A Riccati-Bernoulli Sub-ODE Method for the Resonant Nonlinear Schrödinger Equation with Both Spatio-Temporal Dispersions and Inter-Modal
This work uses the Riccati-Bernoulli sub-ODE method in constructing various new optical soliton solutions
to the resonant nonlinear Schrodinger equation with both Spatio-temporal dispersion and inter-modal dispersion. Actually, the proposed method is effective tool to solve many other nonlinear partial differential equations in mathematical physics. Moreover this method can give a new infinite sequence of solutions. These solutions are expressed by hyperbolic functions, trigonometric functions and rational functions. Finally, with the aid of Matlab release 15, some graphical simulations were designed to see the behavior of these solutions.
M.A.E. Abdelrahman and M. Kunik, The interaction of waves for the ultra-relativistic Euler equations, J. Math. Anal. Appl. 409 (2014) 1140–1158.
M.A.E Abdelrahman and M. Kunik, The ultra-relativistic Euler equations, Math. Meth. Appl. Sci.,38 (2015), 1247-1264. 8 MathLAB Journal Vol 4 (2019) http://purkh.com/index.php/mathlab
M.A.E Abdelrahman, Global solutions for the ultra-relativistic Euler equations, Nonlinear Analysis, 155 (2017), 140-162.
M.A.E Abdelrahman, On the shallow water equations, Z. Naturforsch., 72(9)a (2017), 873-879.
M.A.E Abdelrahman, Numerical investigation of the wave-front tracking algorithm for the full ultra-relativistic Euler equations, International Journal of Nonlinear Sciences and Numerical Simulation, DOI:https://doi.org/10.1515/ijnsns-2017-0121.
P. Razborova, B. Ahmed and A. Biswas, Solitons, shock waves and conservation laws of Rosenau-KdV-RLW equation with power law nonlinearity. Appl. Math. Inf. Sci., 8(2) (2014), 485-491.
A. Biswas and M. Mirzazadeh, Dark optical solitons with power law nonlinearity using G′/G -expansion, Optik,125 (2014), 4603-4608.
M. Younis, S. Ali and S.A. Mahmood, Solitons for compound KdV Burgers equation with variable coefcients and power law nonlinearity. Nonlinear Dyn., 81 (2015), 1191-1196.
A.H. Bhrawy, An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system. Appl. Math. Comput., 247 (2014) , 30-46.
M.A.E. Abdelrahman and M.A. Sohaly, On the new wave solutions to the MCH equation, Indian Journal of Physics (2018), https://doi.org/10.1007/s12648-018-1354-6.
M.A.E. Abdelrahman and M.A. Sohaly, Solitary waves for the nonlinear Schrödinger problem with the probability distribution function in stochastic input case. Eur. Phys. J. Plus., (2017).
M.A.E. Abdelrahman and M.A. Sohaly, The development of the deterministic nonlinear PDEs in particle physics to stochastic case, Results in Physics, 9 (2018), 344-350.
W. Malfliet, Solitary wave solutions of nonlinear wave equation, Am. J. Phys., 60 (1992), 650-654.
W. Malfliet, W. Hereman, The tanh method: Exact solutions of nonlinear evolution and wave equations, Phys.Scr.,54 (1996), 563-568.
A. M. Wazwaz, The tanh method for travelling wave solutions of nonlinear equations, Appl. Math. Comput., 154 (2004), 714-723.
C. Q. Dai and J. F. Zhang, Jacobian elliptic function method for nonlinear differential difference equations, Chaos Solutions Fractals, 27 (2006), 1042-1049.
E. Fan and J .Zhang, Applications of the Jacobi elliptic function method to special-type nonlinear equations, Phys. Lett. A., 305 (2002), 383-392.
S. Liu, Z. Fu, S. Liu, Q.Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A., 289 (2001), 69-74.
 J. H. He, X. H. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons Fractals, 30 (2006), 700-708.
H. Aminikhad, H. Moosaei, M. Hajipour, Exact solutions for nonlinear partial differential equations via Exp-function method, Numer. Methods Partial Differ. Equations, 26 (2009), 1427-1433. 9 MathLAB Journal Vol 4 (2019) http://purkh.com/index.php/mathlab
A. M. Wazwaz, Exact solutions to the double sinh-Gordon equation by the tanh method and a variable separated ODE. method, Comput. Math. Appl., 50 (2005), 1685-1696.
A. M. Wazwaz, A sine-cosine method for handling nonlinear wave equations, Math. Comput. Modelling, 40 (2004),499-508.
C. Yan, A simple transformation for nonlinear waves, Phys. Lett. A., 224 (1996), 77-84.
E. Fan, H.Zhang, A note on the homogeneous balance method, Phys. Lett. A., 246 (1998), 403-406.
M. L. Wang, Exct solutions for a compound KdV-Burgers equation, Phys. Lett. A., 213 (1996), 279-287.
Y. J. Ren, H. Q. Zhang, A generalized F-expansion method to find abundant families of Jacobi elliptic function solutions of the (2+1)-dimensional Nizhnik-Novikov-Veselov equation, Chaos Solitons Fractals, 27 (2006), 959-979.
J. L. Zhang, M. L. Wang, Y. M. Wang, Z. D. Fang, The improved F-expansion method and its applications, Phys. Lett. A., 350 (2006), 103-109.
 E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A., 277 (2000), 212-218.
A. M. Wazwaz, The extended tanh method for abundant solitary wave solutions of nonlinear wave equations, Appl. Math. Comput., 187 (2007), 1131-1142.
M. L. Wang, J. L. Zhang, X. Z. Li, The (G′G)expansion method and travelling wave solutions of nonlinearevolutions equations in mathematical physics, Phys. Lett. A., 372 (2008), 417-423.
S. Zhang, J. L. Tong, W.Wang, A generalized (G′G)expansion method for the mKdv equation with variable coefficients, Phys. Lett. A., 372 (2008), 2254-2257.
Q. Zhou, C. Wei, H. Zhang, J. Lu, H. Yu, P. Yao, Q. Zhu, Exact solutions to the resonant nonlinear Schrödinger equation with both Spatio-temporal and dispersions, Proc. Rom. Acad. Ser. A., 17 (4) (2016), 307-313.
A. Biswas, M.K. Ullah, Q. Zhou, S.P. Moshokoa, H. Triki, M. Belic, Resonant optical solitons with quadratic-cubic nonlinearity by semi-inverse principle, Optik – Int. J. Light Electron Opt., 145 (2017) 18-21.
H. Bulut, T.A. Sulaiman, H.M. Baskonus, Optical solitons to the resonant nonlinear Schrödinger equation with both Spatio-temporal and inter-modal dispersions under Kerr law nonlinearity. Optik, 163 (2018), 49-55.
M.A.E. Abdelrahman, A note on Riccati-Bernoulli sub-ODE method combined with complex transform method applied to fractional differential equations, Nonlinear Engineering Modeling and Application (2018), [DOI: https://doi.org/10.1515/nleng-2017-0145].
 S.Z. Hassan and M.A.E. Abdelrahman, Solitary wave solutions for some nonlinear time fractional partial differential equations, Pramana-J. Phys., (2018) 91:67.
X. F. Yang, Z. C. Deng, and Y. Wei, A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Adv. Diff. Equa., 1 (2015), 117-133.
D. Kumar, J. Singh, and D. Baleanu, A new analysis for fractional model of regularized long-wave equation arising in ion-acoustic plasma waves, Mathematical Methods in the Applied Sciences, 40 (2017), 5642-5653.
K. Hosseini, P. Mayeli, A. Bekir, and O. Guner, Density-dependent conformable space-time fractional diffusion-reaction equation and its exact solutions, Commun. Theor. Phys. 69 (2018), 1-4.
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