Bright-Dark Mixed N -Soliton Solution of Two-Dimensional Multicomponent Maccari System

Abstract

By virtue of the KP hierarchy reduction technique, we construct the general bright-dark mixed N-soliton solution to the multi-component Mel'nikov system comprised of multiple (say M) short-wave components and one longwave component with all possible combinations of nonlinearities including all-positive, all-negative and mixed types. Firstly, the two-bright-one-dark (2-b-1-d) and one-bright-two-dark (1-b-2-d) mixed N-soliton solutions in short-wave components of the three-component Mel'nikov system are derived in detail. Then we extend our analysis to the M-component Mel'nikov system to obtain its general mixed N-soliton solution. The formula obtained unifies the allbright, all-dark and bright-dark mixed N-soliton solutions. For the collision of two solitons, the asymptotic analysis shows that for a M-component Mel'nikov system with M ≥ 3, inelastic collision takes place, resulting in energy exchange among the short-wave components supporting bright solitons only if the bright solitons appear at least in two short-wave components. Whereas, the dark solitons in the short-wave components and the bright solitons in the long-wave component always undergo elastic collision which just accompanied by a position shift.

Local and nonlocal (2 + 1)-dimensional Maccari systems and their soliton solutions

“…Here by using the Hirota bilinear method [61][62][63] we will obtain one-and two-soliton solutions of 3-component Maccari system (6)- (8). Let…”

“…There are several works on obtaining solutions of the Maccari system (3)- (5). By KP hierarchy reduction method, bright-dark mixed soliton solutions [7,8], multi-dark soliton solutions [9], semi-rational solutions [10] of the Maccari system were 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.…”

Multi-dark soliton solutions for the (2+1)-dimensional multi-component Maccari system

“…Different types of rational solutions including multi-rogue waves, lump solitons and interactions between the two were all given in the Maccari system by bilinear method [35]. Additionally, the bright solitons [40] and bright-dark mixed solitons [26,41] for the Maccari system were all studied by different methods. To the best of our knowledge, multi-dark soliton solutions for the coupled Maccari system (1)- (3) and the corresponding multi-component generalization (4)-(5) have never been reported up to now.…”

Multi-soliton solutions for the coupled modified nonlinear Schrödinger equations via Riemann–Hilbert approach

“…Due to the profound theoretical and practical significance, the study of multi-soliton solutions of nonlinear partial differ-ential equations has always been a hot topic. To date, a variety of approaches have been well established to investigate the multi-soliton solutions of nonlinear partial differential equations, such as the inverse scattering method, [9] the Hirota's bilinear method, [10] the Darboux transformation method, [11][12][13] the Riemann-Hilbert approach, [14] the KP hierarchy reduction method, [15][16][17] the generalized unified method, [18,19] and so forth. The Riemann-Hilbert approach as a very powerful approach in dealing with nonlinear partial differential equations with initial-boundary value conditions has attracted more and more attention in recent years.…”