title: Local and nonlocal $(2+1)$-dimensional Maccari systems and their soliton solutions
authors: AslÄą Pekcan
text: 
  Efficiency gain and bidirectional operation of quantum engines with decoupled internal levels
      Daniel Jonathan
      December 30, 2023
  =============================================================================================
  
  
  
  In this work, by using the Hirota bilinear method, we obtain one- and two-soliton solutions of integrable (2+1)-dimensional 3-component Maccari system which is used as a model describing isolated waves localized in a very small part of space and related to very well-known systems like nonlinear Schrödinger, Fokas, and long wave resonance systems. We represent all local and Ablowitz-Musslimani type nonlocal reductions of this system and obtain new integrable systems. By the help of reduction formulas and soliton solutions of the 3-component Maccari system, we obtain one- and two-soliton solutions of these new integrable local and nonlocal reduced 2-component Maccari systems. We also illustrate our solutions by plotting their graphs for particular values of the parameters.
  
  
  Keywords: Maccari systems, Local and nonlocal reductions, Hirota bilinear method, Soliton solutions
  
  
  
  
  
  
  § INTRODUCTION
  
  
  The generalized (2+1)-dimensional (N+1)-component Maccari system is given by
  
      iu_k,t+u_k,xx+pu_k=0,    k=1,2,…,N,
         p_y=∑_k=1^N(σ_k u_ku̅_k)_x,
  
  where p=p(x,y,t), σ_k=± 1, u_k=u_k(x,y,t) for k=1,2,…,N, and the bar notation is used 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,
         iv_t+v_xx+pv=0,
         p_y=(σ_1uu̅+σ_2vv̅)_x,
  
  where σ_k=± 1, k=1, 2. This (2+1)-dimensional 3-component system was first derived by Maccari <cit.> from Nizhnik-Novikov-Veselov equation which is an S-integrable equation, with a reduction technique based on Fourier decomposition and space-time rescalings.  In <cit.>, it was noted that Maccari system can also be obtained from many nonlinear partial differential equations by following the same technique. This method preserves the integrability that is the Maccari system (<ref>)-(<ref>) is also integrable. Painlevé analysis was used to investigate the integrability of the Maccari system by Uthayakumar et al. in <cit.>.
  
  The Maccari system (<ref>)-(<ref>) is related to some well-known equations. If y=x it reduces to coupled nonlinear Schrödinger (NLS) equation <cit.>. When y=t, we have the coupled long-wave resonance system <cit.>. We get (2+1)-dimensional extension of NLS equation introduced by Fokas <cit.> when u=v̅.
  
  Maccari system has many applications in hydrodynamics, plasma physics, Bose-Einstein condensates, and nonlinear optics. It is a model describing isolated waves which are localized in a very small part of space. There are several works on obtaining solutions of the Maccari system (<ref>)-(<ref>). By KP hierarchy reduction method, bright-dark mixed soliton solutions <cit.>, <cit.>, multi-dark soliton solutions <cit.>, semi-rational solutions <cit.> of the Maccari system were obtained. Solitoff and dromion solutions <cit.>, solutions in terms of Jacobi elliptic functions <cit.>, folded solitary waves and foldons <cit.> were found by using variable separation approach. Half bright, bright, dark, half dark, and combined solitons were given by extended modified auxiliary equation mapping technique in <cit.>. By the Hirota bilinear method, soliton solutions <cit.>, lump and rogue wave solutions <cit.>, and dromions <cit.> were derived. By means of the Bäcklund transformation, peakon and compacton solutions of the Maccari system were obtained in <cit.>.
  
  Recently, there is a huge interest on constructing integrable nonlocal equations and finding various types of solutions to these equations.
  The nonlocal reductions were first introduced by Ablowitz and Musslimani <cit.>-<cit.>. It has been indicated that systems admitting nonlocal reductions have discrete symmetry transformations that leave the systems invariant. In <cit.> we showed that a special case of discrete symmetry transformations gives the nonlocal reductions of the same systems. As examples of (1+1)-dimensional nonlocal integrable equations we can mention some famous equations; nonlocal NLS <cit.>-<cit.>, <cit.>-<cit.>, modified Korteweg-de Vries (mKdV) <cit.>-<cit.>, <cit.>, <cit.>-<cit.>, Fordy-Kulish <cit.>, hydrodynamic type <cit.>, KdV <cit.> equations, and so on. For (2+1)-dimensional nonlocal integrable equations one can check the references <cit.>-<cit.> for Davey-Stewartson equations, <cit.> for Fokas equations, <cit.> and <cit.> for (2+1)-dimensional mKdV equations, <cit.> for (2+1)-dimensional NLS equations, <cit.> for Kadomtsev-Petviashvili equations, <cit.> and <cit.> for (2+1)-dimensional negative AKNS equations, and so on. Possible applications of nonlocal equations have been discussed in <cit.>-<cit.>.
  
  
  As far as we know there is no work about the local and nonlocal reductions of the integrable (2+1)-dimensional 3-component Maccari system
  even there are many works about different type of solutions of this system in the literature. We note that if the nonlocal (or local) reductions of integrable systems are done consistently then the reduced equations are also integrable. Finding new integrable local and nonlocal systems is very important in the integrable systems and soliton theory since they have rich physical structures and features leading to many developments in describing nature. In this work we obtain new integrable local and nonlocal reduced systems from the (2+1)-dimensional 3-component Maccari system.
  
  To be able to analyze all local and nonlocal reductions of the Maccari system we first let t→a/it and consider the following (2+1)-dimensional 3-component Maccari system:
  
      au_t+u_xx+pu=0,
         av_t+v_xx+pv=0,
         p_y=(σ_1uu̅+σ_2vv̅)_x,
  
  where a is an arbitrary constant. For this system we have two different local reductions,
  
      i)  v(x,y,t)= ρ u(x,y,t), 
         ii)  v(x,y,t)=ρu̅(x,y,t),
  
  where ρ is a real constant and u̅ is the complex conjugate of the function u. Both of these reductions give the same (2+1)-dimensional 2-component Maccari system,
  
      au_t+u_xx+pu=0,
         p_y=(σ_1+σ_2ρ^2)(uu̅)_x.
  
  The Maccari system (<ref>)-(<ref>) has two different consistent nonlocal reductions. The first one of these reductions is
  
      i)  v(x,y,t)=ρ u(ε_1x,ε_2y,ε_3t),
  
  where ρ is a real constant and ε_j^2=1, j=1,2,3 with the constraints ε_3=1 and ε_1ε_2σ_1=σ_2ρ^2. This reduction gives the following integrable space reversal nonlocal (2+1)-dimensional 2-component Maccari systems:
  
      au_t(x,y,t)+u_xx(x,y,t)+p(x,y,t)u(x,y,t)=0,
         p_y(x,y,t)=(σ_1u(x,y,t)u̅(x,y,t)+σ_2u(ε_1x,ε_2y,t)u̅(ε_1x,ε_2y,t))_x.
  
  The second nonlocal reduction of the Maccari system is given by
  
      ii)  v(x,y,t)=ρu̅(ε_1x,ε_2y,ε_3t),
  
  yielding seven different nonlocal systems which are nonlocal time (T-), space (S-), and space-time (ST-) reversal Maccari systems represented by
  
      au_t(x,y,t)+u_xx(x,y,t)+p(x,y,t)u(x,y,t)=0,
         p_y(x,y,t)=(σ_1u(x,y,t)u̅(x,y,t)+σ_2u(ε_1x,ε_2y,ε_3t)
      u̅(ε_1x,ε_2y,ε_3t))_x,
  
  where a=a̅ε_3 and ε_1ε_2σ_1=σ_2ρ^2.
  
  In this work we study soliton solutions of local and nonlocal reduced (2+1)-dimensional 2-component Maccari systems. We first obtain one- and two-soliton solutions of the 3-component Maccari system by Hirota bilinear method <cit.>-<cit.> which has been used as a powerful and practical tool to obtain multi-soliton solutions of integrable systems analytically. Then by using local and nonlocal Ablowitz-Musslimani type reductions, we obtain soliton solutions of the integrable local and nonlocal reduced 2-component Maccari systems. We present the graphs of some solutions for particular values of the parameters.
  
  
  The lay out of the paper is as follows. In Section 2, we obtain one- and two-soliton solutions of the 3-component Maccari system by Hirota bilinear method. Then we present local and nonlocal reductions of this system in Section 3 and 4, respectively. In Section 5, we derive soliton solutions of integrable local and nonlocal reduced 2-component Maccari systems. We also give some examples of one- and two-soliton solutions of reduced Maccari systems.
  
  
  
  § HIROTA BILINEAR METHOD FOR 3-COMPONENT MACCARI SYSTEM
  
  
  Here by using the Hirota bilinear method <cit.>-<cit.> we will obtain one- and two-soliton solutions of 3-component Maccari system (<ref>)-(<ref>). Let
  
      u=g/f,   v=h/f,   p=2(ln f)_xx.
  
  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 (<ref>)-(<ref>) as
  
      (aD_t+D_x^2){g· f}=0,
         (aD_t+D_x^2){h· f}=0,
         D_xD_y{f· f}=σ_1gg̅+σ_2hh̅.
  
  Here D is a special differential operator called Hirota D-operator given by
      D_t^nD_x^mD_y^r{F· G}=(∂/∂ t-∂/∂ t')^n(∂/∂ x-∂/∂ x')^m(∂/∂ y-∂/∂ y')^r F(x,y,t)G(x',y',t')|_x'=x,t'=t,y'=y.
  
   We expand the functions g, h, and f in the form of power series as
   
      g=ϵ g_1+ϵ^3 g_3+ϵ^5 g_5+⋯, 
          h=ϵ h_1+ϵ^3 h_3+ϵ^5 h_5+⋯, 
          f=1+ ϵ^2 f_2+ϵ^4 f_4+⋯.
  
  
   We substitute the above expansion into the Hirota bilinear form (<ref>)-(<ref>) and make the coefficients of ϵ^m, m=1,2,… to vanish. Note that choosing the functions g_1 and h_1 as exponential functions determines the other functions g_j, h_j, and f_j as exponential functions. Due to this fact, when we consider N soliton solution we have g_2j+1=h_2j+1=f_2j+2=0 for j≥ N. In the following two sections, one- and two-soliton solutions of (2+1)-dimensional 3-component Maccari system will be given.
  
  
  
   §.§ One-soliton solutions of 3-component Maccari system
  
  
  In order to construct one-soliton solution, we take
  
      g_1=e^θ_1,   h_1=e^θ_2,
  
  in the expansion (<ref>) and make the coefficients of ϵ^m, m=1,2,… to vanish in (<ref>)-(<ref>).
  Here θ_j=k_jx+l_jy+ω_jt+δ_j, j=1,2. By the above choice of g_1 and h_1 we have g_2j+1=h_2j+1=f_2j+2=0 for j≥ 1.
  From the coefficients of ϵ, we get the dispersion relations,
  
      ω_j=-k_j^2/a,  j=1,2.
  
  The coefficient of ϵ^2 yields the function f_2 as
  
      f_2=σ_1e^θ_1+θ̅_1/2(k_1+k̅_1)(l_1+l̅_1)+σ_2e^θ_2+θ̅_2/2(k_2+k̅_2)(l_2+l̅_2).
  
  The coefficient of ϵ^4
  
      f_2,xyf_2-f_2,xf_2,y=0
  
  is satisfied if
  
      (l_1+l̅_1-l_2-l̅_2)(k_1+k̅_1-k_2-k̅_2)=0.
  
  Therefore we have two possibilities:
  
      1)  l_2+l̅_2=l_1+l̅_1,
         2)  k_2+k̅_2=k_1+k̅_1.
  
  The coefficients of ϵ^3 are
  
      a(g_1,tf_2-g_1f_2,t)+g_1,xxf_2-2g_1,xf_2,x+g_1f_2,xx=0,
         a(h_1,tf_2-h_1f_2,t)+h_1,xxf_2-2h_1,xf_2,x+h_1f_2,xx=0.
  
  To satisfy the above equations we must have
  
      a̅=-a,    k_1=k_2,
  
  and so that (<ref>) holds. Without loss of generality take ϵ=1. Hence one-soliton solution of the 3-component Maccari system (<ref>)-(<ref>) for a̅=-a is given by (u(x,y,t),v(x,y,t),p(x,y,t)) where
  
      u(x,y,t)=e^k_1x+ω_1t+l_1y+δ_1/1+e^(k_1+k̅_1)x+(ω_1+ω̅_1)t/2(k_1+k̅_1)
      [σ_1e^(l_1+l̅_1)y+δ_1+δ̅_1/(l_1+l̅_1)+σ_2e^(l_2+l̅_2)y+δ_2+δ̅_2/(l_2+l̅_2)       ],
         v(x,y,t)=e^k_1x+ω_1t+l_2y+δ_2/1+e^(k_1+k̅_1)x+(ω_1+ω̅_1)t/2(k_1+k̅_1)
      [σ_1e^(l_1+l̅_1)y+δ_1+δ̅_1/(l_1+l̅_1)+σ_2e^(l_2+l̅_2)y+δ_2+δ̅_2/(l_2+l̅_2)       ],
  
  and
  
      p(x,y,t)=(k_1+k̅_1)e^(k_1+k̅_1)x+(ω_1+ω̅_1)t[σ_1e^(l_1+l̅_1)y+δ_1
      +δ̅_1/(l_1+l̅_1)+σ_2e^(l_2+l̅_2)y+δ_2+δ̅_2/(l_2+l̅_2)]   /(1+e^(k_1+k̅_1)x+(ω_1+ω̅_1)t/2(k_1+k̅_1)
      [σ_1e^(l_1+l̅_1)y+δ_1+δ̅_1/(l_1+l̅_1)+σ_2e^(l_2+l̅_2)y+δ_2+δ̅_2/(l_2+l̅_2)       ])^2,
  
  where the dispersion relations (<ref>) and the conditions (<ref>) hold. Here k_1, l_1, l_2, δ_1, δ_2 are some arbitrary complex numbers.
  
  
  
  
   §.§ Two-soliton solutions of 3-component Maccari system
  
  
  To construct two-soliton solution, we take
  
      g_1=e^θ_1+e^θ_2,   h_1=e^η_1+e^η_2,
  
  in the expansion (<ref>), where θ_j=k_jx+l_jy+ω_jt+δ_i, η_j=r_jx+m_jt+s_jy+α_j for j=1, 2.
  We make the coefficients of ϵ^m, m=1,2,… to vanish in (<ref>)-(<ref>).
  By the choice of g_1 and h_1 we have g_2j+1=h_2j+1=f_2j+2=0 for j≥ 2.
  From the coefficients of ϵ, we get the dispersion relations,
  
      ω_j=-k_j^2/a,  m_j=-r_j^2/a,  j=1, 2.
  
  The coefficient of ϵ^2 gives
  
      f_2=σ_1∑_i≤ i,j≤ 2e^θ_i+θ̅_jα_ij+σ_2∑_i≤ i,j≤ 2e^η_i+η̅_jβ_ij,
  
  where
  
      α_ij=1/2(k_i+k̅_j)(l_i+l̅_j),  β_ij=1/2(r_i+r̅_j)(s_i+s̅_j),    i≤ i,j≤ 2.
  
  From the coefficients of ϵ^3 we obtain the functions g_3 and h_3 as
  
      g_3=A_1e^θ_1+η_2+η̅_1+A_2e^θ_2+η_1+η̅_1+A_3e^θ_1+η_2+η̅_2+A_4e^θ_2+η_1+η̅_2+
      A_5e^θ_1+θ_2+θ̅_1+A_6e^θ_1+θ_2+θ̅_2,
         h_3=B_1e^θ_2+η_1+θ̅_1+B_2e^θ_1+η_2+θ̅_1+B_3e^θ_2+η_1+θ̅_2+B_4e^θ_1+η_2+θ̅_2+
      B_5e^η_1+η_2+η̅_1+B_6e^η_1+η_2+η̅_2,
  
  where
      A_1=σ_2β_21(k_1-k_2)/(k_1+k̅_1),  A_2=-σ_2β_11(k_1-k_2)/(k_2+k̅_1), 
      A_3=σ_2β_22(k_1-k_2)/(k_1+k̅_2),  A_4=-σ_2β_12(k_1-k_2)/(k_2+k̅_2),
      
         A_5=-σ_1(k_1-k_2)[α_11(k_1+k̅_1)-α_21(k_2+k̅_1)]/(k_1+k̅_1)(k_2+k̅_1), 
      A_6=-σ_1(k_1-k_2)[α_12(k_1+k̅_2)-α_22(k_2+k̅_2)]/(k_1+k̅_2)(k_2+k̅_2),
  
  and
      B_1=σ_1α_21(k_1-k_2)/(k_1+k̅_1),  B_2=-σ_1α_11(k_1-k_2)/(k_2+k̅_1), 
      B_3=σ_1α_22(k_1-k_2)/(k_1+k̅_2),  B_4=-σ_1α_12(k_1-k_2)/(k_2+k̅_2),
      
         B_5=-σ_2(k_1-k_2)[β_11(k_1+k̅_1)-β_21(k_2+k̅_1)]/(k_1+k̅_1)(k_2+k̅_1), 
      B_6=-σ_2(k_1-k_2)[β_12(k_1+k̅_2)-β_22(k_2+k̅_2)]/(k_1+k̅_2)(k_2+k̅_2),
  
  with the following conditions satisfied:
  
      a=-a̅,   r_j=k_j,   j=1, 2.
  
  The coefficient of ϵ^4 yields the function f_4 as
      f_4=   C_1e^θ_1+θ̅_1+η_2+η̅_2+C_2e^θ_2+θ̅_2+η_1+η̅_1+C_3e^η_1+η̅_1+η_2+η̅_2
         +C_4e^θ_1+θ̅_1+θ_2+θ̅_2+C_5e^θ_1+θ̅_2+η_2+η̅_1+C_6e^θ_2+θ̅_1+η_1+η̅_2,
  
  where
      C_1=σ_1σ_2κ/(s_2+s̅_2)(l_1+l̅_1),  C_2=σ_1σ_2κ/(s_1+s̅_1)(l_2+l̅_2), 
      C_3=σ_2^2κ(s_1-s_2)(s̅_1-s̅_2)/(s_1+s̅_1)(s_1+s̅_2)(s_2+s̅_1)(s_2+s̅_2),
      
         C_4=σ_1^2κ(l_1-l_2)(l̅_1-l̅_2)/(l_1+l̅_1)(l_1+l̅_2)(l_2+l̅_1)(l_2+l̅_2), 
      C_5=-σ_1σ_2κ/(s_2+s̅_1)(l_1+l̅_2), C_6=-σ_1σ_2κ/(s_1+s̅_2)(l_2+l̅_1),
  
  for
  
      κ=(k_1-k_2)(k̅_1-k̅_2)/4(k_1+k̅_1)(k_1+k̅_2)(k_2+k̅_1)(k_2+k̅_2).
  
  The coefficients of ϵ^j, j=5, 6, 7, 8 vanish directly. Take also ϵ=1. Since two-soliton solution of the 3-component Maccari system (<ref>)-(<ref>)
  has a long expression, we will not give it here explicitly. It can be expressed as (u(x,y,t),v(x,y,t),p(x,y,t)) where
  
      u=g_1+g_3/1+f_2+f_4,   v=h_1+h_3/1+f_2+f_4,   p=2(ln(1+f_2+f_4))_xx.
  
  Here g_1, h_1 are given in (<ref>), g_3, h_3 in (<ref>), f_2 in (<ref>), and f_4 in (<ref>) with
  the dispersion relations (<ref>) and the constraints (<ref>) satisfied.
  
  
  
  
  § LOCAL REDUCTIONS OF 3-COMPONENT MACCARI SYSTEM
  
  
  (a)  v(x,y,t)=ρ u(x,y,t), ρ is a real constant.
  
  The system (<ref>)-(<ref>) consistently reduces to the local system
  
      au_t+u_xx+pu=0,
         p_y=(σ_1+σ_2ρ^2)(uu̅)_x.
  
  Here there is no restriction on the constant a.
  
  (b)  v(x,y,t)=ρu̅(x,y,t), ρ is a real constant.
  
  In this case the system (<ref>)-(<ref>) consistently reduces to the same local system
  (<ref>) and (<ref>) with the restriction a=a̅.
  
  
  
  
  § NONLOCAL REDUCTIONS OF 3-COMPONENT MACCARI SYSTEM
  
  
  (a)  v(x,y,t)=ρ u(ε_1x,ε_2y,ε_3t)=ρ u^ε, ρ is a real constant, and ε_j=± 1, j=1,2,3.
  
  We apply this reduction to 3-component Maccari system (<ref>)-(<ref>). We get
  
      au_t+u_xx+uD_y^-1[σ_1uu̅+σ_2ρ^2u^εu̅^ε]_x=0,
      
      aρ u_t^ε+ρ u_xx^ε+ρ u^ε D_y^-1[σ_1uu̅+σ_2ρ^2u^εu̅^ε]_x=0.
  
  Here D_y^-1 is the standard anti-derivative with respect to y. Let x→ε_1 x, y→ε_2 y, and t→ε_3 t in (<ref>). We have
  
      aε_3u_t+u_xx+ε_1ε_2uD_y^-1[σ_1u^εu̅^ε+σ_2ρ^2uu̅]_x=0.
  
  For consistency, this equation must be same with (<ref>) provided that
  
      ε_3=1,   ε_1ε_2σ_1=σ_2ρ^2.
  
  Since σ_j=± 1, j=1,2, we have ρ=± 1.
  Then the system (<ref>)-(<ref>) reduces to S-reversal nonlocal systems consistently,
  
      au_t(x,y,t)+u_xx(x,y,t)+p(x,y,t)u(x,y,t)=0,
         p_y(x,y,t)=(σ_1u(x,y,t)u̅(x,y,t)+σ_2u(ε_1x,ε_2y,t)u̅(ε_1x,ε_2y,t))_x.
  
  
  Explicitly, we have three different S-reversal nonlocal systems:
  
  
  (i)  (ε_1,ε_2,ε_3)=(1,-1,1), σ_2=-σ_1, σ_1=± 1.
  
      au_t(x,y,t)+u_xx(x,y,t)+p(x,y,t)u(x,y,t)=0,
         p_y(x,y,t)=(σ_1u(x,y,t)u̅(x,y,t)+σ_2u(x,-y,t)u̅(x,-y,t))_x.
  
  
  
  (ii)  (ε_1,ε_2,ε_3)=(-1,1,1), σ_2=-σ_1, σ_1=± 1.
  
      au_t(x,y,t)+u_xx(x,y,t)+p(x,y,t)u(x,y,t)=0,
         p_y(x,y,t)=(σ_1u(x,y,t)u̅(x,y,t)+σ_2u(-x,y,t)u̅(-x,y,t))_x.
  
  
  (iii)  (ε_1,ε_2,ε_3)=(-1,-1,1), σ_2=σ_1, σ_1=± 1.
  
      au_t(x,y,t)+u_xx(x,y,t)+p(x,y,t)u(x,y,t)=0,
         p_y(x,y,t)=(σ_1u(x,y,t)u̅(x,y,t)+σ_2u(-x,-y,t)u̅(-x,-y,t))_x.
  
  
  
  
  (b)  v(x,y,t)=ρu̅(ε_1x,ε_2y,ε_3t), ρ is a real constant, and ε_k=± 1, k=1,2,3.
  
  When we apply this nonlocal reduction to the system (<ref>)-(<ref>) and making the similar analysis as in part (a), we get the following constraints for consistency:
  
      a=a̅ε_3,  ε_1ε_2σ_1=σ_2ρ^2,
  
  yielding ρ=± 1. The system (<ref>)-(<ref>) 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,
         p_y(x,y,t)=(σ_1u(x,y,t)u̅(x,y,t)+σ_2u(ε_1x,ε_2y,ε_3t)u̅(ε_1x,ε_2y,ε_3t))_x.
  
  Explicitly, here we have seven different nonlocal T-, S-, and ST-reversal 2-component Maccari systems corresponding to
  
  (i)  (ε_1,ε_2,ε_3)=(1,1,-1), σ_2=σ_1, a=-a̅,
  
  (ii)  (ε_1,ε_2,ε_3)=(1,-1,-1), σ_2=-σ_1, a=-a̅,
  
  (iii)  (ε_1,ε_2,ε_3)=(-1,-1,-1), σ_2=σ_1, a=-a̅,
  
  (iv)  (ε_1,ε_2,ε_3)=(-1,1,-1), σ_2=-σ_1, a=-a̅,
  
  (v)  (ε_1,ε_2,ε_3)=(1,-1,1), σ_2=-σ_1, a=a̅,
  
  (vi)  (ε_1,ε_2,ε_3)=(-1,-1,1), σ_2=σ_1, a=a̅,
  
  (vii)  (ε_1,ε_2,ε_3)=(-1,1,1), σ_2=-σ_1, a=a̅.
  
  
  Note that the systems obtained for the cases (v)-(vii) have already been given in the part (a) with no restriction on the constant a.
  
  In <cit.>, 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)u̅(-x,-y,t))_x,
  
  was studied and semi-rational solutions of this system were obtained. The above system is similar but different than the system (<ref>) and (<ref>) corresponding to (vi).
  
  
  
  
  
  
  § SOLITON SOLUTIONS OF THE LOCAL AND NONLOCAL REDUCED SYSTEMS
  
  
  In Sections 3 and 4 we have obtained consistent local and nonlocal reductions of 3-component Maccari system (<ref>)-(<ref>) and
  the constraints to be satisfied. By using the reduction formulas with one- and two-soliton solutions of (<ref>)-(<ref>) and following Type 1 or Type 2 approaches <cit.>, <cit.>, <cit.> we can obtain one- and two-soliton solutions of the local and nonlocal reduced 2-component Maccari systems.
  
  
  
  
  
   §.§ One-soliton solutions of the local reduced systems
  
  
  Both of the local reductions; v(x,y,t)=ρ u(x,y,t) and v(x,y,t)=ρu̅(x,y,t), yield the same reduced local system (<ref>) and (<ref>). The latter
  one puts a condition on the constant a as a=a̅. Note that we have given one-soliton solution of 3-component Maccari system (<ref>)-(<ref>)
  under the constraints (<ref>). Therefore we only consider the first local reduction v(x,y,t)=ρ u(x,y,t) giving
  
      h/f=ρg/f.
  
  When we use Type 1 that is the approach based on equating numerators and denominators separately, we get the following conditions:
  
      l_2=l_1,   e^δ_2=ρ e^δ_1.
  
  Hence one-soliton solution of the local reduced system (<ref>) and (<ref>) is given by the pair (u(x,y,t),p(x,y,t)) where
  
      u(x,y,t)=e^k_1x+ω_1t+l_1y+δ_1/(1+(σ_1+ρ^2σ_2)/2(k_1+k̅_1)(l_1+l̅_1)e^(k_1+k̅_1)x
      +(ω_1+ω̅_1)t+(l_1+l̅_1)y+δ_1+δ̅_1)
  
  and
  
      p(x,y,t)=(k_1+k̅_1)(σ_1+ρ^2σ_2)e^(k_1+k̅_1)x
      +(ω_1+ω̅_1)t+(l_1+l̅_1)y+δ_1+δ̅_1/(l_1+l̅_1)(1+(σ_1+ρ^2σ_2)/2(k_1+k̅_1)(l_1+l̅_1)e^(k_1+k̅_1)x
      +(ω_1+ω̅_1)t+(l_1+l̅_1)y+δ_1+δ̅_1)^2.
  
  Here ω_1=-k_1^2/a, σ_j=± 1 for j=1, 2, and a=-a̅. Let k_1=α_1+iβ_1, ω_1=α_2+iβ_2, a=iβ_3, l_1=α_4+iβ_4, and e^δ_1=α_5+iβ_5 where α_1, α_2, α_4,α_5 ∈ℝ, β_j∈ℝ for 1≤ j ≤ 5. Indeed, α_2=-2α_1β_1/β_3 and β_2=(α_1^2-β_1^2)/β_3.
  Then from the solution (<ref>) we have
  
      |u(x,y,t)|^2=2α_1α_4/(σ_1+ρ^2σ_2)sech^2(α_1x+α_2t+α_4y+ϕ)
  
  and from (<ref>) we get
  
      p(x,y,t)=2α_1^2sech^2(α_1x+α_2t+α_4y+ϕ)
  
  with ϕ=1/2ln|(α_5^2+β_5^2)(σ_1+ρ^2σ_2)/8α_1α_4|. These solutions are nonsingular and bounded if (σ_1+ρ^2σ_2)/α_1α_4>0. They are bell-shaped soliton solutions.
  
  
  Example 1. Take the parameters of one-soliton solution of the local system (<ref>) and (<ref>) as (k_1,l_1,a,ρ)=(1+i/2,1/3, 2i,4) with σ_1=σ_2=e^δ_1=1. Then the pair (|u(x,y,t)|^2,p(x,y,t)) becomes
  
      |u(x,y,t)|^2=2/51sech^2(x+1/3y-1/2t+δ),   p(x,y,t)=2sech^2(x+1/3y-1/2t+1/2+δ),
  
  where δ=1/2ln(51/8). The graphs of the above solutions at t=0 are given in Figure 1.
  
  
  
  
  
  
  
  Figures 2(a) and 2(b) depict the movements of the solitons from |u(x,y,t)|^2 and p(x,y,t) for different times in different positions at the y-axis, respectively.
  
  
  
  
  
  
  
   §.§ One-soliton solutions of the nonlocal reduced systems
  
  
  In this section we present one-soliton solutions of the nonlocal systems obtained from 3-component Maccari system (<ref>)-(<ref>) by two types of nonlocal reductions.
  
  
  (a)  v(x,y,t)=ρ u(ε_1x,ε_2y,ε_3t), ρ=± 1, and ε_j=± 1, j=1,2,3.
  
  
  By this nonlocal reduction we obtain three types of nonlocal systems corresponding to (ε_1,ε_2,ε_3)=(1,-1,1) with σ_2=-σ_1; (ε_1,ε_2,ε_3)=(-1,1,1) with σ_2=-σ_1; and (ε_1,ε_2,ε_3)=(-1,-1,1) with σ_2=σ_1. We analyze these cases separately.
  
  
  (a).(i) (ε_1,ε_2,ε_3)=(1,-1,1), σ_2=-σ_1, σ_1=± 1.
  
  When we use Type 1, the reduction formula yields the conditions,
  
      l_2=-l_1,   e^δ_2=ρ e^δ_1.
  
  Therefore one-soliton solution of the nonlocal system (<ref>) and (<ref>) is the pair (u(x,y,t), p(x,y,t)) where
  
      u(x,y,t)=e^k_1x+ω_1 t+l_1y+δ_1/1+σ_1e^(k_1+k̅_1)x+(ω_1+ω̅_1)t+δ_1+δ̅_1/(k_1+k̅_1)(l_1+l̅_1)cosh((l_1+l̅_1)y)
  
  and p(x,y,t) is
  
      p(x,y,t)=2σ_1(k_1+k̅_1)/(l_1+l̅_1)e^(k_1+k̅_1)x+(ω_1+ω̅_1)t+δ_1+δ̅_1cosh((l_1+l̅_1)y)     /[1+σ_1/(k_1+k̅_1)(l_1+l̅_1)e^(k_1+k̅_1)x+(ω_1+ω̅_1)t+δ_1+δ̅_1cosh((l_1+l̅_1)y)]^2.
  
   Let k_1=α_1+iβ_1, ω_1=α_2+iβ_2, a=iβ_3, l_1=α_4+iβ_4, and e^δ_1=α_5+iβ_5 where α_1, α_2, α_4,α_5 ∈ℝ, β_j∈ℝ for 1 ≤ j≤ 5. Here α_2=-2α_1β_1/β_3 and β_2=(α_1^2-β_1^2)/β_3. Then solutions become
  
      |u(x,y,t)|^2=e^2α_1x+2α_2t+2α_4y(α_5^2+β_5^2)/(1+σ_1(α_5^2+β_5^2)/4α_1α_4e^2α_1x+2α_2tcosh(2α_4y))^2,
  
  and
  
      p(x,y,t)=2α_1(α_5^2+β_5^2)σ_1e^2α_1x+2α_2tcosh(2α_4y)/α_4[1+σ_1/4α_1α_4(α_5^2+β_5^2)
       e^2α_1x+2α_2tcosh(2α_4y)]^2.
  
  The above solutions are nonsingular and bounded if σ_1/α_1α_4>0. Consider the following particular example.
  
  
  Example 2. Choose the parameters of the one-soliton solution (<ref>) as (k_1,l_1,a,σ_1,e^δ_1,ρ)=(1+i/4,1/2,i,1,1,1). Then the solutions of the nonlocal system (<ref>) and (<ref>) become
  
      |u(x,y,t)|^2=e^2x-t+y/(1+1/2e^2x-tcosh(y))^2,  
      p(x,y,t)=4e^2x-tcosh(y)/(1+1/2e^2x-tcosh(y))^2.
  
  Both of the above solutions are nonsingular and bounded. The graphs of (<ref>) at t=0 are given in Figure 3.
  
  
  
  
  
  
  
  Figures 4(a) and 4(b) show the movements of the waves from |u(x,y,t)|^2 and p(x,y,t) for different times in different positions at the y-axis, respectively.
  
  
  
  
  
  (a).(ii) (ε_1,ε_2,ε_3)=(-1,1,1), σ_2=-σ_1, σ_1=± 1.
  
  For this case if we use Type 1 we get k_1=0 giving trivial solutions u(x,y,t)=0 and p(x,y,t)=0 for the nonlocal system (<ref>) and (<ref>).
  Therefore we apply Type 2 approach based on the cross multiplication of the reduction formula. But in this case we obtain l_1=-l̅_1 yielding
  again a trivial solution.
  
  
  (a).(iii) (ε_1,ε_2,ε_3)=(-1,-1,1), σ_2=σ_1, σ_1=± 1.
  
  Here similar to the previous case, Type 1 approach gives a trivial solution. Using Type 2 yields the following constraints:
  
      k_1=k̅_1,   l_2=l̅_1,   e^δ_2=σ_3 e^δ_1,   e^δ_1+δ̅_1=2k_1σ_3(l_1+l̅_1)/ρσ_1,
  
  where σ_3=± 1. Since k_1=k̅_1 and a=-a̅ by (<ref>), we have ω_1=-ω̅_1. Hence one-soliton solution of the nonlocal system (<ref>) and (<ref>) is given by the pair (u(x,y,t),p(x,y,t)) where
  
      u(x,y,t)=e^k_1x+ω_1t+l_1y+δ_1/1+ρσ_3 e^2k_1x+(l_1+l̅_1)y
  
  and
  
      p(x,y,t)=8k_1^2σ_3e^2k_1x+(l_1+l̅_1y)/(1+ρσ_3e^2k_1x+(l_1+l̅_1y))^2.
  
  The above solutions are nonsingular and bounded if ρσ_3>0. Let a=iβ_1, ω_1=ik_1^2/β_1=iβ_2, l_1=α_3+iβ_3, where α_3  ∈ℝ, β_j∈ℝ for j=1,2,3. Let also ρσ_3=e^2δ. Then from (<ref>) and
  (<ref>) we have
  
      |u(x,y,t)|^2=k_1σ_1α_3 sech^2(k_1x+α_3 y+δ)   and    p(x,y,t)=2k_1^2 sech^2(k_1x+α_3 y+δ).
  
  They are bell-shaped soliton solutions.
  
  
  (b)  v(x,y,t)=ρu̅(ε_1x,ε_2y,ε_3t), ρ=± -1, and ε_k=± 1, k=1,2,3.
  
  By this nonlocal reduction we obtain seven different nonlocal reduced systems. As we noted previously the systems corresponding to (v)-(vii) are same with the systems obtained by the reduction v(x,y,t)=ρ u(ε_1x,ε_2y,ε_3) and we have already presented one-soliton solutions of these systems in the previous part (a). Therefore here we will only consider one-soliton solutions of the nonlocal reduced systems corresponding to (i)-(iv).
  
  
  (b).(i) (ε_1,ε_2,ε_3)=(1,1,-1), σ_2=σ_1, σ_1=± 1.
  
  When we use Type 1 with the reduction formula we obtain the following constraints:
  
      k_1=k̅_1,   l_2=l̅_1,   e^δ_2=ρ e^δ̅_1.
  
  In this case we have ω_1=-ω̅_1. Hence one-soliton solution of the nonlocal reduced system (<ref>) and (<ref>) with
  (ε_1,ε_2,ε_3)=(1,1,-1) is given by the pair (u(x,y,t), p(x,y,t)) where
  
      u(x,y,t)=e^k_1x+ω_1t+l_1y+δ_1/1+σ_1/2k_1(l_1+l̅_1)e^2k_1x+(l_1+l̅_1)y+δ_1+δ̅_1
  
  and
  
      p(x,y,t)=4k_1σ_1e^2k_1x+(l_1+l̅_1y+δ_1+δ̅_1)/(l_1+l̅_1)[1+σ_1/2k_1(l_1+l̅_1)e^2k_1x+(l_1+l̅_1)y+δ_1
      +δ̅_1]^2.
  
  These solutions are nonsingular and bounded for σ_1/k_1α_3>0. Let a=iβ_1, ω_1=ik_1^2/β_1=iβ_2, l_1=α_3+iβ_3, e^δ_1=α_4+iβ_4, where α_3, α_4 ∈ℝ, β_j∈ℝ for 1≤ j≤4. Let also σ_1(α_4^2+β_4^2)/4k_1α_3=e^2δ. Then from (<ref>) and (<ref>) we get
  
      |u(x,y,t)|^2=k_1α_3/σ_1sech^2(k_1x+α_3y+δ)  and   p(x,y,t)=2k_1sech^2(k_1x+α_3y+δ).
  
  These are bell-shaped soliton solutions.
  
  
  (b).(ii) (ε_1,ε_2,ε_3)=(1,-1,-1), σ_2=-σ_1, σ_1=± 1.
  
  Using Type 1 yields the following constraints:
  
      k_1=k̅_1,   l_2=-l̅_1,   e^δ_2=ρ e^δ̅_1.
  
  Here we also get ω_1=-ω̅_1 directly. Then we obtain one-soliton solution (u(x,y,t),p(x,y,t)) of the nonlocal reduced system (<ref>) and (<ref>) with
  (ε_1,ε_2,ε_3)=(1,-1,-1) as
  
      u(x,y,t)=e^k_1x+ω_1t+l_1y+δ_1/1+σ_1/2k_1(l_1+l̅_1)e^2k_1x+δ_1+δ̅_1cosh((l_1+l̅_1)y)
  
  and
  
      p(x,y,t)=4k_1σ_1e^2k_1x+δ_1+δ̅_1cosh ((l_1+l̅_1)y) /(l_1+l̅_1)[1+σ_1/2k_1(l_1+l̅_1)e^2k_1x+δ_1+δ̅_1cosh((l_1+l̅_1)y)]^2.
  
  Let a=iβ_1, ω_1=ik_1^2/β_1=iβ_2, l_1=α_3+iβ_3, e^δ_1=α_4+iβ_4, where α_3, α_4 ∈ℝ, β_j∈ℝ for 1≤ j≤ 4. Then from (<ref>) and (<ref>) we obtain
  
      |u(x,y,t)|^2=e^2k_1x+2α_3y(α_4^2+β_4^2)/[1+σ_1(α_4^2+β_4^2)/4k_1α_3e^2k_1xcosh(2α_3y)]^2,  
      p(x,y,t)=k_1σ_1(α_4^2+β_4^2)e^2k_1xcosh(2α_3y)/α_3[1+σ_1(α_4^2+β_4^2)/4k_1α_3e^2k_1xcosh(2α_3y)]^2.
  
  The above solutions are nonsingular and bounded for σ_1/k_1α_3>0. As in the nonlocal case (a).(i), the function
  |u(x,y,t)|^2 defines a solitoff and p(x,y,t) a V-type solitary wave solution.
  
  
  (b).(iii) (ε_1,ε_2,ε_3)=(-1,-1,-1), σ_2=σ_1, σ_1=± 1.
  
  If we use Type 1, we get k_1=-k̅_1 which gives trivial solutions u(x,y,t)=0 and p(x,y,t)=0. Therefore we apply Type 2 approach. Here
  we obtain the following constraints:
  
      l_2=l_1,    e^δ_2=σ_3 e^δ̅_1,   e^δ_1+δ̅_1=(k_1+k̅_1)(l_1+l̅_1)/ρσ_1,
  
  where σ_3=± 1. Hence one-soliton solution of the nonlocal reduced system (<ref>) and (<ref>) with
   (ε_1,ε_2,ε_3)=(-1,-1,-1) is obtained as the pair (u(x,y,t),p(x,y,t)) where
  
      u(x,y,t)=e^k_1x+ω_1t+l_1y+δ_1/1+ρ e^(k_1+k̅_1)x+(ω_1+ω̅_1)t+(l_1+l̅_1)y
  
  and
  
      p(x,y,t)=2ρ(k_1+k̅_1)^2e^(k_1+k̅_1)x+(ω_1+ω̅_1)t+(l_1+l̅_1)y/[1+ρ e^(k_1+k̅_1)x+(ω_1+ω̅_1)t+(l_1+l̅_1)y  ]^2.
  
   The above solutions are nonsingular and bounded if ρ=1. For ρ=1, let k_1=α_1+iβ_1, ω_1=α_2+iβ_2, a=iβ_3, l_1=α_4+iβ_4, where α_1, α_2, α_4 ∈ℝ, β_j∈ℝ for 1≤ j≤ 4. Here α_2=-2α_1β_1/β_3 and β_2=(α_1^2-β_1^2)/β_3. Then from (<ref>) and (<ref>) we get
   
      |u(x,y,t)|^2=α_1α_4 σ_1 sech^2(α_1x+α_2t+α_4y+δ),   p(x,y,t)=2α_1^2sech^2(α_1x+α_2t+α_4y+δ).
  
   The solutions are nonsingular and bounded bell-shaped soliton solutions.
   
   
  
   (b).(iv) (ε_1,ε_2,ε_3)=(-1,1,-1), σ_2=-σ_1, σ_1=± 1.
  
  For this case if we use Type 1 we get k_1=-k̅_1 which gives trivial solutions u(x,y,t)=0 and p(x,y,t)=0. We then apply Type 2 and get l_1=-l̅_1 which also yields trivial solutions.
  
  
  
  
   §.§ Two-soliton solutions of the local reduced systems
  
  
  Similar to one-soliton solution, we only consider the local reduction v(x,y,t)=ρ u(x,y,t) since the second local reduction v(x,y,t)=ρu̅(x,y,t) is consistent if a=a̅ different than the constraint a=-a̅ that we obtain for two-soliton solution of Maccari system.
  
  
  
  (a)  v(x,y,t)=ρ u(x,y,t), ρ is a real constant.
  
  Under this reduction, when we use Type 1 approach with two-soliton solution (<ref>), in addition to the conditions (<ref>) we get the following constraints:
  
      s_j=l_j,   e^α_j=ρ e^δ_j,    j=1, 2.
  
  Example 3. Let us take the parameters of two-soliton solution as (k_1,k_2,l_1,l_2,a)=(2,1/2,1/3,1,2i) with σ_1=σ_2=ρ=e^δ_1=e^δ_2=1. Then the graphs of the two-soliton solutions of the local reduced 2-component Maccari system (<ref>) and (<ref>) at t=0 are given in Figure 5.
  
  
  
  
  
  Figures 6(a) and 6(b) show the interactions of the waves from |u(x,y,t)|^2 and p(x,y,t) for different times in different positions at the y-axis, respectively.
  
  
  
  
  
  
  
   §.§ Two-soliton solutions of the nonlocal reduced systems
  
  
  In this section we obtain two-soliton solutions of nonlocal systems obtained from 3-component Maccari systems by two nonlocal reductions.
  
  
  (a)  v(x,y,t)=ρ u(ε_1x,ε_2y,ε_3t), ρ=± 1, and ε_j=± 1, j=1,2,3.
  
  Under this reduction we will consider soliton solutions of the three different S-reversal nonlocal reduced 2-component Maccari systems.
  
  
  
  (a).(i) (ε_1,ε_2,ε_3)=(1,-1,1), σ_2=-σ_1, σ_1=± 1.
  
  If we use this nonlocal reduction with two-soliton solution (<ref>) and apply Type 1, besides (<ref>) we get the following constraints:
  
      s_j=-l_j,   e^α_j=ρ e^δ_j,    j=1, 2.
  
  Example 4. Let us choose the two-soliton solution parameters as (k_1,k_2,l_1,l_2,a)=(1+i/2,1/4,1/3,1,i)
   with σ_1=e^δ_1=e^δ_2=ρ=1. Then the graphs of the two-soliton solutions of (<ref>) and (<ref>) are given in
   Figure 7.
   
  
  
  
  
  Figures 8(a) and 8(b) depict the interactions of the waves from |u(x,y,t)|^2 and p(x,y,t) for different times in different positions at the y-axis, respectively.
  
  
  
  
  
  (a).(ii) (ε_1,ε_2,ε_3)=(-1,1,1), σ_2=-σ_1, σ_1=± 1.
  
  When we use this nonlocal reduction with (<ref>) and apply Type 1,  in addition to (<ref>) we obtain the following constraints:
  
      k_2=-k_1,    s_1=l_2,   s_2=l_1,   e^α_1=ρ e^δ_2,     e^α_2=ρ e^δ_1.
  
  Example 5. Consider the two-soliton solution parameters as (k_1, l_1,l_2,a)=(1+i/2,1/2,-1,2i)
   with σ_1=e^δ_1=e^δ_2=ρ=1. Then the graphs of the two-soliton solutions of the nonlocal system (<ref>) and (<ref>) are given in
   Figure 9.
   
  
  
  
  
  Figures 10(a) and 10(b) depict the interactions of the waves from |u(x,y,t)|^2 and p(x,y,t) for different times in different positions at the y-axis, respectively.
  
  
  
  
  
  (a).(iii) (ε_1,ε_2,ε_3)=(-1,-1,1), σ_2=σ_1, σ_1=± 1.
  
  For this case, applying Type 1 approach yields the conditions,
  
      k_2=-k_1,    s_1=-l_2,   s_2=-l_1,   e^α_1=ρ e^δ_2,     e^α_2=ρ e^δ_1,
  
  besides the constraints (<ref>). Here we obtain nonsingular solutions very similar to the cases (a).(i) and (a).(ii), hence we will not
  give a particular example for this case.
  
  
  
  
  
  (b)  v(x,y,t)=ρu̅(ε_1x,ε_2y,ε_3t), ρ=± -1, and ε_j=± 1, j=1,2,3.
  
  In this part we will consider soliton solutions of four different nonlocal reduced 2-component Maccari systems.
  
  (b).(i) (ε_1,ε_2,ε_3)=(1,1,-1), σ_2=σ_1, σ_1=± 1.
  
  Here using two-soliton solution (<ref>) with this reductions and applying Type 1 yield the following constraints:
  
      k_j=k̅_j,    s_j=l̅_j,   e^α_j=ρ e^δ̅_j,   j=1, 2.
  
  
  Example 6. Consider the two-soliton solution parameters as (k_1, k_2, l_1,l_2,a)=(1,2,1/2,2,i)
   with σ_1=e^δ_1=e^δ_2=ρ=1. Then the graphs of the two-soliton solutions of the nonlocal system (<ref>) and (<ref>) are given in
   Figure 11.
   
  
  
  
  
  
  
  Here the form of the waves changes as time changes but the their locations does not change. Figures 12(a) and 12(b) show the interactions of the waves from |u(x,y,t)|^2 and p(x,y,t) for different times in different positions at the y-axis, respectively.
  
  
  
  
  
  (b).(ii) (ε_1,ε_2,ε_3)=(1,-1,-1), σ_2=-σ_1, σ_1=± 1.
  
  Applying Type 1 approach, here we obtain the constraints,
  
      k_j=k̅_j,    s_j=-l̅_j,   e^α_j=ρ e^δ̅_j,   j=1, 2.
  
  
  Example 7. Take the two-soliton solution parameters as (k_1, k_2, l_1,l_2,a)=(1,2,1/2,1,i)
   with σ_1=e^δ_1=e^δ_2=ρ=1. Then the graphs of the two-soliton solutions of the nonlocal system (<ref>) and (<ref>) are given in
   Figure 13.
   
  
  
  
  
  
  
  In Figure 13(a) there occurs a bump near the combined solitoffs. The amplitudes of the solitoffs change as time changes but they are localized. This property is also same for combined V-type waves in Figure 13(b). Figures 14(a) and 14(b) show the interactions of the waves from |u(x,y,t)|^2 and p(x,y,t) for different times in different positions at the y-axis, respectively.
  
  
  
  
  
  (b).(iii) (ε_1,ε_2,ε_3)=(-1,-1,-1), σ_2=σ_1, σ_1=± 1.
  
  In this case Type 1 gives trivial solution. Therefore we apply Type 2 approach and obtain the constraints,
  
      s_j=l_j,   e^α_j=σ_3 e^δ_j,   j=1, 2,
  
  and
  
      e^2δ_1=ρσ_1(k_1+k̅_2)(k_1+k̅_1)(l_1+l̅_2)(l_1+l̅_1)/σ_3(l_1-l_2)(k_1-k_2) ,   e^2δ_2=ρσ_1(k_2+k̅_1)(k_2+k̅_2)(l_2+l̅_1)(l_2+l̅_2)/σ_3(l_1-l_2)(k_1-k_2),
  
  where σ_3=± 1. Nonsingular two-soliton solutions obtained here are very similar to the ones given in (b).(i) case. Therefore we
  will not give a particular example corresponding to this case.
  
  
  (b).(iv) (ε_1,ε_2,ε_3)=(-1,1,-1), σ_2=-σ_1, σ_1=± 1.
  
  In this case using both Type 1 and Type 2 approaches gives trivial solutions u(x,y,t)=0 and p(x,y,t)=0.
  
  
  
  
  
  
  
  
  § CONCLUSION
  
  In this work we considered integrable (2+1)-dimensional 3-component Maccari system which has many applications in several areas such as hydrodynamics, plasma physics, Bose-Einstein condensates, and nonlinear optics. Under some special transformations, this system can be reduced to well-known systems like NLS, Fokas, and long wave resonance systems. We obtained one- and two-soliton solutions of the 3-component Maccari system by using the Hirota bilinear method. Then we presented all possible local and nonlocal reductions of the system and obtained several new integrable systems. By using one- and two-soliton solutions of the 3-component Maccari system with reduction formulas, we obtained soliton solutions of the reduced integrable 2-component Maccari systems. Types of the one-soliton solutions obtained for these reduced local and nonlocal systems are solitons, solitoffs, and V-type solitary waves. Two-soliton solutions of reduced 2-component systems define interactions of mentioned type of waves.
  
  As a future work we plan to obtain different types of exact solutions to local and nonlocal reductions of 3-component Maccari system. In recent years, rogue wave (also known as freak wave, killer wave) phenomenon has become a hot topic in the nonlinear science. Rogue waves are large and spontaneous ocean surface waves. They also appear in optics, superfluids, Bose-Einstein condensates, cold matter systems, and so on.  By finding
  higher-order soliton solutions via the Hirota bilinear method and applying long wave limit we can obtain rogue wave solutions <cit.>-<cit.> to the 3-component Maccari system and its local and nonlocal reductions. To study integrability properties of the Maccari system we will consider Lie symmetry analysis of this system <cit.>-<cit.>. Under this analysis we can also obtain new solutions of the 3-component Maccari system and its reductions.
  
  
  
  
  
   
  
  00
  Maccari1997 A. Maccari, Universal and integrable nonlinear evolution systems of equations in (2+1) dimensions, J. Math. Phys. 38, 4151, 1997.
  
  Maccari2020 A. Maccari, The Maccari system as model system for rogue waves, Phys. Lett. A 384, 126740, 2020.
  
  Uthayakumar A. Uthayakumar, K. Nakkeeran, and K. Porsezian, Soliton solutions of new (2+1) dimensional nonlinear partial differential equations,
  Chaos Solitons Fractals 10 (9), 1513–1518, 1999.
  
  
  
  Zakharov V.F. Zakharov, A.B. Shabat,  Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of wave in nonlinear media,
  Soviet Phys. JETP 34 (1), 62–69, 1972.
  
  Craik A.D.D. Craik, Wave interactions and fluid flows, Cambridge University Press, Cambridge, 1985.
  
  Fokas A.S. Fokas, On the simplest integrable equation in 2+1, Inverse Probl. 10, L19–L22, 1994.
  
  
  
  HanChen1 Z. Han, Y. Chen, Bright-dark mixed N-soliton solution of the two-dimensional Maccari system, Chin Phys. Lett. 34 (7), 070202, 2017.
  
  HanChen2 Z. Han, Y. Chen, Bright-dark mixed N-soliton solution of the two-dimensional multicomponent Maccari system, Z. Naturforsch. A 72 (8), 2017.
  
  Xu T. Xu, Y. Chen, and Z. Qiao, Multi-dark soliton solutions for the (2+1)-dimensional
  multi-component Maccari system, Modern Phys. Lett. B  33 (31), 1950390, 2019.
  
  Wazwaz W. Liu, A.-M. Wazwaz, Dynamics of fusion and fission collisions between lumps and line solitons in the Maccari's system, Rom. J. Phys. 64, 111, 2019.
  
  
  HZZ W.-H. Huang, C.-L. Zheng, and J.-F. Zhang, Coherent soliton structures of a new (2+1) dimensional evolution equation, Acta Phys. Sin. (in Chinese) 51, 2676, 2002.
  
  HLM W.-H. Huang, Y.-L. Liu, and Z.-Y. Ma, Doubly periodic propagating wave patterns of (2+1)-dimensional Maccari system, Commun. Theor. Phys. 47, 397–402, 2007.
  
  HuangZhang W.-H. Huang, J.-F. Zhang, Folded localized excitations of the Maccari system, Acta Phys. Polon. B 35 (8), 2004.
  
  CCS N. Cheemaa, S. Chen, and A.R. Seadawy, Propagation of isolated waves of coupled nonlinear (2+1)-dimensional Maccari system in plasma physics, Results Phys. 17, 102987, 2020.
  
  Yuan F. Yuan, J. Rao, K. Porsezian, D. Mihalache, and J. He, Various exact rational solutions of the two-dimensional Maccari's system, Rom. J. Phys. 61 (3-4), 378–399, 2016.
  
  Zhang J.-F. Zhang, Generalized dromion structures of new (2+1)-dimensional nonlinear evolution equation, Commun. Theor. Phys. 35, 267–270, 2001.
  
  ZMH J.-F. Zhang, J.-P. Meng, and W.-H. Huang, Novel class of coherent localized structures for the Maccari system, Commun. Theor. Phys. 40, 443–446, 2003.
  
      AbMu1 M.J. Ablowitz and Z.H. Musslimani, Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett. 110, 064105, 2013.
  
  AbMu2 M.J. Ablowitz and Z.H. Musslimani, Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation, Nonlinearity 29, 915–946, 2016.
  
  AbMu3 M.J. Ablowitz and Z.H. Musslimani, Integrable nonlocal nonlinear equations, Stud. App. Math. 139 (1), 7–59, 2016.
  
  
  Lou1 S.Y. Lou, F. Huang, Alice-Bob physics: Coherent solutions of nonlocal KdV systems, Sci. Rep.  7, 869, 2017.
  
  Lou2 S.Y. Lou, Alice-Bob systems, P̂-T̂-Ĉ symmetry invariant and symmetry breaking soliton solutions,
       J. Math. Phys. 59, 083507, 2018.
  
  magnet T.A. Gadzhimuradov, A.M. Agalarov, Towards a gauge equivalent magnetic structure of the nonlocal nonlinear Schrödinger equation,
  Phys. Rev. A 93, 062124, 2016.
  
  origin M. Gürses, A. Pekcan, and K. Zheltukhin, Discrete symmetries and nonlocal reductions, Phys. Lett. A 384, 120065, 2020.
  
  
  chen K. Chen, X. Deng, S. Lou, and D. Zhang, Solutions of local and nonlocal equations reduced from the AKNS hierarchy, Stud. Appl. Math. 141 (1), 113–141, 2018.
  
  FLAH B.F. Feng, X.D. Luo, M.J. Ablowitz, and Z.H. Musslimani, General soliton solution to a nonlocal nonlinear Schrödinger
  equation with zero and nonzero boundary conditions, Nonlinearity 31 (12), 5385–5409, 2018.
  
  
  gerd V.S. Gerdjikov and A. Saxena, Complete integrability of nonlocal nonlinear Schrödinger equation, J. Math. Phys. 58 (1), 013502, 2017.
  
  
  
  GurPek1  M. Gürses and A. Pekcan, Nonlocal nonlinear Schrödinger equations and their soliton solutions, J. Math. Phys. 59, 051501, 2018.
  
  GurPek3 M. Gürses and A. Pekcan, Integrable nonlocal reductions, Symmetries, Differential Equations and Applications SDEA-III, Istanbul, Turkey, August 2017, in: V.G. Kac, P.J. Olver, P. Winternitz, T. Ozer (Eds), Springer Proceedings in Mathematics and Statistics, 266, 27–52, 2018.
  
  
  huang X. Huang, L. Ling, Soliton solutions for the nonlocal nonlinear Schrödinger equation, Eur. Phys. J. Plus 131, 148, 2016.
  
  
  li  M. Li, T. Xu, Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation
  with the self-induced parity-time-symmetric potential, Phys. Rev. E 91, 033202, 2015.
  
  
  aflm1 M.J. Ablowitz, B.F. Feng, X.D. Luo, and Z.H. Musslimani, Inverse scattering transform for the nonlocal reverse space-time
  nonlinear Schrödinger equation, Theor. Math. Phys. 196 (3), 1241–1267, 2018.
  
  
  Wen X.Y. Wen, Z. Yan, and Y. Yang, Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger
  equation with the self-induced parity-time-symmetric potential, Chaos 26, 063123, 2015.
  
  Sax A. Khare, A. Saxena, Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear
  equations, J. Math. Phys. 56, 032104, 2015.
  
  jianke J. Yang, General N-solitons and their dynamics in several nonlocal nonlinear Schrödinger equations, Phys. Lett A 383 (4), 328–337, 2019.
  
  XLLLZ T. Xu, S. Lan, M. Li, L.-L. Li, and G.-W. Zhang, Mixed soliton solutions of the defocusing nonlocal nonlinear Schrödinger equation, Phys. D
  390, 47–61, 2019.
  
  XCLM T. Xu, Y. Chen, M. Li, and D.-X. Meng, General stationary solutions of the nonlocal nonlinear Schrödinger equation and their relevance
  to the 𝒫𝒯-symmetric system, Chaos 29, 123124, 2019.
  
  Ma1 W.-X. Ma, Inverse scattering for nonlocal reverse-time nonlinear Schrödinger equations, Appl. Math. Lett. 102, 106161, 2020.
  
  
  GurPek2  M. Gürses,  A. Pekcan, Nonlocal nonlinear modified KdV equations and their soliton solutions, Commun. Nonlinear Sci. Numer.
  Simul. 67, 427, 2019.
  
  JZ1 J.L. Ji, Z.N. Zhu, On a nonlocal modified Korteweg-de Vries equation: Integrability, Darboux transformation and soliton solutions,
  Commun. Non. Sci. Numer. Simul. 42, 699, 2017.
  
  JZ2 J.L. Ji, Z.N. Zhu, Soliton solutions of an integrable nonlocal modified Korteweg-de Vries equation through
  inverse scattering transform, J. Math. Anal. Appl. 453, 973, 2017.
  
  ma   L.Y. Ma, S.F. Shen, and Z.N. Zhu, Soliton solution and gauge equivalence for an integrable nonlocal complex modified Korteweg-de Vries
  equation, J. Math. Phys. 58, 103501, 2017.
  
  Shi X. Shi, P. Lv, and C. Qi, Explicit solutions to a nonlocal 2-component complex modified Korteweg-de Vries equation, Appl. Math. Lett.
  100, 106043, 2020.
  
  Luo X.-D. Luo, Inverse scattering transform for the complex reverse space-time nonlocal modified Korteweg-de Vries equation with nonzero boundary conditions and constant phase shift, Chaos 29, 073118, 2019.
  
  Yan G. Zhang, Z. Yan, Inverse scattering transforms and soliton solutions of focusing and defocusing nonlocal mKdV equations with nonzero boundary conditions, Phys. D 402, 132170, 2020.
  
  Gurses M. Gürses, Nonlocal Fordy-Kulish equations on symmetric spaces, Phys. Lett. A 381, 1791–1794, 2017.
  
  GurPekKos M. Gürses, A. Pekcan, and K. Zheltukhin, Nonlocal hydrodynamic type of equations, Commun. Nonlinear Sci. Numer. Simul. 85, 105242, 2020.
  
  GurPek6 M. Gürses, A. Pekcan, Nonlocal KdV equations, Phys. Lett. A 384 (35), 126894, 2020.
  
  
  
  RZFH J. Rao, Y. Zhang, A.S. Fokas, and J. He, Rogue waves of the nonlocal Davey-Stewartson I equation, Nonlinearity 31, 4090–4107, 2018.
  
  XLHCY T. Xu, M. Li, Y. Huang, Y. Chen, and C. Yu, Nonsingular localized wave solutions for the nonlocal Davey-Stewartson I equation
  with zero background, Modern Phys. Lett. B 31 (35), 1750338, 2017.
  
  ZXZhou Z.-X. Zhou, Darboux transformations global explicit solutions for nonlocal Davey-Stewartson I equation, Stud. Appl. Math. 141 (2) 186–204, 2018.
  
  
  ZL Y. Zhang, Y. Liu, Breather and lump solutions for nonlocal Davey-Stewartson II equation, Nonlinear Dyn. 96, 107–113, 2019.
  
  Rao J. Rao, Y. Cheng, and J. He, Rational and semirational solutions of the nonlocal Davey-Stewartson equations, Stud. Appl. Math. 139 (4), 2017.
  
  Mihalache Y. Cao, J. Rao, D. Mihalache, and J. He, Semi-rational solutions for the (2+1)-dimensional nonlocal Fokas system, Appl. Math. Lett. 80, 27–34, 2018.
  
      XZhu X. Zhu, A coupled (2+1)-dimensional mKdV system and its nonlocal reductions, Commun. Nonlinear Sci. Numer. Simul. 91, 105438, 2020.
  Guo N.-N. Li, H.-Q. Hao, and R. Guo, Darboux transformation and the exact solutions of the (2+1)-dimensional nonlocal complex modified Korteweg-de Vries adn Maxwell-Bloch equations, Modern Phys. Lett. B 34 (22), 2050230, 2020.
  
  
  LiuLi Y.-K. Liu, B. Li, Rogue waves in the (2+1)-dimensional nonlinear Schrödinger equation with a parity-time symmetric potential, Chin. Phys. Lett. 34 (1), 010202, 2017.
  
  ZhangChen X. Zhang, Y. Chen, and Y. Zhang, Breather, lump and X soliton solutions to nonlocal KP equation, Comput. Math. Appl. 74, 2341–2347, 2017.
  
  
  
         GurPek4  M. Gürses, A. Pekcan, (2+1)-dimensional local and nonlocal reductions of the negative AKNS system: Soliton solutions, Commun. Nonlinear Sci. Numer. Simul. 71, 161–173, 2019.
  
  GurPek5 M. Gürses, A. Pekcan, (2+1)-Dimensional AKNS(-N) Systems: N=3,4, 	arXiv:1910.11298 [nlin.SI].
  
  Hirota1 R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple
  collisions of solitons, Phys. Rev. Lett. 27, 1192, 1971.
  
  Hirota2 R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, 2004.
  
  Hietarinta J. Hietarinta, Introduction to the Bilinear Method, in: Integrability of
  Nonlinear Systems, eds. Y. Kosman-Schwarzbach, B. Grammaticos, and
  K.M. Tamizhmani, Springer Lecture Notes in Physics 495, 95–103, 1997.
  
  Yu1 J.-P. Yu, Y.-L. Sun, Study of lump solutions to dimensionally reduced generalized KP equations, Nonlinear Dyn. 87, 2755–2763, 2017.
  
  Yu2 J.-P. Yu, Y.-L. Sun, A direct Bäcklund transformation for a (3+1)-dimensional Kadomtsev-Petviashvili-Boussinesq-like equation, Nonlinear Dyn. 90, 2263–2768, 2017.
  
  Yu3 Y.-L. Sun, W.-X. Ma, J.-P. Yu, and C.M. Khalique, Dynamics of lump solitary wave of Kadomtsev-Petviashvili-Boussinesq-like equation, Comput. Math. Appl. 78, 840–847, 2019.
  
  
  RW1 W. Gai-Hua, W. Li-Hong, R. Ji-Guang, and H. Jing-Song, New patterns of the two-dimensional rogue waves: (2 + 1)-dimensional Maccari system, Commun. Theor. Phys. 67, 601, 2017.
  
  RW2 H.M. Baskonus, T.A. Sulaiman, and H. Bulut, On the novel wave behaviors to the coupled nonlinear Maccari’s system with complex structure, Optik 131, 1036, 2017.
  
  RW3 M. Shakeel, S.T. Mohyud-Din, and M.A. Iqbal, Closed form solutions for coupled nonlinear Maccari system, Comput. Math. Appl. 76 (4) 799, 2018.
  
  RW4 N. Chemaa, S. Chen, and A.R. Seadawy, Propagation of isolated waves of coupled nonlinear (2 + 1)-dimensional Maccari system in plasma physics, Results Phys. 17, 102987, 2020.
  RW5 Y. Jiang, D.-Q. Xian, and X.-R. Kang, Homoclinic breather and rogue wave solutions to Maccari equation, Comput. Math. Appl. 79 (7), 1890-–1894, 2020.
  
  
  Kumar1 S. Kumar, M. Niwas, and A.-M. Wazwaz, Lie symmetry analysis, exact analytical solutions, and dynamics of solitons for (2+1)-dimensional NNV equations, Phys. Scr. 95 (9), 095204, 2020.
  
  Kumar2 S. Kumar, S. Rani, Lie symmetry reductions and dynamics of soliton solutions of (2+1)-dimensional Pavlov equation, Pramana 94, Art. No. 116, 2020.
  
  Kumar3 S. Kumar, A. Kumar, and H. Kharbanda, Lie symmetry analysis and generalized invariant solutions of (2+1)-dimensional dispersive long wave (DLW) equations, Phys. Scr. 95 (6), 065207, 2020.
  Kumar4  S. Kumar, A. Kumar, Lie symmetry reductions and group invariant solutions of (2+1)-dimensional modified Veronese web equation, Nonlinear Dyn. 98 (3), 1891–1903, 2019.
  
  Kumar5 S. Kumar, A. Kumar, Dynamical structures of solitons and some new types of exact solutions for the (2+1)-dimensional DJKM equation using Lie symmetry analysis, Modern Phys. Lett. B 34, 2150015, 2020.
  
  
  
  