Example 2 Rational wave

Different from the D’Alembert wave(26),the soliton solution(28)possesses arbitrary velocity{-p/k,-ak2-bkp+3ak-2p2+bp3k-3}but fixed sech2shape.

3. Localization of nonlocal symmetry(20)

Similar to the usual KP equation[21]and the supersymmetric KdV equation,[17]the nonlocal symmetry (residual symmetry)(20)can be localized by introducing auxiliary variables

It is straightforward to verify that the nonlocal symmetry of the cKP3-4 equation(1)becomes a local one for the prolonged system Eqs.(1),(17),(18)with{ua=u, va=v}and Eq.(29).The vector form of the localized symmetry of the prolonged system can be written as

According to the closed prolongation structure (30), one can readily obtain the finite transformation (auto-B¨acklund transformation)theorem by solving the initial value problem

Theorem 3 Auto-B¨acklund transformation theorem

If{u, v, φ, φ1, φ2, φ3, φ4}is a solution of the prolonged system Eqs.(1),(17),and(18)with{ua=u, va=v}and Eq.(29),so is{u(ε), v(ε), φ(ε), φ1(ε), φ2(ε), φ3(ε), φ4(ε)}with

Comparing Theorem 2 and Theorem 3, one can find that for the cKP3-4 equation (1), the transformation (33) is equivalent to Eq.(19)by using the transformation 1+εφ →φ.

4. Symmetry reductions of the cKP3-4 equation

Using the standard Lie point symmetry method or the formal series symmetry approach[22,23]to the cKP3-4 equation,it is straightforward to find the general Lie point symmetry solutions of Eq.(21)are generated by the following three generators,

where α, β,and θ are arbitrary functions of t.

The symmetries K0(α), K1(β), and K0(θ) constitute a special Kac–Moody–Virasoro algebra with the nonzero commutators

From Eq. (37), we know that K0and K1constitute the usual Kac–Moody algebra and K2constitutes the Virasoro algebra if we fix the arbitrary functions α, β, and θ as special exponential functions exp(mt) or polynomial functions tmfor m=0, ±1, ±2, ....

Applying the Lie point symmetries K0(α), K1(β), and K0(θ) to the cKP3-4 equation (1), we can get two nontrivial symmetry reductions.

Reduction 1 θ ／=0

For θ ／=0,we rewrite the arbitrary functions in the form

5. Finite transformation theorem of K0(α)+K1(β)+K2(θ)via direct method

However,the exact solution of the initial value problem Eqs.(48)–(50)is very complicated and quite awkward even for the pure KP(a=0)case.[24]An alternative simple method is to find symmetry group via a direct method[25–28]by using a priori ansatz

where x0=x0(t), y0=y0(t),and τ =τ(t)are three arbitrary functions of t.

To verify the correctness one can directly substitute Eqs.(52)–(54)into Eq.(1). In fact,one can take the arbitrary functions x0, y0,and τ in the forms

with τ, y0, and ζ0being arbitrary functions of t and Φ being an arbitrary function of ζ.

6. Conclusion and discussion

In summary, the cKP3-4 equation (1) is a significant(2+1)-dimensional KdV extension with various interesting integrable properties. In this paper,the Painlev′e property,autoand non-auto-B¨acklund transformations, local and nonlocal symmetries, Kac–Moody–Virasoro symmetry algebra, finite transformations related to the local and nonlocal symmetries,and the Kac–Moody–Virasoro group-invariant reductions are investigated.

Usually,starting from the trivial vacuum solution(u=0),the B¨acklund transformation will lead to one soliton solution.However,for the cKP3-4 equation(1),the trivial vacuum solution and B¨acklund transformations will lead to abundant solutions including rational solutions, arbitrary D’Alembert-type waves, solitons with a fixed form (sech2form) and arbitrary velocity, and solitons and soliton molecules with fixed velocity but arbitrary shapes(special D’Alembert waves).

There are two important(1+1)-dimensional symmetry reductions of the cKP3-4 equation(1). The first type of the reduction equation is Lax-integrable with the fourth-order spectral problem. The second reduction is just the KdV equation.The more about the cKP3-4 equation(1)and its special reduction(42)will be reported in our future studies.