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New Travelling Wave Solution-Based New Riccati Equation for Solving KdV and Modified KdV Equations

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Oct 10, 2020

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Korteweg, Diederik Johannes, and Gustav De Vries. “XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves.” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 39.240: 422–443, 1895. KortewegDiederik Johannes De VriesGustav “XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves.” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 39 240 422 443 1895 10.1080/14786449508620739 Search in Google Scholar

Wadati, Miki, and Morikazu Toda. “The exact N-soliton solution of the Korteweg-de Vries equation.” Journal of the Physical Society of Japan, 32.5: 1403–1411, 1972. WadatiMiki TodaMorikazu “The exact N-soliton solution of the Korteweg-de Vries equation.” Journal of the Physical Society of Japan 32 5 1403 1411 1972 10.1142/9789814354332_0021 Search in Google Scholar

Hirota, R. “Exact solutions of the Korteweg-de Vries equation for multiple solitons.” Physical Review Letters, 27: 1192–1194, 1971. HirotaR “Exact solutions of the Korteweg-de Vries equation for multiple solitons.” Physical Review Letters 27 1192 1194 1971 10.1103/PhysRevLett.27.1192 Search in Google Scholar

Miura, Robert M., Clifford S. Gardner, and Martin D. Kruskal. “Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion.” Journal of Mathematical physics, 9: 1204–1209, 1968. MiuraRobert M. GardnerClifford S. KruskalMartin D. “Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion.” Journal of Mathematical physics 9 1204 1209 1968 10.1063/1.1664701 Search in Google Scholar

Ankiewicz, A., Mahyar Bokaeeyan, and N. Akhmediev. “Shallow-water rogue waves: An approach based on complex solutions of the Korteweg–de Vries equation.” Physical Review E, 99: 050201, 2019. AnkiewiczA. BokaeeyanMahyar AkhmedievN. “Shallow-water rogue waves: An approach based on complex solutions of the Korteweg–de Vries equation.” Physical Review E 99 050201 2019 10.1103/PhysRevE.99.05020131212487 Search in Google Scholar

Ji, Jia-Liang, and Zuo-Nong Zhu. “On a nonlocal modified Korteweg-de Vries equation: Integrability, Darboux transformation and soliton solutions.” Communications in Nonlinear Science and Numerical Simulation, 42: 699–708, 2017. JiJia-Liang ZhuZuo-Nong “On a nonlocal modified Korteweg-de Vries equation: Integrability, Darboux transformation and soliton solutions.” Communications in Nonlinear Science and Numerical Simulation 42 699 708 2017 10.1016/j.cnsns.2016.06.015 Search in Google Scholar

Ma, Wen-Xiu. “The inverse scattering transform and soliton solutions of a combined modified Korteweg–de Vries equation.” Journal of Mathematical Analysis and Applications, 471: 796–811, 2019. MaWen-Xiu “The inverse scattering transform and soliton solutions of a combined modified Korteweg–de Vries equation.” Journal of Mathematical Analysis and Applications 471 796 811 2019 10.1016/j.jmaa.2018.11.014 Search in Google Scholar

Liang, Jin-Fu, and Xun Wang. “Investigation of Interaction Solutions for Modified Korteweg-de Vries Equation by Consistent Riccati Expansion Method.” Mathematical Problems in Engineering 2019 (2019). LiangJin-Fu WangXun “Investigation of Interaction Solutions for Modified Korteweg-de Vries Equation by Consistent Riccati Expansion Method.” Mathematical Problems in Engineering 2019 2019 10.1155/2019/9535294 Search in Google Scholar

Wazzan, Luwai. “A modified tanh–coth method for solving the KdV and the KdV–Burgers’ equations.” Communications in Nonlinear Science and Numerical Simulation, 14: 443–450, 2009. WazzanLuwai “A modified tanh–coth method for solving the KdV and the KdV–Burgers’ equations.” Communications in Nonlinear Science and Numerical Simulation 14 443 450 2009 10.1016/j.cnsns.2007.06.011 Search in Google Scholar

Wang, Mingliang, Xiangzheng Li, and Jinliang Zhang. “The (G′ G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics.” Physics Letters A, 372: 417–423, 2008. WangMingliang LiXiangzheng ZhangJinliang “The (G′ G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics.” Physics Letters A 372 417 423 2008 10.1016/j.physleta.2007.07.051 Search in Google Scholar

Zheng-De, Dai, Liu Zhen-Jiang, and Li Dong-Long. “Exact periodic solitary-wave solution for KdV equation.” Chinese Physics Letters, 25: 1531, 2008. Zheng-DeDai Zhen-JiangLiu Dong-LongLi “Exact periodic solitary-wave solution for KdV equation.” Chinese Physics Letters 25 1531 2008 10.1088/0256-307X/25/5/003 Search in Google Scholar

Lü, DaZhao, Cui, Y. Y, Lu, C., Wei, C. Y,. “Novel composite function solutions of the modified KdV equation.” Applied mathematics and computation, 217: 283–288, 2010. DaZhao CuiY. Y LuC. WeiC. Y “Novel composite function solutions of the modified KdV equation.” Applied mathematics and computation 217 283 288 2010 10.1016/j.amc.2010.05.059 Search in Google Scholar

Wazwaz, A-M. “A sine-cosine method for handlingnonlinear wave equations.” Mathematical and Computer modelling, 40: 499–508, 2004. WazwazA-M “A sine-cosine method for handlingnonlinear wave equations.” Mathematical and Computer modelling 40 499 508 2004 10.1016/j.mcm.2003.12.010 Search in Google Scholar

He, Ji-Huan, and Xu-Hong Wu. “Exp-function method for nonlinear wave equations.” Chaos, Solitons & Fractals, 30: 700–708, 2006. HeJi-Huan WuXu-Hong “Exp-function method for nonlinear wave equations.” Chaos, Solitons & Fractals 30 700 708 2006 10.1016/j.chaos.2006.03.020 Search in Google Scholar

El-Ajou, Ahmad, Oqielat, M. N., Al-Zhour, Z., Kumar, S., Momani, S,. “Solitary solutions for time-fractional nonlinear dispersive PDEs in the sense of conformable fractional derivative.” Chaos: An Interdisciplinary Journal of Nonlinear Science, 29: 093102, 2019. El-AjouAhmad OqielatM. N. Al-ZhourZ. KumarS. MomaniS “Solitary solutions for time-fractional nonlinear dispersive PDEs in the sense of conformable fractional derivative.” Chaos: An Interdisciplinary Journal of Nonlinear Science 29 093102 2019 10.1063/1.510023431575153 Search in Google Scholar

Kumar, Sunil, Kumar, A., Momani, S., Abdelfattah, M., Sooppy, M., “Numerical solutions of nonlinear fractional model arising in the appearance of the stripe patterns in two-dimensional systems.” Advances in Difference Equations 2, 413, 2019. KumarSunil KumarA. MomaniS. AbdelfattahM. SooppyM. “Numerical solutions of nonlinear fractional model arising in the appearance of the stripe patterns in two-dimensional systems.” Advances in Difference Equations 2 413 2019 10.1186/s13662-019-2334-7 Search in Google Scholar

Ghanbari, Behzad, Sunil Kumar, and Ranbir Kumar. “A study of behaviour for immune and tumor cells in immuno-genetic tumour model with non-singular fractional derivative.” Chaos, Solitons & Fractals, 133: 109619, 2020. GhanbariBehzad KumarSunil KumarRanbir “A study of behaviour for immune and tumor cells in immuno-genetic tumour model with non-singular fractional derivative.” Chaos, Solitons & Fractals 133 109619 2020 10.1016/j.chaos.2020.109619 Search in Google Scholar

Kumar, S., Nisar, K. S., Kumar, R., Cattani, C., & Samet, B., “A new Rabotnov fractional-exponential function-based fractional derivative for diffusion equation under external force.” Mathematical Methods in the Applied Sciences. 2019. KumarS. NisarK. S. KumarR. CattaniC. SametB. “A new Rabotnov fractional-exponential function-based fractional derivative for diffusion equation under external force.” Mathematical Methods in the Applied Sciences 2019 10.1002/mma.6208 Search in Google Scholar

Jleli, M., Kumar, S., Kumar, R., & Samet, B. “Analytical approach for time fractional wave equations in the sense of Yang-Abdel-Aty-Cattani via the homotopy perturbation transform method.” Alexandria Engineering Journal (2019). JleliM. KumarS. KumarR. SametB “Analytical approach for time fractional wave equations in the sense of Yang-Abdel-Aty-Cattani via the homotopy perturbation transform method.” Alexandria Engineering Journal 2019 10.1016/j.aej.2019.12.022 Search in Google Scholar

Kumar, S., Kumar, A., Abbas, S., Al Qurashi, M., & Baleanu, D “A modified analytical approach with existence and uniqueness for fractional Cauchy reaction–diffusion equations.” Advances in Difference Equations, (2020): 1–18, 2020. KumarS. KumarA. AbbasS. Al QurashiM. BaleanuD “A modified analytical approach with existence and uniqueness for fractional Cauchy reaction–diffusion equations.” Advances in Difference Equations 2020 1 18 2020 10.1186/s13662-019-2488-3 Search in Google Scholar

Korkmaz, Alper. “Exact solutions of space-time fractional EW and modified EW equations.” Chaos, Solitons & Fractals, 96: 132–138, 2017. KorkmazAlper “Exact solutions of space-time fractional EW and modified EW equations.” Chaos, Solitons & Fractals 96 132 138 2017 10.1016/j.chaos.2017.01.015 Search in Google Scholar

Guner, Ozkan, Ahmet Bekir, and Alper Korkmaz. “Tanh-type and sech-type solitons for some space-time fractional PDE models.” The European Physical Journal Plus, 132: 92, 2017. GunerOzkan BekirAhmet KorkmazAlper “Tanh-type and sech-type solitons for some space-time fractional PDE models.” The European Physical Journal Plus 132 92 2017 10.1140/epjp/i2017-11370-7 Search in Google Scholar

Khater, Mostafa MA, Dianchen Lu, and Raghda AM Attia. “Lump soliton wave solutions for the (2+ 1)-dimensional Konopelchenko–Dubrovsky equation and KdV equation.” Modern Physics Letters B, 33: 1950199, 2019. KhaterMostafa MA LuDianchen AttiaRaghda AM “Lump soliton wave solutions for the (2+ 1)-dimensional Konopelchenko–Dubrovsky equation and KdV equation.” Modern Physics Letters B 33 1950199 2019 10.1142/S0217984919501999 Search in Google Scholar

Khater, Mostafa, Raghda AM Attia, and Dianchen Lu. “Modified auxiliary equation method versus three nonlinear fractional biological models in present explicit wave solutions.” Mathematical and Computational Applications, 24: 1, 2019. KhaterMostafa AttiaRaghda AM LuDianchen “Modified auxiliary equation method versus three nonlinear fractional biological models in present explicit wave solutions.” Mathematical and Computational Applications 24 1 2019 10.3390/mca24010001 Search in Google Scholar

Lu, D., Tariq, K. U., Osman, M. S., Baleanu, D., Younis, M., & Khater, M. M. A. “New analytical wave structures for the (3+ 1)-dimensional Kadomtsev-Petviashvili and the generalized Boussinesq models and their applications.” Results in Physics, 14: 102491, 2019. LuD. TariqK. U. OsmanM. S. BaleanuD. YounisM. KhaterM. M. A “New analytical wave structures for the (3+ 1)-dimensional Kadomtsev-Petviashvili and the generalized Boussinesq models and their applications.” Results in Physics 14 102491 2019 10.1016/j.rinp.2019.102491 Search in Google Scholar

Osman, M. S., Korkmaz, A., Rezazadeh, H., Mirzazadeh, M., Eslami, M., & Zhou, Q “The unified method for conformable time fractional Schrödinger equation with perturbation terms.” Chinese Journal of Physics, 56: 2500–2506, 2018. OsmanM. S. KorkmazA. RezazadehH. MirzazadehM. EslamiM. ZhouQ “The unified method for conformable time fractional Schrödinger equation with perturbation terms.” Chinese Journal of Physics 56 2500 2506 2018 10.1016/j.cjph.2018.06.009 Search in Google Scholar

Kurt, Ali. “New periodic wave solutions of a time fractional integrable shallow water equation.” Applied Ocean Research, 85: 128–135, 2019. KurtAli “New periodic wave solutions of a time fractional integrable shallow water equation.” Applied Ocean Research 85 128 135 2019 10.1016/j.apor.2019.01.029 Search in Google Scholar

Tasbozan, Orkun, Ali Kurt, and Ali Tozar. “New optical solutions of complex Ginzburg–Landau equation arising in semiconductor lasers.” Applied Physics B, 125: 104, 2019. TasbozanOrkun KurtAli TozarAli “New optical solutions of complex Ginzburg–Landau equation arising in semiconductor lasers.” Applied Physics B 125 104 2019 10.1007/s00340-019-7217-9 Search in Google Scholar

Tasbozan, Orkun, et al. “New solutions of fractional Drinfeld-Sokolov-Wilson system in shallow water waves.” Ocean Engineering, 161: 62–68, 2018. TasbozanOrkun “New solutions of fractional Drinfeld-Sokolov-Wilson system in shallow water waves.” Ocean Engineering 161 62 68 2018 10.1016/j.oceaneng.2018.04.075 Search in Google Scholar

Tasbozan, O., Şenol, M., Kurt, A., & Özkan, O. “Sine-Gordon expansion method for exact solutions to conformable time fractional equations in RLW-class.” Journal of King Saud University-Science (2018). TasbozanO. ŞenolM. KurtA. ÖzkanO “Sine-Gordon expansion method for exact solutions to conformable time fractional equations in RLW-class.” Journal of King Saud University-Science 2018 Search in Google Scholar

Raza, Nauman. “New optical solitons in nonlinear negative-index materials with Bohm potential.” Indian Journal of Physics, 93: 657–663, 2019. RazaNauman “New optical solitons in nonlinear negative-index materials with Bohm potential.” Indian Journal of Physics 93 657 663 2019 10.1007/s12648-018-1234-0 Search in Google Scholar

Raza, N., Afzal, U., Butt, A. R., & Rezazadeh, H. “Optical solitons in nematic liquid crystals with Kerr and parabolic law nonlinearities.” Optical and Quantum Electronics, 51: 107, 2019. RazaN. AfzalU. ButtA. R. RezazadehH “Optical solitons in nematic liquid crystals with Kerr and parabolic law nonlinearities.” Optical and Quantum Electronics 51 107 2019 10.1007/s11082-019-1813-0 Search in Google Scholar

Javid, Ahmad, and Nauman Raza. “Singular and dark optical solitons to the well posed Lakshmanan–Porsezian–Daniel model.” Optik, 171: 120–129, 2018. JavidAhmad RazaNauman “Singular and dark optical solitons to the well posed Lakshmanan–Porsezian–Daniel model.” Optik 171 120 129 2018 10.1016/j.ijleo.2018.06.021 Search in Google Scholar

Raza, Nauman, and Ahmad Javid. “Dynamics of optical solitons with Radhakrishnan–Kundu–Lakshmanan model via two reliable integration schemes.” Optik, 178: 557–566, 2019. RazaNauman JavidAhmad “Dynamics of optical solitons with Radhakrishnan–Kundu–Lakshmanan model via two reliable integration schemes.” Optik 178 557 566 2019 10.1016/j.ijleo.2018.09.133 Search in Google Scholar

Rezazadeh, Hadi. “New solitons solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity.” Optik, 167: 218–227, 2018. RezazadehHadi “New solitons solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity.” Optik 167 218 227 2018 10.1016/j.ijleo.2018.04.026 Search in Google Scholar

Dusunceli, Faruk. “New exponential and complex traveling wave solutions to the Konopelchenko-Dubrovsky model.” Advances in Mathematical Physics, 2019, 2019, https://doi.org/10.1155/2019/7801247. DusunceliFaruk “New exponential and complex traveling wave solutions to the Konopelchenko-Dubrovsky model.” Advances in Mathematical Physics 2019 2019 https://doi.org/10.1155/2019/7801247 10.1155/2019/7801247 Search in Google Scholar

Tekiyeh, R. M., Manafian, J., Baskonus, H. M., & Dusunceli, F “Applications of He's semi-inverse variational method and ITEM to the nonlinear long-short wave interaction system.” International Journal of Advanced and Applied Sciences, 4: 93–100, 2018. TekiyehR. M. ManafianJ. BaskonusH. M. DusunceliF “Applications of He's semi-inverse variational method and ITEM to the nonlinear long-short wave interaction system.” International Journal of Advanced and Applied Sciences 4 93 100 2018 Search in Google Scholar

Düşünceli, F., Başkonuş, H. M., Alaattin, E. S. E. N., & Bulut, H. “New mixed-dark soliton solutions to the hyperbolic generalization of the Burgers equation.” Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 2: 503–511, 2019. DüşünceliF. BaşkonuşH. M. AlaattinE. S. E. N. BulutH “New mixed-dark soliton solutions to the hyperbolic generalization of the Burgers equation.” Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 2 503 511 2019 10.25092/baunfbed.585940 Search in Google Scholar

Dusunceli, Faruk. “New Exact Solutions for Generalized (3+ 1) Shallow Water-Like (SWL) Equation.” Applied Mathematics and Nonlinear Sciences, 4: 365–370, 2019. DusunceliFaruk “New Exact Solutions for Generalized (3+ 1) Shallow Water-Like (SWL) Equation.” Applied Mathematics and Nonlinear Sciences 4 365 370 2019 10.2478/AMNS.2019.2.00031 Search in Google Scholar

Rezazadeh, H., Tariq, H., Eslami, M., Mirzazadeh, M., & Zhou, Q “New exact solutions of nonlinear conformable time-fractional Phi-4 equation.” Chinese Journal of Physics, 56: 2805–2816, 2018. RezazadehH. TariqH. EslamiM. MirzazadehM. ZhouQ “New exact solutions of nonlinear conformable time-fractional Phi-4 equation.” Chinese Journal of Physics 56 2805 2816 2018 10.1016/j.cjph.2018.08.001 Search in Google Scholar

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