Biao Xu¹, Jiangli Wang²This email address is being protected from spambots. You need JavaScript enabled to view it., Fang Yuanlu³This email address is being protected from spambots. You need JavaScript enabled to view it.

¹Shijiazhuang University of Applied Technology, Shijiazhuang 050081, China ²Qinhuangdao Open University, Qinhuangdao 066000, China ³Tianjin Vocational College of Bioengineering, Basic teaching department, Tianjin Kaifaquxiqu 300462, China

1. [1] G.-Q. Xu and Z.-B. Li, (2004) “Symbolic computation of the Painlev ´ e test for nonlinear partial differential equations
   using Maple" Computer Physics Communications 161(1-2): 65–75. DOI: 10.1016/j.cpc.2004.04.005.
2. [2] V. F. Zaitsev and A. D. Polyanin. Handbook of exact solutions for ordinary differential equations. Chapman and Hall/CRC, 2002.
3. [3] C. Grossmann, H.-G. Roos, and M. Stynes. Numerical treatment of partial differential equations. 154. Springer, 2007.
4. [4] C. Soize and R. Ghanem, (2021) “Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets" Computer Methods in Applied Mechanics and Engineering 380: DOI: 10.1016/j.cma.2021.113777.
5. [5] J. Pu, W. Peng, and Y. Chen, (2021) “The datadriven localized wave solutions of the derivative nonlinear Schr¨odinger equation by using improved PINN approach" Wave Motion 107: DOI: 10.1016/j.wavemoti.2021.102823.
6. [6] L. Akinyemi, M. ¸Senol, and O. S. Iyiola, (2021) “Exact solutions of the generalized multidimensional mathematical physics models via sub-equation method" Mathematics and Computers in Simulation 182: 211–233. DOI:10.1016/j.matcom.2020.10.017.
7. [7] L. Akinyemi, (2021) “Two improved techniques for the perturbed nonlinear Biswas–Milovic equation and its optical solitons" Optik 243: DOI:10.1016/j.ijleo.2021.167477.
8. [8] X.-Y. Gao, Y.-J. Guo, andW.-R. Shan, (2021) “Beholding the shallow water waves near an ocean beach or in a lake via a Boussinesq-Burgerssystem" Chaos, Solitons and Fractals 147: DOI: 10.1016/j.chaos.2021.110875.
9. [9] A. Yoku¸s, (2021) “Construction of different types of traveling wave solutions of the relativistic wave equation associated with the Schr¨odinger equation" Mathematical Modelling and Numerical Simulation with Applications 1(1): 24–31.
10. [10] J.-H. He and Y. O. El-Dib, (2021) “The reducing rank method to solve third-order Duffing equation with the homotopy perturbation" Numerical Methods for Partial Differential Equations 37(2): 1800–1808. DOI: 10.1002/num.22609.
11. [11] J. Singh, A. Ahmadian, S. Rathore, D. Kumar, D. Baleanu, M. Salimi, and S. Salahshour, (2021) “An efficient computational approach for local fractional Poisson equation in fractal media" Numerical Methods for Partial Differential Equations 37(2): 1439–1448. DOI: 10.1002/num.22589.
12. [12] H. Tajadodi, Z. A. Khan, A. u. Rehman Irshad, J. Gómez-Aguilar, A. Khan, and H. Khan, (2021) “Exact solutions of conformable fractional differential equations" Results in Physics 22: DOI: 10.1016/j.rinp.2021.103916.
13. [13] M. M. Khater, T. A. Nofal, H. Abu-Zinadah, M. S. Lotayif, and D. Lu, (2021) “Novel computational and accurate numerical solutions of the modified Benjamin–Bona–Mahony (BBM) equation arising in the optical illusions field" Alexandria Engineering Journal 60(1): 1797–1806. DOI: 10.1016/j.aej.2020.11.028.
14. [14] M. M. A. Khater, R. A. M. Attia, S. K. Elagan, and F. S. Bayones, (2021) “ANALYTICAL AND SEMI ANALYTICAL SOLUTIONS OF THE INTERNAL WAVES OF DEEP-STRATIFIED FLUIDS" Thermal Science 25(SpecialIssue 2): S227–S232. DOI: 10.2298/TSCI21S2227K.
15. [15] M. M. A. Khater, S. Anwar, K. U. Tariq, and M. S. Mohamed, (2021) “Some optical soliton solutions to the perturbed nonlinear Schrödinger equation by modified Khater method" AIP Advances 11(2): DOI: 10.1063/5.0038671.
16. [16] H. Aminikhah, A. R. Sheikhani, and H. Rezazadeh, (2015) “Exact solutions for the fractional differential equations by using the first integral method" Nonlinear engineering 4(1): 15–22.
17. [17] M. M. Khater, (2021) “Numerical simulations of Zakharov’s (ZK) non-dimensional equation arising in Langmuir and ion-acoustic waves" Modern Physics Letters B 35(31): 2150480.
18. [18] M. M. Khater, (2021) “Diverse bistable dark novel explicit wave solutions of cubic–quintic nonlinear Helmholtz model" Modern Physics Letters B 35(26): 2150441.
19. [19] M. M. Khater, (2021) “Abundant breather and semianalytical investigation: On high-frequency waves’ dynamics
    in the relaxation medium" Modern Physics Letters B 35(22): 2150372.
20. [20] M. M. A. Khater, (2021) “Diverse solitary and Jacobian solutions in a continually laminated fluid with respect to shear flows through the Ostrovsky equation" Modern Physics Letters B 35(13): DOI: 10.1142 /S0217984921502201.
21. [21] M. M. Khater, S. Elagan, M. El-Shorbagy, S. Alfalqi, J. Alzaidi, and N. A. Alshehri, (2021) “Folded novel accurate analytical and semi-analytical solutions of a generalized Calogero-Bogoyavlenskii-Schiff equation" Communications in Theoretical Physics 73(9): DOI: 10.1088/1572-9494/ac049f.
22. [22] M. M. A. Khater and D. Lu, (2021) “Analytical versus numerical solutions of the nonlinear fractional time-space telegraph equation" Modern Physics Letters B 35(19): DOI: 10.1142/S0217984921503243.
23. [23] M. M. Khater, M. S. Mohamed, and R. A. Attia, (2021) “On semi analytical and numerical simulations for a mathematical biological model; the time-fractional nonlinear Kolmogorov– Petrovskii–Piskunov (KPP) equation" Chaos, Solitons and Fractals 144: DOI: 10.1016/j.chaos.2021.110676.
24. [24] M. M. A. Khater and B. Ghanbari, (2021) “On the solitary wave solutions and physical characterization of gas diffusion in a homogeneous medium via some efficient techniques" European Physical Journal Plus 136(4): DOI: 10.1140/epjp/s13360-021-01457-1.
25. [25] M. M. Khater, K. S. Nisar, and M. S. Mohamed, (2021) “Numerical investigation for the fractional nonlinear spacetime telegraph equation via the trigonometric Quintic B-spline scheme" Mathematical Methods in the Applied Sciences 44(6): 4598–4606.
26. [26] M. M. Khater, A. Mousa, M. El-Shorbagy, and R. A. Attia, (2021) “Analytical and semi-analytical solutions for Phi-four equation through three recent schemes" Results in Physics 22: DOI: 10.1016/j.rinp.2021.103954.
27. [27] H. Aminikhah, A. R. Sheikhani, and H. Rezazadeh, (2016) “Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method" Boletim da Sociedade Paranaense de Matematica 34(2): 213–229. DOI: 10.5269/bspm.v34i2.25501.
28. [28] M. M. Khater, A. E.-S. Ahmed, S. Alfalqi, J. Alzaidi, S. Elbendary, and A. M. Alabdali, (2021) “Computational and approximate solutions of complex nonlinear Fokas–Lenells equation arising in optical fiber" Results in Physics 25: DOI: 10.1016/j.rinp.2021.104322.
29. [29] M. M. Khater, A. E.-S. Ahmed, and M. El-Shorbagy, (2021) “Abundant stable computational solutions of Atangana–Baleanu fractional nonlinear HIV-1 infection of CD4+ T-cells of immunodeficiency syndrome" Results in Physics 22: DOI: 10.1016/j.rinp.2021.103890.
30. [30] J. Zhang, D. Lu, S. A. Salama, and M. M. A. Khater,(2022) “Accurate demonstrating of the interactions of two long waves with different dispersion relations: Generalized Hirota-Satsuma couple KdV equation" AIP Advances 12(2): DOI: 10.1063/5.0084588.
31. [31] L. Akinyemi, M. Mirzazadeh, and K. Hosseini, (2022) “Solitons and other solutions of perturbed nonlinear Biswas–Milovic equation with Kudryashov’s law of refractive index" Nonlinear Analysis: Modelling and Control 27(3): 479–495. DOI: 10.15388/namc.2022.27.26374.
32. [32] S. Abbagari, A. Houwe, L. Akinyemi, Y. Saliou, and T. B. Bouetou, (2022) “Modulation instability gain and discrete soliton interaction in gyrotropic molecular chain" Chaos, Solitons and Fractals 160: DOI: 10.1016/j.chaos.2022.112255.
33. [33] H. Ahmad, M. N. Alam, and M. Omri, (2021) “New computational results for a prototype of an excitable system" Results in Physics 28: DOI: 10.1016/j.rinp.2021.104666.
34. [34] G. Akram, M. Sadaf, and I. Zainab, (2022) “Observations of fractional effects of β-derivative and M-truncated derivative for space time fractional Phi-4 equation via two analytical techniques" Chaos, Solitons and Fractals 154: DOI: 10.1016/j.chaos.2021.111645.
35. [35] G. Akram, M. Sadaf, and M. A. U. Khan, (2023) “Soliton solutions of the resonant nonlinear Schr ¨odinger equation using modified auxiliary equation method with three different nonlinearities" Mathematics and Computers in Simulation 206: 1–20. DOI: 10.1016/j.matcom.2022.10.032.
36. [36] H. Ahmad, T. A. Khan, P. S. Stanimirovic, W. Shatanawi, and T. Botmart, (2022) “New approach on conventional solutions to nonlinear partial differential equations describing physical phenomena" Results in Physics 41: DOI: 10.1016/j.rinp.2022.105936.
37. [37] Hamood-Ur-Rehman, M. I. Asjad, M. Inc, and I. Iqbal, (2022) “Exact solutions for new coupled Konno–Oono equation via Sardar subequation method" Optical and Quantum Electronics 54(12): DOI: 10.1007/s11082-022-04208-3.
38. [38] M. S. M. Shehata, H. Rezazadeh, E. H. M. Zahran, E. Tala-Tebue, and A. Bekir, (2019) “New Optical Soliton Solutions of the Perturbed Fokas-Lenells Equation" Communications in Theoretical Physics 71(11): 1275–1280. DOI: 10.1088/0253-6102/71/11/1275.
39. [39] W. A. Faridi, M. I. Asjad, and S. M. Eldin, (2022) “Exact Fractional Solution by Nucci’s Reduction Approach and New Analytical Propagating Optical Soliton Structures in Fiber-Optics" Fractal and Fractional 6(11): DOI: 10.3390/fractalfract6110654.
40. [40] M. Jaradat, A. Batool, A. R. Butt, and N. Raza, (2022) “New solitary wave and computational solitons for Kundu–Eckhaus equation" Results in Physics 43: DOI: 10.1016/j.rinp.2022.106084.
41. [41] T. A. Alrebdi, N. Raza, S. Arshed, and A.-H. Abdel-Aty, (2022) “New solitary wave patterns of Fokas-System arising in monomode fiber communication systems" Optical and Quantum Electronics 54(11): DOI: 10.1007/s11082-022-04062-3.
42. [42] J. Dikwa, A. Houwe, S. Abbagari, L. Akinyemi, and M. Inc, (2022) “Modulated waves patterns in the photovoltaic photorefractive crystal" Optical and Quantum Electronics 54(12): DOI: 10.1007/s11082-022-04224-3.
43. [43] A. Houwe, Y. Saliou, P. Djorwe, S. Abbagari, L. Akinyemi, and S. Y. Doka, (2022) “Modulation instability gain and modulated wave shape incited by the acoustic longitudinal vibrations in molecular chain model" Physica Scripta 97(8): DOI: 10.1088/1402-4896/ac7a6b.
44. [44] H. S. Ali, M. Habib, M. M. Miah, M. M. Miah, and M. A. Akbar, (2023) “Diverse solitary wave solutions of fractional order Hirota-Satsuma coupled KdV system using two expansion methods" Alexandria Engineering Journal 66: 1001–1014. DOI: 10.1016/j.aej.2022.12.021.
45. [45] M. A. Akbar, F. A. Abdullah, and M. M. Haque, (2023) “Analytical soliton solutions of the perturbed fractional nonlinear Schr¨odinger equation with space–time beta derivative by some techniques" Results in Physics 44: DOI: 10.1016/j.rinp.2022.106170.
46. [46] S. Kumar, B. Mohan, and R. Kumar, (2022) “Lump, soliton, and interaction solutions to a generalized twomode higher-order nonlinear evolution equation in plasma physics" Nonlinear Dynamics 110(1): 693–704. DOI: 10.1007/s11071-022-07647-5.
47. [47] S. K. Mohanty, S. Kumar, A. N. Dev, M. K. Deka, D. V. Churikov, and O. V. Kravchenko, (2022) “An efficient technique of [Formula presented]–expansion method for modified KdV and Burgers equations with variable coefficients" Results in Physics 37: DOI: 10.1016/j.rinp.2022.105504.
48. [48] A. R. Seadawy, S. T. R. Rizvi, S. Ahmed, and T. Batool, (2023) “Propagation of W-shaped and M-shaped solitons with multi-peak interaction for ultrashort light pulse in fibers" Optical and Quantum Electronics 55(3): DOI: 10.1007/s11082-022-04478-x.
49. [49] H. Rezazadeh, D. Kumar, T. A. Sulaiman, and H. Bulut, (2019) “New complex hyperbolic and trigonometric solutions for the generalized conformable fractional Gardner equation" Modern Physics Letters B 33(17): DOI: 10.1142/S0217984919501963.
50. [50] S. T. R. Rizvi, A. R. Seadawy, S. K. Naqvi, and S. O. Abbas, (2023) “Study of mixed derivative nonlinear Schr¨odinger equation for rogue and lump waves, breathers and their interaction solutions with Kerr law" Optical and Quantum Electronics 55(2): DOI: 10.1007/s11082-022-04415-y.
51. [51] Z. Zhao, L. He, and A.-M.Wazwaz, (2023) “Dynamics of lump chains for the BKP equation describing propagation of nonlinear waves" Chinese Physics B:
52. [52] Z. Zhao, J. Yue, and L. He, (2022) “New type of multiple lump and rogue wave solutions of the (2+1)-dimensional Bogoyavlenskii–Kadomtsev–Petviashvili equation" Applied Mathematics Letters 133: DOI: 10.1016/j.aml.2022.108294.
53. [53] Z. Zhao, (2019) “Conservation laws and nonlocally related systems of the Hunter–Saxton equation for liquid crystal" Analysis and Mathematical Physics 9(4):2311–2327. DOI: 10.1007/s13324-019-00337-3.
54. [54] Z. Zhao and L. He, (2021) “Lie symmetry, nonlocal symmetry analysis, and interaction of solutions of a (2+1)-dimensional KdV–mKdV equation" Theoretical and Mathematical Physics 206(2): 142–162.
55. [55] Z. Zhao and L. He, (2021) “Resonance Y-type soliton and hybrid solutions of a (2+1)-dimensional asymmetrical Nizhnik–Novikov–Veselov equation" Applied Mathematics Letters 122: DOI: 10.1016/j.aml.2021.107497.
56. [56] M. Eslami and H. Rezazadeh, (2016) “The first integral method for Wu–Zhang system with conformable timefractional
    derivative" Calcolo 53(3): 475–485. DOI: 10.1007/s10092-015-0158-8.
57. [57] M. Hashemi and Z. Balmeh, (2018) “On invariant analysis and conservation laws of the time fractional variant Boussinesq and coupled Boussinesq-Burger’s equations" European Physical Journal Plus 133(10): DOI: 10.1140/epjp/i2018-12289-1.
58. [58] F.-L. Xia, F. Jarad, M. S. Hashemi, and M. B. Riaz, (2022) “A reduction technique to solve the generalized nonlinear dispersive mK(m,n) equation with new local derivative" Results in Physics 38: DOI: 10.1016/j.rinp.2022.105512.
59. [59] M. Hashemi, (2021) “A novel approach to find exact solutions of fractional evolution equations with non-singular kernel derivative" Chaos, Solitons and Fractals 152:DOI: 10.1016/j.chaos.2021.111367.
60. [60] R. Johnson and S. Thompson, (1978) “A solution of the inverse scattering problem for the Kadomtsev-Petviashvili equation by the method of separation of variables" Physics Letters A 66(4): 279–281. DOI: 10.1016/0375-9601(78)90236-0.
61. [61] W.-X. Ma, (2015) “Lump solutions to the Kadomtsev-Petviashvili equation" Physics Letters, Section A: General, Atomic and Solid State Physics 379(36): 1975–1978. DOI: 10.1016/j.physleta.2015.06.061.
62. [62] M. El-Sabbagh and A. Ali, (2005) “New exact solutions for (3+1)-dimensional Kadomtsev-Petviashvili equation and generalized (2+1)-dimensional Boussinesq equation" International Journal of Nonlinear Sciences and Numerical Simulation 6(2): 151–162. DOI: 10.1515/IJNSNS.2005.6.2.151.
63. [63] X. Yong, W.-X. Ma, Y. Huang, and Y. Liu, (2018) “Lump solutions to the Kadomtsev–Petviashvili I equation with a self-consistent source" Computers and Mathematics with Applications 75(9): 3414–3419. DOI: 10.1016/j.camwa.2018.02.007.
64. [64] W.-X. Ma, X. Yong, and X. Lü, (2021) “Soliton solutions to the B-type Kadomtsev–Petviashvili equation under general dispersion relations"Wave Motion 103: DOI: 10.1016/j.wavemoti.2021.102719.
65. [65] J.-W. Xia, Y.-W. Zhao, and X. Lü, (2020) “Predictability, fast calculation and simulation for the interaction solutions to the cylindrical Kadomtsev-Petviashvili equation" Communications in Nonlinear Science and Numerical Simulation 90: DOI: 10.1016/j.cnsns.2020.105260.
66. [66] C. Wang and H. Fang, (2020) “General high-order localized waves to the Bogoyavlenskii–Kadomtsev–Petviashvili equation" Nonlinear Dynamics 100(1): 583–599. DOI: 10.1007/s11071-020-05499-5.
67. [67] H. F. Ismael, W.-X. Ma, and H. Bulut, (2021) “Dynamics of soliton and mixed lump-soliton waves to a generalized Bogoyavlensky-Konopelchenko equation" Physica Scripta 96(3): DOI: 10.1088/1402-4896/abdc55.
68. [68] L. Cheng, Y. Zhang, W.-X. Ma, and J.-Y. Ge, (2021) “Wronskian and lump wave solutions to an extended second KP equation" Mathematics and Computers in Simulation 187: 720–731. DOI: 10.1016/j.matcom.2021.03.024.
69. [69] Y.-L. Wang, Y.-T. Gao, S.-L. Jia, G.-F. Deng, and W.-Q. Hu, (2017) “Solitons for a (2 + 1)-dimensional variablecoefficient Bogoyavlensky-Konopelchenko equation in a fluid" Modern Physics Letters B 31(25): DOI: 10.1142/S0217984917502165.
70. [70] F. Calogero and A. Degasperis, (1976) “Nonlinear evolution equations solvable by the inverse spectral transform.-I" Il Nuovo Cimento B Series 11 32(2): 201–242. DOI: 10.1007/BF02727634.
71. [71] A. Atangana, D. Baleanu, and A. Alsaedi, (2015) “New properties of conformable derivative" Open Mathematics 13(1): 889–898. DOI: 10.1515/math-2015-0081.
72. [72] H. Rezazadeh, (2018) “New solitons solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity" Optik 167: 218–227. DOI: 10.1016/j.ijleo.2018.04.026.