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New Exact Solutions of the (2 + 1)-Dimensional AKNS Equation

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DOI: 10.4236/jamp.2015.311167    4,385 Downloads   4,788 Views  
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N-soliton solutions and the bilinear form of the (2 + 1)-dimensional AKNS equation are obtained by using the Hirota method. Moreover, the double Wronskian solution and generalized double Wronskian solution are constructed through the Wronskian technique. Furthermore, rational solutions, Matveev solutions and complexitons of the (2 + 1)-dimensional AKNS equation are given through a matrix method for constructing double Wronskian entries. The three solutions are new.

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Sun, Y. (2015) New Exact Solutions of the (2 + 1)-Dimensional AKNS Equation. Journal of Applied Mathematics and Physics, 3, 1391-1405. doi: 10.4236/jamp.2015.311167.


[1] Gardner, C.S., Greene, J.M., Kruskal, M.D. and Miura, R.M. (1967) Method for Solving the Korteweg de Vries Equation. Physical Review Letters, 19, 1095-1097.
[2] Matveev, V.B. and Salle, M.A. (1991) Darboux Transformations and Solitons. Springer-Verlag, Berlin.
[3] Hirota, R. (1971) Exact Solution of the Korteweg-de Vries Equation for Multiple Collisions of Solitons. Physical Review Letters, 27, 1192-1194.
[4] Freeman, N.C. and Nimmo, J.J.C. (1983) Soliton Solutions of the KdV and KP Equations: The Wronskian Technique. Physics Letters A, 95, 1-3.
[5] Nimmo, J.J.C. and Freeman, N.C. (1983) A Method of Obtaining the N-Soliton Solution of the Boussinesq Equation in Terms of a Wronskian. Physics Letters A, 95, 4-6.
[6] Hu, X.B. and Wang, H.Y. (2006) Construction of dKP and BKP Equations with Self-Consistent Sources. Inverse Problems, 22, 1903-1920.
[7] Hu, X.B. and Wang, H.Y. (2007) New Type of Kadomtsev-Petviashvili Equation with Self-Consistent Soureces and Its Blinear Bäcklund Transformation. Inverse Problems, 23, 1433-1444.
[8] Satsuma, J. (1979) A Wronskian Representation of N-Soliton Solutions of Nonlinear Evolution Equations. Journal of the Physical Society of Japan, 46, 359-360.
[9] Matveev, V.B. (1992) Generalized Wronskian Formula for Solutions of the KdV Equation: First Applications. Physics Letters A, 166, 205-208.
[10] Ma, W.X. (2002) Complexiton Solutions to the Korteweg-de Vries Equation. Physics Letters A, 301, 35-44.
[11] Ma, W.X. (2004) Wronskians, Generalized Wronskians and Solutions to the Korteweg-de Vries Equation. Chaos, Solitons & Fractals, 19, 163-170.
[12] Ma, W.X. and Maruno, K. (2004) Complexiton Solutions of the Toda Lattice Equation. Physica A, 343, 219-237.
[13] Ma, W.X. and You, Y. (2005) Solving the Korteweg-de Vries Equation by Its Bilinear Form: Wronskian Solutions. Transactions of the American Mathematical Society, 357, 1753-1778.
[14] Ma, W.X. (2005) Complexiton Solutions to Integrable Equations. Nonlinear Analysis: Theory, Methods & Applications, 63, 2461-2471.
[15] Ma, W.X. (2005) Complexiton Solutions of the Korteweg-de Vries Equation with Self-Consistent Sources. Chaos, Solitons & Fractals, 26, 1453-1458.
[16] Ma, W.X., Li, C.X. and He, J.S. (2009) A Second Wronskian Formulation of the Boussinesq Equation. Nonlinear Analysis: Theory, Methods & Applications, 70, 4245-4258.
[17] Ablowitz, M.J., Kaup, D.J., Newell, A.C. and Segur, H. (1974) The Inverse Scattering Transform-Fourier Analysis for Nonlinear Problems. Studies in Applied Mathematics, 53, 249-315.
[18] Liu, Q.M. (1990) Double Wronskian Solution of the AKNS and Classical Boussinesq Hierarchies. Journal of the Physical Society of Japan, 59, 3520-3527.
[19] Chen, D.Y., Zhang, D.J. and Bo, J.B. (2008) New Double Wronskian Solutions of the AKNS Equation. Science in China Series A: Mathematics, 51, 55-69.
[20] Lou, S.Y. and Hu, X.B. (1997) Infinitely Many Lax Pair and Symmetry Constraints of the KP Equation. Journal of Mathematical Physics, 38, 6407-6427.
[21] Lou, S.Y., Chen, C.L. and Tang, X.Y. (2002) (2 + 1)-Dimensional (M + N)-Component AKNS System: Painlevé Integrability, Infinitely Many Symmetries, Similarity Reductions and Exact Solutions. Journal of Mathematical Physics, 43, 4078-4109.

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