The Multi-Soliton Solutions to The KdV Equation by Hirota Method

Lixin MA


The Hirota bilinear method is used to solve the KdV model. As a result, the exact expression of multi-soliton solutions of the KdV equation is obtained.


Nonlinear partial differential equations; The KdV equation; Hirota bilinear method; Multi-soliton solution

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Parkes, E. J., & Duffy, B. R. (1996). An automated tanh-functionmethod for finding solitary wave solutions to nonlinear evolution equations. Computer Physics Communications, 98(3), 288-300.

Fan, E. (2000). Extended tanh-function method and its applications to nonlinear equations. Physics Letters A, 277(4-5), 212-218.

Liu, S., Fu, Z., Liu, S., & Zhao, Q. (2001). Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Physics Letters A, 289(1-2), 69-74.

Fu, Z., Liu, S., Liu, S., & Zhao, Q. (2001). New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Physics Letters A, 290(1-2), 72-76.

Wang, M., & Zhou, Y. (2003). The periodic wave solutions for the Klein-Gordon-Schr¨odinger equations. Physics Letters A, 318(1-2), 84-92.3

Zhou, Y., Wang, M., & T. Miao, T. (2004). The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations. Physics Letters A, 323(1-2), 77-88.

Wang, M., & Li, X. (2005). Extended -expansion method and periodic wave solutions for the generalized Zakharov equations. Physics Letters A, 343(1-3), 48-54.

Wang, M., & Li, X. (2005). Applications of -expansion to periodic wave solutions for a new Hamiltonian amplitude equation. Chaos, Solitons and Fractals, 24(5), 1257-1268.

Wazwaz, A.-M. (2003). A study on nonlinear dispersive partial differential equations of compact and noncompact solutions. Applied Mathematics and Computation, 135(2-3), 399-409.

Wazwaz, A.-M. (2003). A construction of compact and noncompact solutions for nonlinear dispersive equations of even order. Applied Mathematics and Computation, 135(2-3), 411-424.

Wang, M. (1995). Solitary wave solutions for variant Boussinesq equations. Physics Letters A, 199(3-4), 169-172.

Wang, M. (1996). Exact solutions for a compound KdV-Burgers equation. Physics Letters A, 213(5-6), 279-287.

Wang, M., Zhou, Y., & Li, Z. B. (1996). Application of a homogeneous balance method to exact solutions of nonlinear evolutionequations in mathematical physics. Physics Letters A, 216, 67-75.

Hirota, R., & Satsuma, J. (1997). Nonlinear evolution equations generated from the Bäcklund transformation for the Boussinesq equa-tion. Prog Theor Phys, 157, 797-807.

Hirota, R. (1971). Exact solution of the KdV equation formultiple collisions of solitons. Phys Rev Lett, 27, 1192-1194.

Hu, X. B., & Clarkson, P. A. (1995). Rational solutions of a differential-difference KdV equation,the Toda equation and the discrete KdV equation. J Phys A: Math Gen, 28, 5009-5016.

Khatera, A. H., Hassanb, M. M., & Temsaha, R. S. (2007). Co-noidal wave solutions for a class of fifth-order KdV equations. Mathematics and Computers in Simulation, 70, 221-226.

Wazwaz, A. M. (2007). Analytic study on the gen-eralized fifth-order KdV equation: New solitons and periodic solutions. Communications in Nonlinear Science and Numerical Simulation, 12, 1172-1180.




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