Subscribe to our Newsletter and get informed about new publication regulary and special discounts for subscribers!

IJARM > IJARM Volume 6 > AKNS Formalism and Exact Solutions of KdV and...
< Back to Volume

AKNS Formalism and Exact Solutions of KdV and Modified KdV Equations with Variable-Coefficients

Full Text PDF


We apply the AKNS hierarchy to derive the generalized KdV equation andthe generalized modified KdV equation with variable-coefficients. We system-atically derive new exact solutions for them. The solutions turn out to beexpressible in terms of doubly-periodic Jacobian elliptic functions.


International Journal of Advanced Research in Mathematics (Volume 6)
S. Das and D. Ghosh, "AKNS Formalism and Exact Solutions of KdV and Modified KdV Equations with Variable-Coefficients", International Journal of Advanced Research in Mathematics, Vol. 6, pp. 32-41, 2016
Online since:
September 2016

[1] J. H. He, Variational iteration method-a kind of non-linear analytical technique: some examples, Inter. J. Nonlin. Mech. 34(4) (1999) 699-708.

[2] R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27(18) (1971) 1192.

[3] M. R. Miura, Backlund Transformation, Springer-Verlag, Berlin, (1978).

[4] Z. Yan, H. Zhang, Nonlinear wave agenda of similar reduction with damping term, Physics Journals. 49(11) (2000) 2113-2117.

[5] M. J. Ablowitz, P. A. Clarkson, Soliton, nonlinear evolution equation and inverse scattering, Cambridge University press, New York, (1991).

[6] P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure. Appl. Math. 21(5) (1968) 467-490.

[7] C. Yan, A simple transformation for nonlinear waves, Phys. Lett. A. 224(1) (1996) 77-84.

[8] S. Yu et al., Solitary wave solutions to approximate fully nonlinear double sine-Gordon equation, International Journal of Nonlinear Science. 3(3) (2007) 163-169.

[9] J. F. Zhang, Homogeneous balance method and chaotic and fractal solutions for the Nizhnik- Novikov-Veselov equation, Phys. Lett. A. 313(5) (2003) 401-407.

[10] P. G. Drazin, R. S. Johnson, Solitons: An Introduction, Cambridge University press, London, (1983).

[11] M. Lakshmanan, S. Rajasekar, Nonlinear dynamics : Integrability, Chaos and Patterns, Advanced Texts in Physics, Springer-Verlag, Berlin, (2003).

[12] J. P. Wang, A list of 1+ 1 dimensional integrable equations and their properties, Journal of Nonlinear Math. Phys. 9 (2002) 213-233.

[13] V.B. Matveev, Generalized Wronskian formula for solutions of the KdV equations: first applications, Phys. Lett. A. 166(3-4) (1992) 205-208.

[14] M.R. Gupta, Exact inverse scattering solution of a non-linear evolution equation in a non-uniform medium, Phys. Lett. A. 72(6) (1979) 420-422.

[15] M.J. Ablowitz et al., The Inverse Scattering Transform-Fourier Analysis for Nonlinear Problems, Stud. Appl. Math. 53(4) (1974) 249-315.

[16] M.J. Ablowitz et al., Nonlinear-evolution equations of physical significance, Phys. Rev. Lett. 31(2) (1973) 125.

[17] M.J. Ablowitz, J.F. Ladik, Nonlinear differential− difference equations, Journal of Mathematical Physics. 16(3) (1975) 598-603.

[18] W. Hua, D.J. Zhang, Strong Symmetries of Non-Isospectral Ablowitz-Ladik Equations, Chin. Phys. Lett. 28(2) (2011) 020203.

[19] H.H. Chen, C.S. Liu, Solitons in nonuniform media, Phys. Rev. Lett. 37(11) (1976) 693.

[20] R. Hirota, J. Satsuma, N-soliton solution of the K-dV equation with loss and nonuniformity terms, Journal of the Physical Society of Japan. 41 (1976) 2141.

[21] W.L. Chan. L. Kam-Shun, Nonpropagating solitons of the variable coefficient and nonisospectral Korteweg-de Vries equation, J. Math. Phys. 30(11) (1989) 2521-2526.

[22] J.B. Bi, Y.P. Sun, D.Y. Chen, Soliton Solutions for Nonisospectral AKNS Equation by Hirota's Method, Communications in Theoretical Physics. 45(3) (2006) 398.

[23] Z. Qiao, C. Cao, W. Strampp, Category of nonlinear evolution equtions, algebraic structure, and r-matrix, Unpublished paper.

[24] Z.T. Fu et al., New exact solutions to KdV equations with variable coefficients or forcing, Applied Mathematics and Mechanics. 25(1) (2004) 73-79.

[25] A. Biswas, Solitary wave solution for KdV equation with power-law nonlinearity and timedependent coefficients, Nonlinear Dynamics. 58(1-2) (2009) 345-348.

[26] A. Biswas, Solitary wave solution for the generalized KdV equation with time-dependent damping and dispersion, Communications in Nonlinear Science and Numerical Simulation. 14(9) (2009) 3503-3506.

[27] H. Ma, A. Deng, Y. Wang, Exact solution of a KdV equation with variable coefficients, Computers and Mathematics with Applications. 61(8) (2011) 2278-2280.

[28] S. Zhang, Application of Exp-function method to a KdV equation with variable coefficients, Phys. Lett. A. 365(5) (2007) 448-453.

[29] S.M. Yu, L.X. Tian, Generalized soliton solutions to generalized KdV equation with variable coefficients by Exp-function method, Journal of Physics: Conference Series. 96(1) (2008) 012022.

[30] S.F. Deng, Exact solutions for a nonisospectral and variable-coefficient kdv equation, Communications in Theoretical Physics. 43(6) (2005) 961.

[31] K. Pradhan, P. K. Panigrahi, Parametrically controlling solitary wave dynamics in the modified Korteweg-de Vries equation, Journal of Physics A: Mathematical and General. 39(20) (2006) L343.

[32] J. Li et al., Symbolic computation on integrable properties of a variable-coefficient Korteweg-de Vries equation from arterial mechanics and Bose-Einstein condensates, Physica Scripta. 75(3) (2007) 278.

[33] J. Li et al., Lax pair, Bäcklund transformation and N-soliton-like solution for a variablecoefficient Gardner equation from nonlinear lattice, plasma physics and ocean dynamics with symbolic computation, Journal of Mathematical Analysis and Applications. 336(2) (2007).

[34] X.L. Gai et al., On a variable-coefficient Korteweg-de Vries model in fluid-filled elastic tubes, Journal of Physics A: Mathematical and Theoretical. 43(45) (2010) 455205.

[35] Y. Jiang et al., Soliton solutions for a variable-coefficient Korteweg? de Vries equation in fluids and plasmas, Physica Scripta. 82(5) (2010) 055008.

[36] A.G. Johnpillai, C.M. Khalique, Group analysis of KdV equation with time dependent coefficients, Applied Mathematics and Computation. 216(12) (2010) 3761-3771.

[37] C.A.G. Sierra, On a KdV equation with higher-order nonlinearity: Traveling wave solutions, Journal of computational and applied mathematics. 235(17) (2011) 5330-5332.

[38] Y. Qin et al., Bell polynomial approach and N-soliton solutions for a coupled KdV-mKdV system, Communications in Theoretical Physics. 58(1) (2012) 73.

[39] Y. Jiang et al., Soliton solutions and integrability for the generalized variable-coefficient extended Korteweg-de Vries equation in fluids, Applied Mathematics Letters. 26(4) (2013) 402-407.

[40] Y.F. Xiao, H.L. Xue, The new multi-order exact solutions of some nonlinear evolution equations, Journal of Atomic and Molecular Sciences. 3 (2012) 136-151.

Show More Hide
Cited By:

[1] V. Serkin, T. Belyaeva, "Novel conditions for soliton breathers of the complex modified Korteweg–de Vries equation with variable coefficients", Optik, Vol. 172, p. 1117, 2018