The First Integral Method for Solving Maccari’s System

Abstract

In this paper, we investigate the first integral method for solving the solutions of Maccari's system. This idea can obtain some exact solutions of this system based on the theory of Commutative algebra.

New solitary wave solutions of Maccari system

“…In this section, we seek the exact solutions of Maccari system (Maccari, 1996;Rostamy et al, 2011;Jabbari et al, 2011;Zhao, 2008;Zhang, 2007;Wazwaz, 2012;Rostamy and Zabihi, 2012;Abdou et al, 2012;Bekir, 2009;El-Ganaini, 2012;Ye and Zhang, 2011;Manafian and Zamanpour, 2013;Lee and Sakthivel, 2013;Akbari, 2013;Kumar and Chand, 2013;Malik et al, 2012;Li-Na and Qing, 2005;Jun-Ting and Lun-Xun, 2007;Huang et al, 2007;Mirzazadeh, 2015;Ahmed et al, 2013;Hafez et al, 2015;MohyudDin and Shakeel, 2013) by using extended trial equation method.…”

“…The integrability property is clearly indicated by exhibiting the corresponding Lax pair that is obtained by implementing the reduction technique to the Lax pair of Kadomtsev-Petviashvili equation (Maccari, 1996). Maccari system is a kind of NLEEs that are frequently introduced to define the motion of the isolated waves, localized in a small part of space, in many fields such as quantum field theory, hydrodynamics, plasma physics, and nonlinear optics (Rostamy et al, 2011).…”

“…Many authors have used to find exact solutions a lot of methods of Maccari system such as exp À ΦðξÞ À Á Àexpansion method, extended F-expansion method, projective Ricatti equation method, first integral method, the tanh-coth method, the sech-csch method, the sec-tan ansatz method, the rational sinh-cosh ansatz method, G 0 =G À Á Àexpansion method, mapping method, lie symmetry approach, He's semi-inverse method, exp-function method, Kudryashov method, the modified simplest equation method, the extended homogeneous balance method, bifurcation theory, extended sine-Gordon equation method, variable separation approach, sine-cosine method and the generalized algebraic method (Maccari, 1996;Rostamy et al, 2011;Jabbari et al, 2011;Zhao, 2008;Zhang, 2007;Wazwaz, 2012;Rostamy and Zabihi, 2012;Abdou et al, 2012;Bekir, 2009;El-Ganaini, 2012;Ye and Zhang, 2011;Manafian and Zamanpour, 2013;Lee and Sakthivel, 2013;Akbari, 2013;Kumar and Chand, 2013;Malik et al, 2012;LiNa and Qing, 2005;Jun-Ting and Lun-Xun, 2007;Huang et al, 2007;Mirzazadeh, 2015;Ahmed et al, 2013;Hafez et al, 2015;Mohyud-Din and Shakeel, 2013).…”

Exact solutions of the stochastic Maccari system forced by multiplicative noise

“…Numerous authors have investigated the exact solutions to the SMS () using a variety of techniques, such as the multilinear variable separation approach [21], modified trial equation [22, 23], generalized Kudryashov [23], mapping method [24, 25], generalized Riccati relation [26], $$ exp(-\Phi (\eta ))$$‐expansion [27], $$(G^{\prime }/G)$$‐expansion [28, 29], the extended modified auxiliary equation mapping method [30], modified $$exp(-\Phi (\eta ))$$‐expansion [31] , He's semi‐inverse [29, 32], Kudryashov [33], first integral [34].…”

Application of Tanh Method to Complex Coupled Nonlinear Evolution Equations

“…The complex coupled Maccari equations is a nonlinear evolution equations described the motion of the isolated waves, localized in a small part of space, in many fields such as hydrodynamic, plasma physics, nonlinear optics and others [22][23][24][25][26][27][28]. Complex coupled Maccari equation derived from the Kadomtsev-Petviashvili equation (the best known two-dimensional generalizations of the KdV equation) and can be written in the form i u t + u xx + uv = 0,…”