Abstract
Problem statement: To obtain new exact traveling wave soliton solutions for the threewave interaction equation in a dispersive medium and a non zero phase mismatch. Approach: The tanh method is usually used to find a traveling wave analytic soliton solutions for one nonlinear wave and evolution partial differential equation. Here, we generalize this method to solve a system of nonlinear evolution partial differential equations, then we use this generalization to find new family of exact traveling wave soliton solutions for the nonlinear three-wave interaction equation. Results: We were able to generalize the tanh method and apply this generalization to the (TWI) system of (PDE’s). We derive a system of algebraic Eq. 28-32 and introduced some interested sets of solutions for this system, these sets of solutions leads us to write explicit analytic new family of soliton solutions for the three-wave interaction equation. Conclusion: The generalization of the tanh method is proved its efficiency in obtaining exact solutions for nonlinear evolution partial differential equations. This method also can be used similarly to obtain exact solutions for another interested nonlinear evolution system of partial differential equations.
Highlights
Obtaining an exact solution for a nonlinear equation is considered an interesting problem for mathematicians, so, what if we have a system of nonlinear equations? As an example on those systems is the nonlinear Three-Wave Interaction (TWI) system of Partial Differential Equations (PDE’s), which represents a mathematical model for three interacting optics waves? This system describes many physical phenomena, such as, the resonant quadratic nonlinear interaction of three optics waves (Ibragimov and Struthers, 1997), the second harmonic generation process which produces the first coherent or laser light source (Rushchitskii, 1996; Kumar et al, 2008) and the study of the model in x2 materials (Chen et al, 2004)
Many analytic solutions for the (TWI) system were found, such as, the solution of this system when it includes the phase mismatch (Δk) (Ibragimov et al, 2001), the solution of this system when it includes the second order dispersion (Werner and Drummond, 1994; Menyuk et al, 1994; Tahar, 2007), the solution of this system when it doesn’t include the second order dispersion by the Inverse Scattering Transform method (IST) (Ibragimov et al, 1998; Batiha, 2007) and the exact soliton solution found by some ansatz introduced by (Huang, 2000)
In this study we introduce a direct generalization of the tanh method (Wazwaz, 2004) to solve a nonlinear evolution system of (PDE’s), we apply this generalization and get new families of soliton solutions for the (TWI) system of (PDE’s) which includes nonzero quadratic dispersion coefficients {g1, g2, g3} and a nonzero phase mismatch (Δk) in Eq 14
Summary
Obtaining an exact solution for a nonlinear equation is considered an interesting problem for mathematicians, so, what if we have a system of nonlinear equations? As an example on those systems is the nonlinear Three-Wave Interaction (TWI) system of Partial Differential Equations (PDE’s), which represents a mathematical model for three interacting optics waves? This system describes many physical phenomena, such as, the resonant quadratic nonlinear interaction of three optics waves (Ibragimov and Struthers, 1997), the second harmonic generation process which produces the first coherent or laser light source (Rushchitskii, 1996; Kumar et al, 2008) and the study of the model in x2 materials (Chen et al, 2004). Many analytic solutions for the (TWI) system were found, such as, the solution of this system when it includes the phase mismatch (Δk) (Ibragimov et al, 2001), the solution of this system when it includes the second order dispersion (Werner and Drummond, 1994; Menyuk et al, 1994; Tahar, 2007), the solution of this system when it doesn’t include the second order dispersion by the Inverse Scattering Transform method (IST) (Ibragimov et al, 1998; Batiha, 2007) and the exact soliton solution found by some ansatz introduced by (Huang, 2000). In this study we introduce a direct generalization of the tanh method (Wazwaz, 2004) to solve a nonlinear evolution system of (PDE’s), we apply this generalization and get new families of soliton solutions for the (TWI) system of (PDE’s) which includes nonzero quadratic dispersion coefficients {g1, g2, g3} and a nonzero phase mismatch (Δk) in Eq 14. In all of the obtained solutions we mentioned that replacing the tanh function with the coth function will keep our solutions valid
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