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Reductions and New Exact Solutions of ZK, Gardner KP, and Modified KP Equations via Generalized Double Reduction Theorem

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  • R. Naz
  • Z. Ali
  • I. Naeem

Abstract

We study here the Lie symmetries, conservation laws, reductions, and new exact solutions of (2 + 1) dimensional Zakharov‐Kuznetsov (ZK), Gardner Kadomtsev‐Petviashvili (GKP), and Modified Kadomtsev‐Petviashvili (MKP) equations. The multiplier approach yields three conservation laws for ZK equation. We find the Lie symmetries associated with the conserved vectors, and three different cases arise. The generalized double reduction theorem is then applied to reduce the third‐order ZK equation to a second‐order ordinary differential equation (ODE) and implicit solutions are established. We use the Sine‐Cosine method for the reduced second‐order ODE to obtain new explicit solutions of ZK equation. The Lie symmetries, conservation laws, reductions, and exact solutions via generalized double reduction theorem are computed for the GKP and MKP equations. Moreover, for the GKP equation, some new explicit solutions are constructed by applying the first integral method to the reduced equations.

Suggested Citation

  • R. Naz & Z. Ali & I. Naeem, 2013. "Reductions and New Exact Solutions of ZK, Gardner KP, and Modified KP Equations via Generalized Double Reduction Theorem," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:340564
    DOI: 10.1155/2013/340564
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    References listed on IDEAS

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    1. Zhao, Xiaoshan & Xu, Wei & Jia, Huabing & Zhou, Hongxian, 2007. "Solitary wave solutions for the modified Kadomtsev–Petviashvili equation," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 465-475.
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    Cited by:

    1. Rehana Naz & Igor Leite Freire & Imran Naeem, 2014. "Comparison of Different Approaches to Construct First Integrals for Ordinary Differential Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).

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