Local and nonlocal (2+1) -dimensional Maccari systems and their soliton solutions
NNonlocal (2 + 1)-dimensional Maccari systems
Aslı Pekcan ∗ Department of Mathematics, Faculty of ScienceHacettepe University, 06800 Ankara - Turkey
Abstract
We obtain one-soliton solution of (2 + 1)-dimensional 3-component Maccari system byHirota method. Then we find local and nonlocal reductions of this system. By using theAblowitz-Musslimani reduction formulas we obtain one-soliton solutions of the local andnonlocal reduced 2-component Maccari systems.Keywords: Maccari system, Nonlocal reduction, Hirota method, One-soliton solutions
The generalized (2 + 1)-dimensional ( N + 1)-component Maccari system is given by iu k,t + u k,xx + pu k = 0 , k = 1 , , . . . , N, (1) p y = N (cid:88) k =1 ( σ k u k ¯ u k ) x , (2)where p = p ( x, y, t ), σ k = ± u k = u k ( x, y, t ) for k = 1 , , . . . , N , and the bar notation isused for complex conjugation. Here u k denote N different complex short wave amplitudes and p denotes the real long wave amplitude. The system for N = 2 is iu t + u xx + pu = 0 , (3) iv t + v xx + pv = 0 , (4) p y = ( σ u ¯ u + σ v ¯ v ) x , (5)where σ k = ± k = 1 ,
2. This (2 + 1)-dimensional 3-component system was first derived byMaccari [1] from Nizhnik-Novikov-Veselov equation which is an S-integrable equation, witha reduction technique based on Fourier decomposition and space-time rescalings. In [2], itwas noted that Maccari system can also be obtained from many nonlinear partial differentialequations by following the same technique. This method preserves the integrability that isthe Maccari system (3)-(5) is also integrable. Painlev´e analysis was used to investigate theintegrability of the Maccari system by Uthayakumar et al. in [3]. ∗ [email protected] a r X i v : . [ n li n . S I] O c t he Maccari system (3)-(5) is related to some well-known equations. If y = x it reduces tocoupled nonlinear Schr¨odinger (NLS) equation [4]. When y = t , we have the coupled long-waveresonance system [5]. We get (2 + 1)-dimensional extension of NLS equation introduced byFokas [6] when u = ¯ v .Maccari system has many applications in hydrodynamics, plasma physics, Bose-Einstein con-densates, nonlinear optics, and so on since it is a model describing isolated waves which arelocalized in a very small part of space. There are several works on obtaining solutions of theMaccari system (3)-(5). By KP hierarchy reduction method, bright-dark mixed soliton solu-tions [7], [8], multi-dark soliton solutions [9], semi-rational solutions [10] of the Maccari systemwere obtained. Solitoff and dromion solutions [11], solutions in terms of Jacobi elliptic functions[12], folded solitary waves and foldons [13] were found by using variable separation approach.Half bright, bright, dark, half dark, and combined solitons were given by extended modifiedauxiliary equation mapping technique in [14]. By Hirota bilinear method, soliton solutions [3],lump and rogue wave solutions [15], and dromions [16] were derived. By means of the B¨acklundtransformation, peakon and compacton solutions of the Maccari system were obtained in [17].Recently, there is a huge interest on constructing integrable nonlocal equations and findingvarious types of solutions to these equations. The nonlocal reductions were first introducedby Ablowitz and Musslimani [18]-[20]. It has been indicated that systems admitting nonlocalreductions have discrete symmetry transformations that leave the systems invariant. In [21] weshowed that a special case of discrete symmetry transformations are the nonlocal reductionsof the same systems. As examples of (1 + 1)-dimensional nonlocal integrable equations we canmention some famous equations; nonlocal NLS [18]-[20], [22]-[35], modified Korteweg-de Vries(mKdV) [19]-[22], [26], [36]-[42], Fordy-Kulish [43], hydrodynamic type [44], KdV [45] equations,and so on. About (2 + 1)-dimensional nonlocal integrable equations one can check, [46]-[50] forDavey-Stewartson equations, [51] for Fokas equations, [52] and [53] for (2 + 1)-dimensionalmKdV equations, [54] for (2 + 1)-dimensional NLS equations, [55] for Kadomtsev-Petviashviliequations, [56] and [57] for (2 + 1)-dimensional negative AKNS equations, and so on.If the nonlocal (or local) reductions of integrable systems are done consistently the reducedequations are also integrable. In this work we find the integrable reductions of the (2 + 1)-dimensional 3-component Maccari system. To be able to analyze all reductions we will let t → ai t and consider the following Maccari system: au t + u xx + pu = 0 , (6) av t + v xx + pv = 0 , (7) p y = ( σ u ¯ u + σ v ¯ v ) x , (8)where a is an arbitrary constant.In Section 2, we first obtain one-soliton solution of the 3-component Maccari system byHirota method. Then we present local and nonlocal reductions of this system in Section 3and 4, respectively. In Section 5, by using one-soliton solution of the Maccari system withthe reduction formulas we derive soliton solutions of integrable local and nonlocal reduced2-component Maccari systems. 2 One-soliton solutions of -component Maccari system Here by using the Hirota method we obtain one-soliton solutions of 3-component Maccarisystem (6)-(8). Let u = gf , v = hf , p = 2(ln f ) xx . (9)Here g ( x, y, t ), h ( x, y, t ) are complex-valued functions and f ( x, y, t ) is a real-valued function.We obtain the Hirota bilinear form of the system (6)-(8) as( aD t + D x ) { g · f } = 0 , (10)( aD t + D x ) { h · f } = 0 , (11) D x D y { f · f } = σ g ¯ g + σ h ¯ h. (12)Now for one-soliton solution we insert g = (cid:15)g , h = (cid:15)h , and f = 1 + (cid:15) f where g = e θ , h = e θ , (13)for θ j = k j x + l j y + ω j t + δ j , j = 1 ,
2, into (10)-(12). From the coefficients of (cid:15) , we get ω j = − k j a , j = 1 , . (14)The coefficient of (cid:15) yields the function f as f = σ e θ +¯ θ k + ¯ k )( l + ¯ l ) + σ e θ +¯ θ k + ¯ k )( l + ¯ l ) . (15)The coefficient of (cid:15) f ,xy f − f ,x f ,y = 0 (16)is satisfied if ( l + ¯ l − l − ¯ l )( k + ¯ k − k − ¯ k ) = 0 . (17)Therefore we have two possibilities: 1) l + ¯ l = l + ¯ l , (18)2) k + ¯ k = k + ¯ k . (19)The coefficients of (cid:15) are a ( g ,t f − g f ,t ) + g ,xx f − g ,x f ,x + g f ,xx = 0 , (20) a ( h ,t f − h f ,t ) + h ,xx f − h ,x f ,x + h f ,xx = 0 . (21)For both possibilities (18) and (19), to satisfy the above equations we must have¯ a = − a, k = k . (22)Hence one-soliton solution of the 3-component Maccari system (6)-(8) for ¯ a = − a is u ( x, y, t ) = e k x + ω t + l y + δ e ( k k x +( ω ω t k +¯ k ) [ σ e ( l l y + δ δ ( l +¯ l ) + σ e ( l l y + δ δ ( l +¯ l ) ] , (23) v ( x, y, t ) = e k x + ω t + l y + δ e ( k k x +( ω ω t k +¯ k ) [ σ e ( l l y + δ δ ( l +¯ l ) + σ e ( l l y + δ δ ( l +¯ l ) ] , (24)3nd p ( x, y, t ) = ( k + ¯ k ) e ( k +¯ k ) x +( ω +¯ ω ) t [ σ e ( l l y + δ δ ( l +¯ l ) + σ e ( l l y + δ δ ( l +¯ l ) ](1 + e ( k k x +( ω ω t k +¯ k ) [ σ e ( l l y + δ δ ( l +¯ l ) + σ e ( l l y + δ δ ( l +¯ l ) ]) , (25)where the dispersion relation (14) and the condition (22) hold. Here k , l , l , δ , δ are somearbitrary complex numbers. -component Maccari system (a) v ( x, y, t ) = ρu ( x, y, t ), ρ is a real constant.The system (6)-(8) consistently reduces to the local system au t + u xx + pu = 0 , (26) p y = ( σ + σ ρ )( u ¯ u ) x . (27)Here there is no restriction on the constant a . (b) v ( x, y, t ) = ρ ¯ u ( x, y, t ), ρ is a real constant.In this case the system (6)-(8) consistently reduces to the same local system (26) and (27)with the restriction a = ¯ a . -component Maccari system (a) v ( x, y, t ) = ρu ( ε x, ε y, ε t ), ρ is a real constant, and ε k = ± k = 1 , , ε = 1 , ε ε σ = σ ρ . (28)Since σ j = ± j = 1 ,
2, we have ρ = ±
1. Then the system (6)-(8) reduces to the space (S-)reversal nonlocal systems consistently, au t ( x, y, t ) + u xx ( x, y, t ) + p ( x, y, t ) u ( x, y, t ) = 0 , (29) p y ( x, y, t ) = ( σ u ( x, y, t )¯ u ( x, y, t ) + σ u ( ε x, ε y, t )¯ u ( ε x, ε y, t )) x . (30)Explicitly, we have three types of S-reversal nonlocal systems: (i) ( ε , ε , ε ) = (1 , − , , σ = − σ , σ = ± au t ( x, y, t ) + u xx ( x, y, t ) + p ( x, y, t ) u ( x, y, t ) = 0 , (31) p y ( x, y, t ) = ( σ u ( x, y, t )¯ u ( x, y, t ) + σ u ( x, − y, t )¯ u ( x, − y, t )) x . (32)4 ii) ( ε , ε , ε ) = ( − , , , σ = − σ , σ = ± au t ( x, y, t ) + u xx ( x, y, t ) + p ( x, y, t ) u ( x, y, t ) = 0 , (33) p y ( x, y, t ) = ( σ u ( x, y, t )¯ u ( x, y, t ) + σ u ( − x, y, t )¯ u ( − x, y, t )) x . (34) (iii) ( ε , ε , ε ) = ( − , − , , σ = σ , σ = ± au t ( x, y, t ) + u xx ( x, y, t ) + p ( x, y, t ) u ( x, y, t ) = 0 , (35) p y ( x, y, t ) = ( σ u ( x, y, t )¯ u ( x, y, t ) + σ u ( − x, − y, t )¯ u ( − x, − y, t )) x . (36) (b) v ( x, y, t ) = ρ ¯ u ( ε x, ε y, ε t ), ρ is a real constant, and ε k = ± k = 1 , , a = ¯ aε , ε ε σ = σ ρ , (37)yielding ρ = ±
1. The system (6)-(8) reduces to the following nonlocal systems: au t ( x, y, t ) + u xx ( x, y, t ) + p ( x, y, t ) u ( x, y, t ) = 0 , (38) p y ( x, y, t ) = ( σ u ( x, y, t )¯ u ( x, y, t ) + σ u ( ε x, ε y, ε t )¯ u ( ε x, ε y, ε t )) x . (39)Explicitly, here we have seven nonlocal time (T-), space (S-), and space-time (ST-) reversalMaccari systems with two components (38) and (39) corresponding to (i) ( ε , ε , ε ) = (1 , , − , σ = σ , a = − ¯ a , (ii) ( ε , ε , ε ) = (1 , − , − , σ = − σ , a = − ¯ a , (iii) ( ε , ε , ε ) = ( − , − , − , σ = σ , a = − ¯ a , (iv) ( ε , ε , ε ) = ( − , , − , σ = − σ , a = − ¯ a , (v) ( ε , ε , ε ) = (1 , − , , σ = − σ , a = ¯ a , (vi) ( ε , ε , ε ) = ( − , − , , σ = σ , a = ¯ a , (vii) ( ε , ε , ε ) = ( − , , , σ = − σ , a = ¯ a .Note that the equations obtained for the cases (v)-(vii) are already given in the part (a). In[51], the (2 + 1)-dimensional nonlocal Fokas system given by iu t ( x, y, t ) + u xx ( x, y, t ) + u ( x, y, t ) p ( x, y, t ) = 0 ,p y = ( u ( x, y, t )¯ u ( − x, − y, t )) x , (40)was studied and semi-rational solutions of this system were obtained. The above system is sim-ilar but different than the system (38) and (39) corresponding to (vi).5 Soliton solutions of the local and nonlocal reduced sys-tems
In Sections 3 and 4 we have obtained consistent local and nonlocal reductions of 3-componentMaccari system (6)-(8) and the constraints to be satisfied. By using the reduction formulaswith one-soliton solutions of (6)-(8) and following Type 1 or Type 2 approaches [25], [26], [36]we can obtain one-soliton solutions of the local and nonlocal reduced 2-component Maccarisystems.
Both of the local reductions; v ( x, y, t ) = ρu ( x, y, t ) and v ( x, y, t ) = ρ ¯ u ( x, y, t ), yield the samesystem (26) and (27). The latter one puts a condition on the constant a as a = ¯ a . Note that wehave given one-soliton solution of 3-component Maccari system (6)-(8) under the constraints(22). Therefore we consider the first local reduction v ( x, y, t ) = ρu ( x, y, t ) giving hf = ρ gf . (41)When we use Type 1 that is based on equating numerators and denominators separately, weget the following conditions: l = l , e δ = ρe δ . (42)Hence one-soliton solution of the local reduced system (26) and (27) is given by the pair( u ( x, y, t ) , p ( x, y, t )) where u ( x, y, t ) = e k x + ω t + l y + δ (1 + ( σ + ρ σ )2( k +¯ k )( l +¯ l ) e ( k +¯ k ) x +( ω +¯ ω ) t +( l +¯ l ) y + δ +¯ δ ) (43)and p ( x, y, t ) = ( k + ¯ k ) sech ( ψ + δ ) (44)with ψ = 12 ( k + ¯ k ) x + 12 ( ω + ¯ ω ) t + 12 ( l + ¯ l ) y,δ = 12 ln (cid:12)(cid:12)(cid:12) σ + ρ σ k + ¯ k )( l + ¯ l ) (cid:12)(cid:12)(cid:12) . (45)Here ω = − k a , σ j = ± j = 1 ,
2, and a = − ¯ a by (22). Let k = α + iβ , ω = α + iβ , a = iβ , l = α + iβ , and e δ = α + iβ where α , α , α , α ∈ R , β j ∈ R for j = 1 , , , , α = − α β β and β = ( α − β ) β . Then from the solution (43) we have | u ( x, y, t ) | = 2 α α ( σ + ρ σ ) sech ( α x + α t + α y + δ ) , (46)and from (44) we get p ( x, y, t ) = 4 α sech ( α x + α t + α y + δ ) , (47)with ( α + β )( σ + ρ σ )8 α α = e δ . This pair of solutions is nonsingular and bounded if ( σ + ρ σ ) α α > .2 One-soliton solutions of the nonlocal reduced systems In this section we present one-soliton solutions of the nonlocal systems reduced from 3-componentMaccari system (6)-(8) by two types of nonlocal reductions. (a) v ( x, y, t ) = ρu ( ε x, ε y, ε t ), ρ = ±
1, and ε k = ± k = 1 , , ε , ε , ε ) =(1 , − ,
1) with σ = − σ ; ( ε , ε , ε ) = ( − , ,
1) with σ = − σ ; and ( ε , ε , ε ) = ( − , − , σ = σ . We analyze these cases separately. (a).(i) ( ε , ε , ε ) = (1 , − , , σ = − σ , σ = ± l = − l , e δ = ρe δ . (48)Therefore one-soliton solution of the nonlocal system (31) and (32) is the pair ( u ( x, y, t ) , p ( x, y, t ))where u ( x, y, t ) = e k x + ω t + l y + δ σ e ( k k x +( ω ω t + δ δ ( k +¯ k )( l +¯ l ) cosh(( l + ¯ l ) y ) , (49)and p ( x, y, t ) is p ( x, y, t ) = σ ( k +¯ k )( l +¯ l ) e ( k +¯ k ) x +( ω +¯ ω ) t + δ +¯ δ cosh(( l + ¯ l ) y )[1 + σ ( k +¯ k )( l +¯ l ) e ( k +¯ k ) x +( ω +¯ ω ) t + δ +¯ δ cosh(( l + ¯ l ) y )] . (50)Let k = α + iβ , ω = α + iβ , a = iβ , l = α + iβ , and e δ = α + iβ where α , α , α , α ∈ R , β j ∈ R for j = 1 , , , ,
5. Here α = − α β β and β = ( α − β ) β . Then the pair of the solutionsbecomes | u ( x, y, t ) | = e α x +2 α t +2 α y ( α + β )(1 + σ ( α + β )4 α α e α x +2 α t cosh(2 α y )) , (51)and p ( x, y, t ) = 2 α ( α + β ) σ e α x +2 α t cosh(2 α y ) α [1 + σ α α ( α + β ) e α x +2 α t cosh(2 α y )] . (52)The above solutions are nonsingular and bounded if σ α α >
0. Consider the following particularexample.
Example 1.
Take the parameters of the one-soliton solution (49) as ( k , l , a, σ , e δ , ρ ) =(1 + i , , i, , , | u ( x, y, t ) | = e x − t + y (1 + e x − t cosh( y )) , (53)and p ( x, y, t ) = 4 e x − t cosh( y )(1 + e x − t cosh( y )) . (54)7oth of the above functions are nonsingular and bounded. The solution (53) is a solitoff and(54) is a V-type solitary wave. The graph of the above solutions at t = 0 are given in Figure 1. (a) (b) Figure 1: One-soliton solution of the nonlocal system (31) and (32) at t = 0 with the parameters k = 1 + i , l = , a = i, σ = e δ = ρ = 1. (a) Solitoff solution | u ( x, y, t ) | , (b) V-type solitarywave solution p ( x, y, t ). (a).(ii) ( ε , ε , ε ) = ( − , , , σ = − σ , σ = ± k = 0 giving trivial solutions u ( x, y, t ) = 0 and p ( x, y, t ) = 0 for the nonlocal system (33) and (34). Therefore we apply Type 2 approachbased on the cross multiplication of the reduction formula. But in this case we obtain l = − ¯ l yielding again a trivial solution. (a).(iii) ( ε , ε , ε ) = ( − , − , , σ = σ , σ = ± k = ¯ k , l = ¯ l , e δ = σ e δ , e δ +¯ δ = 2 k σ ( l + ¯ l ) ρσ , (55)where σ = ±
1. Since k = ¯ k and a = − ¯ a by (22), we have ω = − ¯ ω . Hence one-solitonsolution of the nonlocal system (35) and (36) is given by the pair ( u ( x, y, t ) , p ( x, y, t )) where u ( x, y, t ) = e k x + ω t + l y + δ ρσ e k x +( l +¯ l ) y (56)and p ( x, y, t ) = 8 k σ e k x +( l +¯ l y ) (1 + ρσ e k x +( l +¯ l y ) ) . (57)8he above solutions are nonsingular and bounded if ρσ >
0. Let a = iβ , ω = i k β = iβ , l = α + iβ , where α ∈ R , β j ∈ R for j = 1 , ,
3. Let also ρσ = e δ . Then from (56) and (57)we have | u ( x, y, t ) | = k σ α sech ( k x + α y + δ ) and p ( x, y, t ) = 2 k sech ( k x + α y + δ ) . (58)They are bell-shaped soliton solutions. (b) v ( x, y, t ) = ρ ¯ u ( ε x, ε y, ε t ) = ρ ¯ u ε , ρ = ± −
1, and ε k = ± k = 1 , , v ( x, y, t ) = ρu ( ε x, ε y, ε ) and we presented one-soliton solutions of these systemsin the previous part. Therefore here we will only consider one-soliton solutions of the nonlocalreduced systems corresponding to (i)-(iv). (b).(i) ( ε , ε , ε ) = (1 , , − σ = σ , σ = ± k = ¯ k , l = ¯ l , e δ = ρe ¯ δ . (59)In this case we have ω = − ¯ ω . Hence one-soliton solution of the nonlocal reduced system (38)and (39) with ( ε , ε , ε ) = (1 , , −
1) is given by the pair ( u ( x, y, t ) , p ( x, y, t )) where u ( x, y, t ) = e k x + ω t + l y + δ σ k ( l +¯ l ) e k x +( l +¯ l ) y + δ +¯ δ (60)and p ( x, y, t ) = 4 k σ e k x +( l +¯ l y + δ +¯ δ ) ( l + ¯ l )[1 + σ k ( l +¯ l ) e k x +( l +¯ l ) y + δ +¯ δ ] . (61)These solutions are nonsingular and bounded for σ k α >
0. Let a = iβ , ω = i k β = iβ , l = α + iβ , e δ = α + iβ , where α , α ∈ R , β j ∈ R for j = 1 , , ,
4. Let also σ ( α + β )4 k α = e δ .Then from (60) and (61) we get | u ( x, y, t ) | = k α σ sech ( k x + α y + δ ) and p ( x, y, t ) = 2 k sech ( k x + α y + δ ) . (62)These are bell-shaped soliton solutions. 9 b).(ii) ( ε , ε , ε ) = (1 , − , − σ = − σ , σ = ± k = ¯ k , l = − ¯ l , e δ = ρe ¯ δ . (63)Here also we get ω = − ¯ ω . Then we obtain one-soliton solution of the nonlocal reduced system(38) and (39) with ( ε , ε , ε ) = (1 , − , −
1) as u ( x, y, t ) = e k x + ω t + l y + δ σ k ( l +¯ l ) e k x + δ +¯ δ cosh(( l + ¯ l ) y ) (64)and p ( x, y, t ) = 4 k σ e k x + δ +¯ δ cosh(( l + ¯ l ) y )( l + ¯ l )[1 + σ k ( l +¯ l ) e k x + δ +¯ δ cosh(( l + ¯ l ) y )] . (65)Let a = iβ , ω = i k β = iβ , l = α + iβ , e δ = α + iβ , where α , α ∈ R , β j ∈ R for j = 1 , , ,
4. Then from (64) and (65) we obtain | u ( x, y, t ) | = e k x +2 α y ( α + β )[1 + σ ( α + β )4 k α e k x cosh(2 α y )] , p ( x, y, t ) = k σ ( α + β ) e k x cosh(2 α y ) α [1 + σ ( α + β )4 k α e k x cosh(2 α y )] . (66)The above solutions are nonsingular and bounded for σ k α >
0. As in the nonlocal case (a).(i),the function | u ( x, y, t ) | defines a solitoff and p ( x, y, t ) a V-type solitary wave solution. (b).(iii) ( ε , ε , ε ) = ( − , − , − σ = σ , σ = ± k = − ¯ k which gives trivial solution u ( x, y, t ) = 0 and p ( x, y, t ) = 0.Therefore we apply Type 2 approach. Here we obtain the following constraints: l = l , e δ = σ e ¯ δ , e δ +¯ δ = ( k + ¯ k )( l + ¯ l ) ρσ , (67)where σ = ±
1. Therefore one-soliton solution of the nonlocal reduced system (38) and (39)with ( ε , ε , ε ) = ( − , − , −
1) is obtained as the pair ( u ( x, y, t ) , p ( x, y, t )) where u ( x, y, t ) = e k x + ω t + l y + δ ρe ( k +¯ k ) x +( ω +¯ ω ) t +( l +¯ l ) y (68)and p ( x, y, t ) = 2 ρ ( k + ¯ k ) e ( k +¯ k ) x +( ω +¯ ω ) t +( l +¯ l ) y [1 + ρe ( k +¯ k ) x +( ω +¯ ω ) t +( l +¯ l ) y ] . (69)The above solutions are nonsingular and bounded if ρ = 1. For ρ = 1, let k = α + iβ , ω = α + iβ , a = iβ , l = α + iβ , where α , α , α ∈ R , β j ∈ R for j = 1 , , ,
4. Here α = − α β β and β = ( α − β ) β . Then from (68) and (69) we get | u ( x, y, t ) | = α α σ sech ( α x + α t + α y + δ ) , p ( x, y, t ) = 2 α sech ( α x + α t + α y + δ ) . (70)10onsider the following particular example. Example 2.
Choose the parameters of the one-soliton solution (68) as ( k , l , a, σ , e δ , ρ ) =(1 + i , , i, , √ , p ( x, y, t ) will be similar to | u ( x, y, t ) | .Therefore here we will only present the solution (70). Then the solution | u ( x, y, t ) | of thenonlocal system (38) and (39) with ( ε , ε , ε ) = ( − , − , −
1) becomes | u ( x, y, t ) | = 12 sech ( x − t + 12 y ) . (71)This is a nonsingular and bounded bell-shaped soliton solution. The graph of the above solutionat t = 0 is given in Figure 2.Figure 2: Bell-shaped one-soliton solution | u ( x, y, t ) | for (38) and (39) with ( ε , ε , ε ) =( − , − , −
1) at t = 0 with the parameters k = 1 + i , l = , a = i, σ = √ , e δ = ρ = 1. (b).(iv) ( ε , ε , ε ) = ( − , , − σ = − σ , σ = ± k = − ¯ k which gives trivial solution u ( x, y, t ) = 0 and p ( x, y, t ) = 0. We then apply Type 2 approach and get l = − ¯ l which also yields a trivialsolution. In this work we considered (2 + 1)-dimensional 3-component Maccari system which is an in-tegrable system. We obtained one-soliton solution of this system by using the Hirota method.Then we presented local and nonlocal reductions of the Maccari system. We also gave one-soliton solutions of the reduced integrable 2-component Maccari systems. The Maccari systemhas many special type of wave interactions. Once can obtain two- and three-soliton solutionsof the system and analyze the wave interactions for the nonlocal reductions of this system.11 eferences [1] A. Maccari, Universal and integrable nonlinear evolution systems of equations in (2+1)dimensions, J. Math. Phys. , 4151, 1997.[2] A. 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