Painlevé property, local and nonlocal symmetries and symmetry reductions for a (2+1)-dimensional integrable KdV equation
aa r X i v : . [ n li n . S I] J u l Painlev´e property, local and nonlocal symmetries and symmetryreductions for a (2+1)-dimensional integrable KdV equation
Xiao-bo Wang, Man Jia and S. Y. Lou ∗ School of Physical Science and Technology, Ningbo University, Ningbo, 315211, China
July 7, 2020
Abstract
The Painlev´e property for a (2+1)-dimensional Korteweg-de Vries (KdV) extension, the combinedKP3 (Kadomtsev- Petviashvili) and KP4 (cKP3-4) is proved by using Kruskal’s simplification. Thetruncated Painlev´e expansion is used to find the Schwartz form, the B¨acklund/Levi transformationsand the residual nonlocal symmetry. The residual symmetry is localized to find its finite B¨acklundtransformation. The local point symmetries of the model constitute a centerless Kac-Moody-Virasoroalgebra. The local point symmetries are used to find the related group invariant reductions including anew Lax integrable model with a fourth order spectral problem. The finite transformation theorem orthe Lie point symmetry group is obtained by using a direct method.
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
PACS:05.45.Yv, 02.30.Ik, 47.20.Ky, 52.35.Mw, 52.35.Sb
Recently, a novel (2+1)-dimensional Korteweg-de Vries (KdV) extension, the combined KP3 (Kadomtsev-Petviashvili) and KP4 (cKP3-4) equation u xt = a [(6 uu x + u xxx ) x − u yy ] + b (2 vu x + v xxx + 4 uu y ) x − v yy , (1) u y = v x . is proposed by one of the present authors (Lou) [1]. KdV equation [2] and its (2+1)-dimensional extensionssuch as the KP equation [3], the Nizhnik-Novikov-Veselov (NNV) equation[4, 5, 6], the asymmetric NNVequation (ANNV) [7, 8, 9] and the Ito equation are fundamental nonlinear integrable models in mathematicalphysics [10].The Lax integrability of the cKP3-4 equation is guaranteed by the existence of the Lax pair [1] ( w x = v y ) ψ y = i( ψ xx + uψ ) , i ≡ √− , (2) ψ t = 2i bψ xxxx + 4 aψ xxx + 4i buψ xx + 2(3 au + 2i bu x + bv ) ψ x − i(3 av + bw − bu + 3 a i u x − bu xx + i bv x ) ψ, (3)and the dual Lax pair φ y = − i( φ xx + uφ ) , (4) φ t = − bφ xxxx + 4 aφ xxx − buφ xx + 2(3 au − bu x + bv ) φ x +i(3 av + bw − bu − a i u x − bu xx − i bv x ) φ. (5) ∗ Corresponding author:[email protected]. Data Availability Statement: The data that support the findings of thisstudy are available from the corresponding author upon reasonable request.
1n Ref. [1], the multiple solitons of the model (1) are obtained by using Hirota’s bilinear approach.Applying the velocity resonant mechanism [11, 12, 13] to the multiple soliton solutions, the soliton moleculeswith arbitrary number of solitons are also found in [1]. It is further discovered that the model permits theexistence of the arbitrary D’Alembert type waves which implies that there are one special type of solitonsand soliton molecules with arbitrary shapes but fixed model dependent velocity.In this paper, we investigate other significant properties such as the Painlev´e property (PP), Schwartzform, B¨aclund transformations, infinitely many local and nonlocal symmetries, Kac-Moody-Virasoro sym-metry algebras, group invariant solutions and symmetry reductions for the cKP3-4 equation (1). To studythe PP of a nonlinear partial differential equation system, there are some equivalent ways such as theWeiss-Tabor-Carnevale (WTC) approach [14], Kruskal’s simplification, Conte’s invariant form [15] and Lou’sextended method [16]. In the section 2 of this paper, the PP of (1) is tested by using the Kruskal’s sim-plification. Using the truncated Panlev´e expansion, one can find many interesting results for integrablesystems 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 fromthe 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 waveand/or a Painlev´e wave [20, 21]. In the section 3, the nonlocal symmetry (the residual symmetry) is localizedby introducing a prolonged system. Whence a nonlocal symmetry is localized, it is straightforward to findits finite transformation which is equivalent to the B¨acklund/Levi transformation. In section 4, it is foundthat similar to the usual KP equation, the general Lie point symmetries of the cKP3-4 equation possess alsothree 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 typeof (1+1)-dimensional reduction equation is Lax integrable with fourth order spectral problem. The secondtype of symmetry reduction equation is just the usual KdV equation. In section 5, we study the finite trans-formation theorem of the general Lie point symmetries via a simple direct method instead of the traditionalcomplicated method by solving an initial value problem. The last section includes a short summary andsome discussions. According to the standard WTC approach, if the model (1) is Painlev´e integrable, all the possiblesolutions of the model can be written as u = ∞ X j =0 u j φ j − α , v = ∞ X j =0 v j φ j − β , (6)with four arbitrary functions among u j and v j in addition to the fifth arbitrary function, the arbitrarysingular manifold φ , where α and β should be positive integers. In other words, all the solutions of themodel 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 ∼ u φ − α and v ∼ v φ − β into (1), and comparing the leading terms for φ ∼
0, we get the only possible branch with u = − φ x , v = − φ x φ y , α = β = 2 . (7)Substituting (6) with (7) into (1) yields the recursion relation on the functions { u j , v j } (cid:18) J J ( j − f y − ( j − f x (cid:19) (cid:18) u j v j (cid:19) ≡ J (cid:18) u j v j (cid:19) = (cid:18) F ( u , u , . . . , u j − , v , v , . . . , v j − ) F ( u , u , . . . , u j − , v , v , . . . , v j − ) (cid:19) , (8)where J = ( j − j − (cid:2) bf x f y − a ( j + 1)( j − f x (cid:3) , J = − bj ( j − j − j − f x , F and F aredependent only on u , u , . . . , u j − , v , v , . . . , v j − and the derivatives of φ with respect to x, y and t .The determinant of the matrix J readsdet J = f x ( j + 1)( j − j − j − j − af x + bf y ) . (9)2rom (8) and (9), we have (cid:18) u j v j (cid:19) = J − (cid:18) F F (cid:19) (10)for j = − , , , j = − , , , { φ, u , u , u , u } , can be included in the formal solution (6) if all the resonant conditions from (8)are satisfied. To verify these resonant conditions, one can use the Kruskal’s simplification for the singularmanifold φ ∼ φ ∼ x + ψ ( y, t ) ∼
0. Under the Krudkal’s simplification, it is straightforward tofind u = − , v = − ψ y , u = v = 0 , u = − ψ yy ,v = 3 a b (cid:0) ψ y − u (cid:1) + 12 ψ y − u ψ y + 12 b ψ t , v = − ψ yy ψ y + u y ,v = u ψ y − ψ yyy , v = u ψ y + 13 u y , v = u ψ y + 14 u y (11)while ψ, u , u , u and u are arbitrary functions of y and t . Thus the model (1) is not only Lax integrablebut also Painlev´e integrable.Because the cKP3-4 equation (1) is Painlev´e integrable, we can use the truncated Painelev´e expansion, u = u φ + u φ + u , v = v φ + v φ + v , (12)to find other interesting properties of the cKP3-4 equation (1).It is known that using the relations (12) with u = v = 0, the cKP3-4 equation can be bilinearized [1]to [ D x D τ + a (3 D y − D x )] f · f = 0 , (13)and [ a (2 bD x D y − D x D t + 3 D x D τ ) + bD y D τ ] f · f = 0 , (14)with help of the auxiliary variable τ , where the Hirota’s bilinear operators D z , z = x, y, t, τ are defined by D nz f · g = ( ∂ z − ∂ z ′ ) n f ( z ) g ( z ′ ) | z ′ = z . (15)After introducing M¨obious transformation ( φ → c + c φb + b φ with c b = c b ) invariants, g = φ t φ x , h = φ y φ x , S = φ xxx φ x − φ xx φ x , w = a ( S x − h y − hh x ) − g x , (16)and substituting (12) with u v = 0 into (1), one can directly obtain the auto and/or non-auto B¨acklundtransformation (BT) theorem and the residual symmetry theorem: Theorem 1.
B¨acklund transformation theorem. If φ is a solution of the Schwartz cKP3-4 equation h xx ( h − xx w x ) x = b (cid:2) S x (3 h x h xxx − h xx ) + S x ( hh xxx − h x h xx ) − hh xx S xxx − h xx h xxxxx + h xxx h xxxx − (3 h x h y + hh xy + h yy ) h xxx + 5 h y h xx + h xx (4 h x h xy + hh xxy + h yyx ) (cid:3) , (17)then both u a = h + aw x bh xx − S − φ xx φ x − h x S s + hS xx + h xxxx − hh xy − h yy +3 h x h y h xx ,v a = g + bh +3 ah b − b ( a + bh ) S − a +2 bhb u a − h xx − a + bh ) φ xx φ x − h x φ xx φ x (18)and ( u b = u a − φ x φ + φ xx φ ,v b = v a − φ x φ y φ + φ xy φ , (19)3re 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 thefields { u, v } = { u a , v a } are related to the singular manifold φ by (18), then { σ u , σ v } = { φ xx , φ xy } (20)is a nonlocal symmetry (residual symmetry) of the cKP3-4 equation (1). In other words, (20) solve thesymmetry equations, the linearized equations of (1) σ uxt = a [(6 σ u u x + 6 uσ ux + σ uxxx ) x − σ uyy ] + b (cid:0) vσ ux + 2 σ v u x + σ vxxx + 4 σ u u y + 4 uσ uy (cid:1) x − σ vyy , (21) σ uy = σ vx . From (17), one can find that when b = 0, the Schwartz cKP3-4 is reduced back to the usual Schwartz KPequation w = 0 . The BT (18) is a non-auto BT because it changes a solution of the Schwartz cKP3-4 equation (17) to that ofthe usual cKP3-4 equation (1). The BT (19) may be considered as a non-auto BT if u a and v a are replacedby (18). The BT (19) may also be considered as an auto-BT which changes one solution { u a , v a } to another { u b , v b } for the same equation (1).From the auto-BT (19) and the trivial seed solution { u a = 0 , v a = 0 } , one can obtain some interestingexact solutions. Substituting { u a = 0 , v a = 0 } into (18), we have h aw x bh xx − S − φ xx φ x − h x S s + hS xx + h xxxx − hh xy − h yy + 3 h x h y h xx = 0 , (22) g + bh + 3 ah b − b ( a + bh ) S − h xx − a + bh ) φ xx φ x − h x φ xx φ x . (23)After solving the over determined system (17), (22) and (23), one can find various exact solutions from theBT (19) with { u a = 0 , v a = 0 } . Here, we discuss only for the travelling wave solutions of the system (17),(22) and (23). For the travelling wave, φ = Φ( kx + py + ωt ), the Schwartz equation (17) becomes an identitywhile (22) and (23) becomes( ak + bp )[Φ ξξξξξ Φ ξ − (5Φ ξξξξ Φ ξξ + 4Φ ξξξ )Φ ξ + 17Φ ξ Φ ξξ Φ ξξξ − ξξ ] = 0 , (24)(3 akp + bp + k ω )Φ ξ − k ( ak + bp )(4Φ ξ Φ ξξξ − ξξ ) = 0 . (25)Here we list three special solution examples of the cKP3-4 equation (1) related to (24) and (25). Example 1.
D’Alembert type arbitrary travelling waves moving in one direction with a fixed model dependentvelocity.
Φ = Φ( ξ ) , ξ = b x − a t − aby, p = − ak/b, ω = − ka /b ,u = − bv/a = 2 b [ln(Φ)] ξξ , (26)where Φ is an arbitrary function of ξ = b x − a t − aby .Because of the arbitrariness of Φ, the localized excitations with special fixed model dependent velocity {− a /b , − a /b } possess rich structures including kink shapes, plateau shapes, molecule forms, few cycleforms, periodic solitons, etc. in addition to the usual sech form [1]. Example 2.
Rational wave.
Φ = kx + py − p k − (3 ak + bp ) t + ξ ,u = kv/p = − k (3 akp t + bp t − k x − k py − ξ k ) (27)with arbitrary constants k, p and ξ . Example 3.
Soliton solution.
Φ = 1 + exp( ξ ) , ξ = kx + py − k ( − ak − bk p + 3 akp + bp ) t + ξ = kv/p = k (cid:18) ξ (cid:19) , (28)with arbitrary constants k, p and ξ .Different from the D’Alembert wave (26), the soliton solution (28) possesses arbitrary velocity {− p/k, − ak − bkp + 3 ak − p + bp k − } but fixed sech shape. (20) Similar to the usual KP equation [21] and the supersymmetric KdV equation [22], the nonlocal symmetry(residual symmetry) (20) can be localized by introducing auxiliary variables φ = φ x , φ = φ y , φ = φ x , φ = φ x . (29)It is straightforward to verify that the nonlocal symmetry of the cKP3-4 equation (1) becomes a local onefor the prolonged system (1), (17), (18) with { u a = u, v a = v } and (29). The vector form of the localizedsymmetry of the prolonged system can be written as V = 2 φ ∂ u + 2 φ ∂ v − φ ∂ φ − φφ ∂ φ − φφ ∂ φ − φ + φφ ) ∂ φ − φ φ + φφ ) ∂ φ . (30)According to the closed prolongation structure (30), one can readily obtain the finite transformation (autoB¨acklund transformation) theorem by solving the initial value problemd u ( ǫ )d ǫ = 2 φ ( ǫ ) , d v ( ǫ )d ǫ = 2 φ ( ǫ ) , d φ ( ǫ )d ǫ = − φ ( ǫ ) , d φ ( ǫ )d ǫ = − φ ( ǫ ) φ ( ǫ ) , d φ ( ǫ )d ǫ = − φ ( ǫ ) φ ( ǫ ) , d φ ( ǫ )d ǫ = − φ ( ǫ ) + φ ( ǫ ) φ ( ǫ )] , d φ ( ǫ )d ǫ = − φ ( ǫ ) φ ( ǫ ) + φ ( ǫ ) φ ( ǫ )] , (31) { u ( ǫ ) , v ( ǫ ) , φ ( ǫ ) , φ ( ǫ ) , φ ( ǫ ) , φ ( ǫ ) , φ ( ǫ ) }| ǫ =0 = { u, v, φ, φ , φ , φ , φ } . Theorem 3.
Auto B¨acklund transformation theorem. If { u, v, φ, φ , φ , φ , φ } is a solution of theprolonged system (1), (17), (18) with { u a = u, v a = v } and (29), so is { u ( ǫ ) , v ( ǫ ) , φ ( ǫ ) , φ ( ǫ ) , φ ( ǫ ) , φ ( ǫ ) ,φ ( ǫ ) } with φ ( ǫ ) = φ ǫφ , φ ( ǫ ) = φ (1 + ǫφ ) , φ ( ǫ ) = φ (1 + ǫφ ) ,φ ( ǫ ) = φ (1 + ǫφ ) − ǫφ (1 + ǫφ ) , φ ( ǫ ) = φ (1 + ǫφ ) − ǫφ φ (1 + ǫφ ) , (32) u ( ǫ ) = u + 2 ǫφ ǫφ − ǫ φ (1 + ǫφ ) , v ( ǫ ) = v + 2 ǫφ ǫφ − ǫ φ φ (1 + ǫφ ) . (33)Comparing the theorem 2 and the theorem 3, one can find that for the cKP3-4 equation (1), the transfor-mation (33) is equivalent to (19) by using the transformation 1 + ǫφ → φ . Using the standard Lie point symmetry method or the formal series symmetry approach [23, 24] to thecKP3-4 equation, it is straightforward to find the general Lie point symmetry solutions of (21) are generatedby the following three generators, (cid:18) σ u σ v (cid:19) = K ( α ) = (cid:18) αu x αv x + b α t (cid:19) , (34) (cid:18) σ u σ v (cid:19) = K ( β ) = (cid:18) βu y + b β t βv y − a b β t (cid:19) , (35)5nd K ( θ ) = θu t + θ t b ( ay + bx ) u x + θ t ( yu ) y + a b θ t + y b θ tt θv t + θ t b ( ay + bx ) v x − θ t ( yv ) y + au + bv b θ t + bx − ay b θ tt − a b θ t ! , (36)where α, β and θ are arbitrary functions of t .The symmetries K ( α ) , K ( β ) and K ( θ ) constitute a special Kac-Moody-Virasoro algebra with thenonzero commutators[ K ( θ ) , K ( α )] = K ( θα t ) , [ K ( θ ) , K ( β )] = K ( θβ t ) , [ K ( θ ) , K ( θ )] = K ( θ θ t − θ t θ ) , (37)where the commutator [ F, G ] with F = (cid:0) F ( u, v ) , F ( u, v ) (cid:1) T and G = (cid:0) G ( u, v ) , G ( u, v ) (cid:1) T , where thesuperscript T means the transposition of matrix, is defined by[ F, G ] ≡ (cid:18) F ′ u F ′ v F ′ u F ′ v (cid:19) G − (cid:18) G ′ u G ′ v G ′ u G ′ v (cid:19) F, and F ′ u , F ′ v , F ′ u , F ′ v , G ′ u , G ′ v , G ′ u and G ′ v are partial linearized operators, say, F ′ u G ≡ dd ǫ F ( u + ǫG , v ) (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 . From (37), we know that K and K constitute the usual Kac-Moody algebra and K constitutes theVirasoro algebra if we fix the arbitrary functions α, β and θ as special exponential functions exp( mt ) orpolynomial functions t m for m = 0 , ± , ± , . . . . Applying the Lie point symmetries K ( α ) , K ( β ) and K ( θ ) to the cKP3-4 equation (1), we can get twonontrivial symmetry reductions. Reduction 1: θ = 0 . For θ = 0, we rewrite the arbitrary functions in the form θ ≡ ρ = 0 , α ≡ ρ α t , β ≡ ρ β t , θ t ≡ ρβ t . (38)Under the new definitions (38), the group invariant condition K ( α ) + K ( β ) + K ( θ ) = 0yields the first type of group invariant solutions in the form u = U ( ξ, η ) ρ − ρ t y bρ − a b − β t ρ b , (39) v = V ( ξ, η ) ρ − aU ( ξ, η ) bρ + (2 ay − bx ) ρ t b ρ + 7 a b − ρα t b + 3 aρβ t b , (40)where U ( ξ, η ) ≡ U and V ( ξ, η ) ≡ V are group invariant functions of the group invariant variables ξ and η with ξ = xρ − aybρ − α + aθ b , η = yρ − β . (41)Substituting (39) and (40) into (1), we can find the group invariant reduction equations for the groupinvariant functions U and V , U η = V ξ ,V ηη = ( V ξξξ + 4 U V ξ + 2 V U ξ ) ξ . (42)It is interesting that the reduction system (42) is Lax integrable with the fourth order spectral problem λ Ψ = 2Ψ ξξξξ + 4 U Ψ ξξ + 2(2 U ξ − i V )Ψ ξ − (cid:18)Z V η d ξ − U − U ξξ + i V ξ (cid:19) Ψ , (43)Ψ η = i(Ψ ξξ + U Ψ) . (44)6 eduction 2: β = 0 . For β = 0 case, the group invariant condition K ( α ) + K ( β ) = 0and v x = u y yield the usual KdV reduction( U T + U XXX + 6
U U X ) X = 0 , X = 1 √ β (cid:18) αβ y − x (cid:19) , T = Z β − / ( aβ − bα )d t (45)with u = U ( X, T ) β − β t y bβ + α (3 aβ − bα )6 β ( aβ − bα ) , (46) v = − αU ( X, T ) β − β t x bβ + (cid:20) (3 aβ + bα ) β t b β − α t bβ (cid:21) y − α (3 aβ − bα )6 β ( aβ − bα ) . (47) K ( α ) + K ( β ) + K ( θ ) via directmethod The finit transformation of K ( α ) + K ( β ) + K ( θ ) may be obtained by solving the initial value problem( { x ′ , y ′ , t ′ , u ′ , v ′ } = { x ′ ( ǫ ) , y ′ ( ǫ ) , t ′ ( ǫ ) , u ′ ( ǫ ) , v ′ ( ǫ ) } , { α ′ , β ′ , θ ′ } = { α ( t ′ ) , β ( t ′ ) , θ ( t ′ ) } ),d t ′ d ǫ = θ ′ , d y ′ d ǫ = (cid:18) β ′ + 12 θ ′ t ′ y ′ (cid:19) , d x ′ d ǫ = (cid:20) α ′ + 14 b θ ′ t ′ ( ay ′ + bx ′ ) (cid:21) , (48)d u ′ d ǫ = − b β ′ t ′ − u ′ θ ′ t ′ − b θ ′ t ′ t ′ y ′ − a b θ ′ t ′ , (49)d v ′ d ǫ = − b α ′ t ′ + 3 a b β ′ t ′ − a b θ ′ t ′ u ′ − θ ′ t ′ v ′ + 9 a b θ ′ t ′ + 2 ay ′ − bx ′ b θ ′ t ′ t ′ , (50) t ′ (0) = t, y ′ (0) = y, x ′ (0) = x, u ′ (0) = u, v ′ (0) = v. However, the exact solution of the initial value problem (48)–(50) is very complicated and quite awkwardeven for the pure KP ( a = 0) case [25]. An alternative simple method is to find symmetry group via a directmethod [26, 27, 28, 29] by using a priori ansatz u = A + BU ( x ′ , y ′ , t ′ ) , v = A + B U ( x ′ , y ′ , t ′ ) + B V ( x ′ , y ′ , t ′ ) , (51)to determine the functions { A, B, A , B , B , x ′ , y ′ , t ′ } = { A, B, A , B , B , x ′ , y ′ , t ′ } ( x, y, t ) suchthat both { u, v } and { U, V } are solutions of the cKP3-4 equation (1).Substituting the ansatz (51) into (1) and requiring U and V satisfying the same cKP3-4 equation(1) but with different variables { x ′ , y ′ , t ′ } , one can readily determine all the undetermined functions { A, B, A , B , B , x ′ , y ′ , t ′ } . The final result can be summarized to the following finite transfor-mation theorem. Theorem 4.
Finite transformation theorem. If { U, V } = { U ( x, y, t ) , V ( x, y, t ) } is a solution of (1), sois { u, v } with u = √ τ t U ( x ′ , y ′ , t ′ ) + τ tt y bτ t + y t b √ τ t + 3 a b (cid:0) √ τ t − (cid:1) , (52) v = a √ τ t b ( √ τ t − U ( x ′ , y ′ , t ′ ) + q τ t V ( x ′ , y ′ , t ′ ) + τ tt ( bx − ay )8 b τ t + a b (cid:18) − √ τ t − q τ t (cid:19) + bx t − ay t b √ τ t − ay t b √ τ t , (53) x ′ = √ τ t x − ab ( √ τ t − √ τ t ) y + x , y ′ = √ τ t y + y , t ′ = τ, (54)where x = x ( t ) , y = y ( t ) and τ = τ ( t ) are three arbitrary functions of t .7o verify the correctness one can directly substitute (52)–(54) into (1). In fact, one can take the arbitraryfunctions x , y and τ in the forms τ = t + ǫθ, x = ǫα, y = ǫβ, (55)where θ, α and β are arbitrary functions of t . Substituting (55) into (52) and (53) yields (cid:18) uv (cid:19) = (cid:18) UV (cid:19) + ǫ [ K ( θ ) + K ( α ) + K ( β )] + O ( ǫ ) , (56)which means theorem 4 is just the finite transformation theorem of the symmetry K ( θ ) + K ( α ) + K ( β ).Applying theorem 4 to the D’Alembert wave (26), we get a new solution u = 2 b √ τ t [ln(Φ)] ζζ + τ tt y bτ t + y t b √ τ t + 3 a b (cid:0) √ τ t − (cid:1) , (57) ζ = b √ τ t ( bx − ay ) + ζ (58)with τ, y and ζ being arbitrary functions of t and Φ being an arbitrary function of ζ . In summary, the cKP3-4 equation (1) is a significant (2+1)-dimensional KdV extension with variousinteresting integrable properties. In this paper, the Painlev´e property, auto- and nonauto- B¨acklund trans-formations, local and nonlocal symmetries, Kac-Moody-Virasoro symmetry algebra, finite transformationsrelated to the local and nonlocal symmetries, and the Kac-Moody-Virasoro group invariant reductions areinvestigated.Usually, starting from the trivial vacuum solution ( u = 0), the B¨acklund transformation will lead toone soliton solution. However, for the cKP3-4 equation (1), the trivial vacuum solution and B¨acklundtransformations will lead to abundant solutions including rational solutions, arbitrary D’Alembert typewaves, solitons with a fixed form (sech form) and arbitrary velocity, and solitons and soliton molecules withfixed velocity but arbitrary shapes (special D’Alembert waves).There are two important (1+1)-dimensional symmetry reductions of the cKP3-4 equation (1). The firsttype of reduction equation is Lax integrable with fourth order spectral problem. The second reduction is justthe KdV equation. The more about the cKP3-4 equation (1) and its special reduction (42) will be reportedin our future studies. Acknowledgements
The work was sponsored by the National Natural Science Foundations of China (Nos.11975131,11435005)and K. C. Wong Magna Fund in Ningbo University.
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