On the complex properties of the first equation of the Kadomtsev-Petviashvili hierarchy

[1] Houwe A., Sabi’u J., Hammouch Z., Doka S.Y., Solitary pulses of a conformable nonlinear differential equation governing wave propagation in low-pass electrical transmission line, Physica Scripta, 95(4), 045203, 2020. HouweA. Sabi’uJ. HammouchZ. DokaS.Y. Solitary pulses of a conformable nonlinear differential equation governing wave propagation in low-pass electrical transmission line Physica Scripta 95 4 045203 2020 Search in Google Scholar

[2] Gasmi B., Kessi A., Hammouch Z., Various optical solutions to the (1+1)-Telegraph equation with space-time conformable derivatives, International Journal of Nonlinear Analysis and Applications, 12, 767–780, 2021. GasmiB. KessiA. HammouchZ. Various optical solutions to the (1+1)-Telegraph equation with space-time conformable derivatives International Journal of Nonlinear Analysis and Applications 12 767 780 2021 Search in Google Scholar

[3] Wazwaz A.M., Multi-front waves for extended form of modified Kadomtsev-Petviashvili equation, Applied Mathematics and Mechanics, 32, 875–880, 2011. WazwazA.M. Multi-front waves for extended form of modified Kadomtsev-Petviashvili equation Applied Mathematics and Mechanics 32 875 880 2011 Search in Google Scholar

[4] Hamou A.A., Hammouch Z., Azroul E., Agarwal P., Monotone iterative technique for solving finite difference systems of time fractional parabolic equations with initial/periodic conditions, Applied Numerical Mathematics, 181, 561–593, 2022. HamouA.A. HammouchZ. AzroulE. AgarwalP. Monotone iterative technique for solving finite difference systems of time fractional parabolic equations with initial/periodic conditions Applied Numerical Mathematics 181 561 593 2022 Search in Google Scholar

[5] Leble S.B., Ustinov N.V., Darboux transforms, deep reductions and solitons, Journal of Physics A: Mathematical and General, 26, 5007, 1993. LebleS.B. UstinovN.V. Darboux transforms, deep reductions and solitons Journal of Physics A: Mathematical and General 26 5007 1993 Search in Google Scholar

[6] Parkes E.J., Duffy B.R., An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Computer Physics Communications, 98(3), 288–300, 1996. ParkesE.J. DuffyB.R. An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations Computer Physics Communications 98 3 288 300 1996 Search in Google Scholar

[7] Abdusalam H.A., On an improved complex tanh-function method, International Journal of Nonlinear Sciences and Numerical Simulation, 6(2), 99–106, 2005. AbdusalamH.A. On an improved complex tanh-function method International Journal of Nonlinear Sciences and Numerical Simulation 6 2 99 106 2005 Search in Google Scholar

[8] Yan C., A simple transformation for nonlinear waves, Physics Letters A, 224(1–2), 77–84, 1996. YanC. A simple transformation for nonlinear waves Physics Letters A 224 1–2 77 84 1996 Search in Google Scholar

[9] Guirao J.L.G., Baskonus H.M., Kumar A., Rawat M.S., Yel G., Complex patterns to the (3+1)-dimensional B-type Kadomtsev-Petviashvili-Boussinesq equation, Symmetry, 12(1), 17, 2020. GuiraoJ.L.G. BaskonusH.M. KumarA. RawatM.S. YelG. Complex patterns to the (3+1)-dimensional B-type Kadomtsev-Petviashvili-Boussinesq equation Symmetry 12 1 17 2020 Search in Google Scholar

[10] Baskonus H.M., Bulut H., Sulaiman T.A., New complex hyperbolic structures to the Lonngren-Wave equation by using sine-Gordon expansion method, Applied Mathematics and Nonlinear Sciences, 4(1), 129–138, 2019. BaskonusH.M. BulutH. SulaimanT.A. New complex hyperbolic structures to the Lonngren-Wave equation by using sine-Gordon expansion method Applied Mathematics and Nonlinear Sciences 4 1 129 138 2019 Search in Google Scholar

[11] Al-Sekhary A.A., Gepreel K.A., Exact solutions for nonlinear integro-partial differential equations using the (G′G,1G) \left( {\frac{{G'}}{G},\frac{1}{G}} \right) -expansion method, International Journal of Applied Engineering Research, 14(10), 2449–2461, 2019. Al-SekharyA.A. GepreelK.A. Exact solutions for nonlinear integro-partial differential equations using the (G′G,1G) \left( {\frac{{G'}}{G},\frac{1}{G}} \right) -expansion method International Journal of Applied Engineering Research 14 10 2449 2461 2019 Search in Google Scholar

[12] Bulut H., Ismael H.F., Exploring new features for the perturbed Chen-Lee-Liu model via (m + 1/G’)-expansion method, Proceedings of the Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan, 48(1), 164–173, 2022. BulutH. IsmaelH.F. Exploring new features for the perturbed Chen-Lee-Liu model via (m + 1/G’)-expansion method Proceedings of the Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan 48 1 164 173 2022 Search in Google Scholar

[13] Khalique C.M., Adem K.R., Exact solutions of the (2+1)-dimensional Zakharov-Kuznetsov modified equal width equation using Lie group analysis, Mathematical and Computer Modelling, 54(1–2), 184–189, 2011. KhaliqueC.M. AdemK.R. Exact solutions of the (2+1)-dimensional Zakharov-Kuznetsov modified equal width equation using Lie group analysis Mathematical and Computer Modelling 54 1–2 184 189 2011 Search in Google Scholar

[14] Elboree M.K., The Jacobi elliptic function method and its application for two component BKP hierarchy equations, Computers Mathematics with Applications, 62, 4402–4414, 2011. ElboreeM.K. The Jacobi elliptic function method and its application for two component BKP hierarchy equations Computers Mathematics with Applications 62 4402 4414 2011 Search in Google Scholar

[15] Tala-Tebue E., Zayed E.M.E., New Jacobi elliptic function solutions, solitons and other solutions for the (2+1)-dimensional nonlinear electrical transmission line equation, The European Physical Journal Plus, 133(314), 1–7, 2018. Tala-TebueE. ZayedE.M.E. New Jacobi elliptic function solutions, solitons and other solutions for the (2+1)-dimensional nonlinear electrical transmission line equation The European Physical Journal Plus 133 314 1 7 2018 Search in Google Scholar

[16] Zhang S., Xia T., A generalized new auxiliary equation method and its applications to nonlinear partial differential equations, Physics Letters A, 363(5–6), 356–360, 2007. ZhangS. XiaT. A generalized new auxiliary equation method and its applications to nonlinear partial differential equations Physics Letters A 363 5–6 356 360 2007 Search in Google Scholar

[17] Gepreel K.A., Exact solutions for nonlinear integral member of Kadomtsev-Petviashvili hierarchy differential equations using the modified (w/g)-expansion method, Computers and Mathematics with Applications, 72(9), 2072–2083, 2016. GepreelK.A. Exact solutions for nonlinear integral member of Kadomtsev-Petviashvili hierarchy differential equations using the modified (w/g)-expansion method Computers and Mathematics with Applications 72 9 2072 2083 2016 Search in Google Scholar

[18] Gepreel K.A., Nofal T.A., Alasmari A.A., Exact solutions for nonlinear integro-partial differential equations using the generalized Kudryashov method, Journal of the Egyptian Mathematical Society, 25(4), 438–444, 2017. GepreelK.A. NofalT.A. AlasmariA.A. Exact solutions for nonlinear integro-partial differential equations using the generalized Kudryashov method Journal of the Egyptian Mathematical Society 25 4 438 444 2017 Search in Google Scholar

[19] Sirendaoreji, Jiong S., Auxiliary equation method for solving nonlinear partial differential equations, Physics Letters A, 309(5–6), 387–396, 2003. Sirendaoreji JiongS. Auxiliary equation method for solving nonlinear partial differential equations Physics Letters A 309 5–6 387 396 2003 Search in Google Scholar

[20] Gao W., Ghanbari B., Günerhan H., Baskonus H.M., Some mixed trigonometric complex soliton solutions to the perturbed nonlinear Schrödinger equation, Modern Physics Letters B, 34(03), 2050034, 2020. GaoW. GhanbariB. GünerhanH. BaskonusH.M. Some mixed trigonometric complex soliton solutions to the perturbed nonlinear Schrödinger equation Modern Physics Letters B 34 03 2050034 2020 Search in Google Scholar

[21] Gao W., Rezazadeh H., Pinar Z., Baskonus H.M., Sarwar S., Yel G., Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique, Optical and Quantum Electronics, 52(52), 1–13, 2020. GaoW. RezazadehH. PinarZ. BaskonusH.M. SarwarS. YelG. Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique Optical and Quantum Electronics 52 52 1 13 2020 Search in Google Scholar

[22] Batwa S., Ma W.X., A study of lump-type and interaction solutions to the (3+1)-dimensional Jimbo-Miwa-like equation, Computers and Mathematics with Applications, 76(7), 1576–1582, 2018. BatwaS. MaW.X. A study of lump-type and interaction solutions to the (3+1)-dimensional Jimbo-Miwa-like equation Computers and Mathematics with Applications 76 7 1576 1582 2018 Search in Google Scholar

[23] Ma W.X., Lump-type solutions to the (3+1)-dimensional Jimbo-Miwa equation, International Journal of Nonlinear Sciences and Numerical Simulation, 17(7–8), 355–359, 2016. MaW.X. Lump-type solutions to the (3+1)-dimensional Jimbo-Miwa equation International Journal of Nonlinear Sciences and Numerical Simulation 17 7–8 355 359 2016 Search in Google Scholar

[24] Yong X., Li X., Huang Y., General lump-type solutions of the (3+1)-dimensional Jimbo-Miwa equation, Applied Mathematics Letters, 86, 222–228, 2018. YongX. LiX. HuangY. General lump-type solutions of the (3+1)-dimensional Jimbo-Miwa equation Applied Mathematics Letters 86 222 228 2018 Search in Google Scholar

[25] Yue Y., Huang L., Chen Y., Localized waves and interaction solutions to an extended (3+1)-dimensional Jimbo-Miwa equation, Applied Mathematics Letters, 89, 70–77, 2019. YueY. HuangL. ChenY. Localized waves and interaction solutions to an extended (3+1)-dimensional Jimbo-Miwa equation Applied Mathematics Letters 89 70 77 2019 Search in Google Scholar

[26] Ma W.X., Lee J.H., A transformed rational function method and exact solutions to the (3+1)-dimensional Jimbo-Miwa equation, Chaos Solitons and Fractals, 42(3), 1356–1363, 2009. MaW.X. LeeJ.H. A transformed rational function method and exact solutions to the (3+1)-dimensional Jimbo-Miwa equation Chaos Solitons and Fractals 42 3 1356 1363 2009 Search in Google Scholar

[27] Du X.X., Tian B., Yin Y., Lump mixed lump-kink, breather and rogue waves for a B-type Kadomtsev-Petviashvili equation, Waves in Random and Complex Media, 31(1), 101–116, 2021. DuX.X. TianB. YinY. Lump mixed lump-kink, breather and rogue waves for a B-type Kadomtsev-Petviashvili equation Waves in Random and Complex Media 31 1 101 116 2021 Search in Google Scholar

[28] Yildirim Y., Bright, dark and singular optical solitons to Kundu-Eckhaus equation having four-wave mixing in the context of birefringent fibers by using of trial equation methodology, Optik, 182, 393–399, 2019. YildirimY. Bright, dark and singular optical solitons to Kundu-Eckhaus equation having four-wave mixing in the context of birefringent fibers by using of trial equation methodology Optik 182 393 399 2019 Search in Google Scholar

[29] Li Y.Z., Liu J.G., Multiple periodic-soliton solutions of the (3+1)-dimensional generalised shallow water equation, Pramana, 90(71), 1–11, 2018. LiY.Z. LiuJ.G. Multiple periodic-soliton solutions of the (3+1)-dimensional generalised shallow water equation Pramana 90 71 1 11 2018 Search in Google Scholar

[30] Zhang Y., Dong H., Zhang X., Yang H., Rational solutions and lump solutions to the generalized (3+1)-dimensional Shallow Water-like equation, Computers and Mathematics with Applications, 73(2), 246–252, 2017. ZhangY. DongH. ZhangX. YangH. Rational solutions and lump solutions to the generalized (3+1)-dimensional Shallow Water-like equation Computers and Mathematics with Applications 73 2 246 252 2017 Search in Google Scholar

[31] Zhao Z., Dai Z., Han S., The EHTA for nonlinear evolution equations, Applied Mathematics and Computation, 217(8), 4306–4310, 2010. ZhaoZ. DaiZ. HanS. The EHTA for nonlinear evolution equations Applied Mathematics and Computation 217 8 4306 4310 2010 Search in Google Scholar

[32] Zhao Z., Han B., Lump solutions of a (3+1)-dimensional B-type KP equation and its dimensionally reduced equations, Analysis and Mathematical Physics, 9, 119–130, 2019. ZhaoZ. HanB. Lump solutions of a (3+1)-dimensional B-type KP equation and its dimensionally reduced equations Analysis and Mathematical Physics 9 119 130 2019 Search in Google Scholar

[33] Zhao Z., He L., Multiple lump solutions of the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation, Applied Mathematics Letters, 95, 114–121, 2019. ZhaoZ. HeL. Multiple lump solutions of the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation Applied Mathematics Letters 95 114 121 2019 Search in Google Scholar

[34] Fan E., Extended tanh-function method and its applications to nonlinear equations, Physics Letters A, 277(4–5), 212–218, 2000. FanE. Extended tanh-function method and its applications to nonlinear equations Physics Letters A 277 4–5 212 218 2000 Search in Google Scholar

[35] Jimbo M., Miwa T., Solitons and infinite dimensional Lie algebras, Publications of the Research Institute for Mathematical Sciences, 19(3), 943–1001, 1983. JimboM. MiwaT. Solitons and infinite dimensional Lie algebras Publications of the Research Institute for Mathematical Sciences 19 3 943 1001 1983 Search in Google Scholar

[36] Wazwaz A.M., New (3+1)-dimensional nonlinear evolution equations with mKdV equation constituting its main part: Multiple soliton solutions, Chaos Solitons and Fractals, 76, 93–97, 2015. WazwazA.M. New (3+1)-dimensional nonlinear evolution equations with mKdV equation constituting its main part: Multiple soliton solutions Chaos Solitons and Fractals 76 93 97 2015 Search in Google Scholar

[37] Kadomtsev B.B., Petviashvili V.I., On the stability of solitary waves in weakly dispersive media, Soviet Physics Doklady, 15, 539–541, 1970. KadomtsevB.B. PetviashviliV.I. On the stability of solitary waves in weakly dispersive media Soviet Physics Doklady 15 539 541 1970 Search in Google Scholar

[38] Feng Z, Wang X., The first integral method to the two-dimensional Burgers-Korteweg-de Vries equation, Physics Letters A, 308(2–3), 173–178, 2003. FengZ WangX. The first integral method to the two-dimensional Burgers-Korteweg-de Vries equation Physics Letters A 308 2–3 173 178 2003 Search in Google Scholar

[39] Zhang Y.J., A class of integro-differential equations constrained from the KP hierarchy, Journal of Physics A: Mathematical and General, 27(24), 8149–8160, 1994. ZhangY.J. A class of integro-differential equations constrained from the KP hierarchy Journal of Physics A: Mathematical and General 27 24 8149 8160 1994 Search in Google Scholar

[40] Chen S., Dark and composite rogue waves in the coupled Hirota equations, Physics Letters A, 378(38–39), 2851–2856, 2014. ChenS. Dark and composite rogue waves in the coupled Hirota equations Physics Letters A 378 38–39 2851 2856 2014 Search in Google Scholar

[41] Ma W.X., Zhu Z., Solving the (3+1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm, Applied Mathematics and Computation, 218(24), 11871–11879, 2012. MaW.X. ZhuZ. Solving the (3+1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm Applied Mathematics and Computation 218 24 11871 11879 2012 Search in Google Scholar

[42] Chen Y., Wang Q., Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions solutions to (1+1)-dimensional dispersive long wave equation, Chaos Solitons and Fractals, 24(3), 745–757, 2005. ChenY. WangQ. Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions solutions to (1+1)-dimensional dispersive long wave equation Chaos Solitons and Fractals 24 3 745 757 2005 Search in Google Scholar

[43] Kumar S., Kumar A., Lie symmetry reductions and group invariant solutions of (2+1)-dimensional modified Veronese web equation, Nonlinear Dynamics, 98, 1891–1903, 2019. KumarS. KumarA. Lie symmetry reductions and group invariant solutions of (2+1)-dimensional modified Veronese web equation Nonlinear Dynamics 98 1891 1903 2019 Search in Google Scholar

[44] Mahmud A.A., Baskonus H.M., Tanriverdi T., Muhamad K.A., Optical solitary waves and soliton solutions of the (3+1)-dimensional generalized Kadomtsev-Petviashvili-Benjamin-Bona-Mahony equation, Computational Mathematics and Mathematical Physics, 63(6), 1085–1102, 2023. MahmudA.A. BaskonusH.M. TanriverdiT. MuhamadK.A. Optical solitary waves and soliton solutions of the (3+1)-dimensional generalized Kadomtsev-Petviashvili-Benjamin-Bona-Mahony equation Computational Mathematics and Mathematical Physics 63 6 1085 1102 2023 Search in Google Scholar

[45] Kumar S., Kumar A., Kharbanda H., Lie symmetry analysis and generalized invariant solutions of (2+1)-dimensional dispersive long wave (DLW) equations, Physica Scripta, 95(6), 065207, 2020. KumarS. KumarA. KharbandaH. Lie symmetry analysis and generalized invariant solutions of (2+1)-dimensional dispersive long wave (DLW) equations Physica Scripta 95 6 065207 2020 Search in Google Scholar

[46] Muhamad K.A., Tanriverdi T., Mahmud A.A., Baskonus H.M., Interaction characteristics of the Riemann wave propagation in the (2+1)-dimensional generalized breaking soliton system, International Journal of Computer Mathematics, 100(6), 1340–1355, 2023. MuhamadK.A. TanriverdiT. MahmudA.A. BaskonusH.M. Interaction characteristics of the Riemann wave propagation in the (2+1)-dimensional generalized breaking soliton system International Journal of Computer Mathematics 100 6 1340 1355 2023 Search in Google Scholar

[47] Kumar A., Kumar S., Kharbanda H., Closed-form invariant solutions from the Lie symmetry analysis and dynamics of solitonic profiles for (2+1)-dimensional modified Heisenberg ferromagnetic system, Modern Physics Letters B, 36(07), 2150609, 2022. KumarA. KumarS. KharbandaH. Closed-form invariant solutions from the Lie symmetry analysis and dynamics of solitonic profiles for (2+1)-dimensional modified Heisenberg ferromagnetic system Modern Physics Letters B 36 07 2150609 2022 Search in Google Scholar

[48] Kumar S., Ma W.X., Kumar A., Lie symmetries optimal system and group-invariant solutions of the (3+1)-dimensional generalized KP equation, Chinese Journal of Physics, 69, 1–23, 2021. KumarS. MaW.X. KumarA. Lie symmetries optimal system and group-invariant solutions of the (3+1)-dimensional generalized KP equation Chinese Journal of Physics 69 1 23 2021 Search in Google Scholar

[49] Baskonus H.M., Mahmud A.A., Muhamad K.A., Tanriverdi T., A study on Caudrey-Dodd-Gibbon-Sawada-Kotera partial differential equation, Mathematical Methods in the Applied Sciences, 45(14), 8737–8753, 2022. BaskonusH.M. MahmudA.A. MuhamadK.A. TanriverdiT. A study on Caudrey-Dodd-Gibbon-Sawada-Kotera partial differential equation Mathematical Methods in the Applied Sciences 45 14 8737 8753 2022 Search in Google Scholar

[50] Kumar S., Kumar A., Newly generated optical wave solutions and dynamical behaviors of the highly nonlinear coupled Davey-Stewartson Fokas system in monomode optical fibers, Optical and Quantum Electronics, 55(566), 1–33, 2023. KumarS. KumarA. Newly generated optical wave solutions and dynamical behaviors of the highly nonlinear coupled Davey-Stewartson Fokas system in monomode optical fibers Optical and Quantum Electronics 55 566 1 33 2023 Search in Google Scholar

[51] Kumar S., Kumar A., Dynamical behaviors and abundant optical soliton solutions of the cold bosonic atoms in a zig-zag optical lattice model using two integral schemes, Mathematics and Computers in Simulation, 201, 254–274, 2022. KumarS. KumarA. Dynamical behaviors and abundant optical soliton solutions of the cold bosonic atoms in a zig-zag optical lattice model using two integral schemes Mathematics and Computers in Simulation 201 254 274 2022 Search in Google Scholar

[52] Kumar S., Kumar D., Kumar A., Lie symmetry analysis for obtaining the abundant exact solutions, optimal system and dynamics of solitons for a higher-dimensional Fokas equation, Chaos Solitons and Fractals, 142, 110507, 2021. KumarS. KumarD. KumarA. Lie symmetry analysis for obtaining the abundant exact solutions, optimal system and dynamics of solitons for a higher-dimensional Fokas equation Chaos Solitons and Fractals 142 110507 2021 Search in Google Scholar

[53] Baskonus H.M., Mahmud A.A., Muhamad K.A., Tanriverdi T., Gao W., Studying on Kudryashov-Sinelshchikov dynamical equation arising in mixtures liquid and gas bubbles, Thermal Science, 26(2 Part B), 1229–1244, 2022. BaskonusH.M. MahmudA.A. MuhamadK.A. TanriverdiT. GaoW. Studying on Kudryashov-Sinelshchikov dynamical equation arising in mixtures liquid and gas bubbles Thermal Science 26 2 Part B 1229 1244 2022 Search in Google Scholar

[54] Wazwaz A.M., Kadomtsev-Petviashvili hierarchy: N-soliton solutions and distinct dispersion relations, Applied Mathematics Letters, 52, 74–79, 2016. WazwazA.M. Kadomtsev-Petviashvili hierarchy: N-soliton solutions and distinct dispersion relations Applied Mathematics Letters 52 74 79 2016 Search in Google Scholar

[55] Wazwaz A.M., Partial Differential Equations and Solitary Waves Theory, Springer, Berlin, Germany, 2009. WazwazA.M. Partial Differential Equations and Solitary Waves Theory Springer Berlin, Germany 2009 Search in Google Scholar

[56] Yan L., Baskonus H.M., Cattani C., Gao W., Extractions of the gravitational potential and high-frequency wave perturbation properties of nonlinear (3+1)-dimensional Vakhnenko-Parkes equation via novel approach, Mathematical Methods in the Applied Sciences, DOI: 10.1002/mma.8726, 2024. YanL. BaskonusH.M. CattaniC. GaoW. Extractions of the gravitational potential and high-frequency wave perturbation properties of nonlinear (3+1)-dimensional Vakhnenko-Parkes equation via novel approach Mathematical Methods in the Applied Sciences 10.1002/mma.8726 2024 Open DOISearch in Google Scholar

[57] Yan L., Yel G., Kumar A., Baskonus H.M., Gao W., Newly developed analytical scheme and its applications to the some nonlinear partial differential equations with the conformable derivative, Fractal and Fractional, 5(4), 238, 2021. YanL. YelG. KumarA. BaskonusH.M. GaoW. Newly developed analytical scheme and its applications to the some nonlinear partial differential equations with the conformable derivative Fractal and Fractional 5 4 238 2021 Search in Google Scholar