THERMAL SCIENCE
International Scientific Journal
FRACTAL SOLITARY WAVE SOLUTIONS AND VARIATIONAL PRINCIPLE OF THE FRACTAL GENERAL KADOMTSEV-PETVIASHVILI EQUATION
ABSTRACT
This work examines the fractal generalized Kadomtsev-Petviashvili equation, which describes the evolution of non-linear long waves of small amplitude. The fractal traveling wave transformation and the fractal semi-inverse method are employed to derive a fractal variational principle, which was found to be a strong minimum according to the He-Weierstrass function. The solution of the two examples is presented in the form of images. This paper demonstrates that the fractal dimension affects the waveform of the generalized Kadomtsev-Petviashvili equation.
KEYWORDS
PAPER SUBMITTED: 2023-08-11
PAPER REVISED: 2024-02-23
PAPER ACCEPTED: 2024-03-01
PUBLISHED ONLINE: 2025-07-06
THERMAL SCIENCE YEAR
2025, VOLUME
29, ISSUE
Issue 3, PAGES [1775 - 1782]
- Yi, C., Li, Y. S., The Constraint of the Kadomtsev-Petviashvili Equation and its Special Solutions, Physics Letters A, 157 (1991), 1, pp. 22-26
- Wazwaz, A. M., Kadomtsev-Petviashvili Hierarchy: N-Soliton Solutions and Distinct Dispersion, Applied Mathematics Letters, 52 (2016), Feb., pp. 74-79
- Ablowitz, M. J., et al., Whitham Modulation Theory for the Kadomtsev-Petviashvili Equation, Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, 473 (2017), 20160695
- Ma, Y. L., Li, B. Q., Rogue Wave Solutions, Soliton and Rogue Wave Mixed Solution for a Generalized (3+1)-Dimensional Kadomtsev-Petviashvili Equation in Fluids, Modern Physics Letters B, 32 (2018), 29, 1850358
- Fogaça, D., et al., Kadomtsev-Petviashvili Equation in Relativistic Fluid Dynamics, Communications in Non-linear Science and Numerical Simulation, 18 (2013), 2, pp. 221-235
- Ma, W. X., Lump Solutions to the Kadomtsev-Petviashvili Equation, Physics Letters A, 379 (2015), 36, pp. 1975-1978
- Ma, Y. L., et al., New Extended Kadomtsev-Petviashvili Equation: Multiple Soliton Solutions, Breather, Lump and Interaction Solutions, Non-linear Dynamics, 104 (2021), Apr., pp. 1581-1594
- Qin, C. Y., et al., Rogue Waves, Bright-Dark Solitons and Traveling Wave Solutions of the (3+1)-Dimensional Generalized Kadomtsev-Petviashvili Equation, Computers & Mathematics With Applications, 75 (2018), 12, pp. 4221-4231
- Ma, Y., et al., New Extended Kadomtsev-Petviashvili Equation: Multiple Soliton Solutions, Breather, Lump and Interaction Solutions, Non-linear Dynamics, 104 (2021), Apr., pp. 1-14
- Duan, W. S., Weakly Two-Dimensional Dust Acoustic Waves, Physics of Plasmas, 8 (2001), 8, pp. 3583-3586
- Kalamvokas P., et al., A Semi-Periodic Initial-Value Problem for the Kadomtsev-Petviashvili II Equation, Non-linearity, 36 (2023), 10
- Alves, C. O., Ji, C., Existence and Concentration of Nontrivial Solitary Waves for a Generalized Kadomtsev-Petviashvili Equation in R2, Journal of Differential Equations, 368 (2023), Sept., pp. 141-172
- Wang, Y., Lu, X., Backlund Transformation and Interaction Solutions of a Generalized Kadomtsev-Petviashvili Equation with Variable Coefficients, Chinese Journal of Physics, 89 (2024), Jun., pp. 37-45
- Wazwaz, A. M., Xu, G. Q., Kadomtsev-Petviashvili Hierarchy: Two Integrable Equations with Time-Dependent Coefficients, Non-linear Dynamics, 100 (2020), June, pp. 3711-3716
- He, J.-H., et al., A Fractal Modification of Chen-Lee-Liu Equation and Its Fractal Variational Principle, International Journal of Modern Physics B, 35 (2021), 21, 2150214
- He, J.-H., et al., A New Fractional Derivative and its Application to Explanation of Polar Bear Hairs, Journal of King Saud University Science, 28 (2016), 2, pp. 190-192
- Shen, Y., He, J.-H., Variational Principle for a Generalized KdV Equation in a Fractal Space, Fractals, 28 (2020), 4, 20500693
- Sun, J. S., Variational Principle for fractal High-Order Long Water-Wave Equation, Thermal Science, 27 (2023), 3A, pp. 1899-1905
- He, J.-H., Fractal Calculus and its Geometrical Explanation, Results in Physics, 10 (2018), Sept., pp. 272-276
- Shang, C. H., Yi, H. A., A Fractal-Fractional Model on Impact Stress of Crusher Drum, Thermal Science, 27 (2023), 3A, pp. 2119-2125
- Zeng, H. J., et al., Thermal Performance of Fractal Metasurface and Its Mathematical Model, Thermal Science, 28 (2024), 3A, pp. 2379-2383
- He, J.-H., et al., Forced Non-Linear Oscillator in a Fractal Space, Facta Universitatis Series: Mechanical Engineering, 20 (2022), 1, pp. 1-20
- He, C. H., Liu, C., A Modified Frequency-Amplitude Formulation for Fractal Vibration Systems, Fractals, 30 (2022), 3, 2250046
- Mei, Y., et al., Fractal Space Based Dimensionless Analysis of the Surface Settlement Induced by the Shield Tunneling, Facta Universitatis series Mechanical Engineering, 21 (2023), 4, pp. 737-749
- He, C. H., Liu, C., Fractal Dimensions of a Porous Concrete and Its Effect on the Concrete's Strength, Facta Universitatis Series: Mechanical Engineering, 21 (2023), 1, pp. 137-150
- Zuo, Y. T., Variational Principle for a Fractal Lubrication Problem, Fractals, 32 (2024), 5, pp. 1-6
- Guan, Y. Z., et al., Variational Formulations for a Coupled Fractal-Fractional KdV System, Fractals, 32 (2024), 3, 24500543
- Wang, Y., et al., Variational Principles for Fractal Boussinesq-like B (m, n) Equation, Fractals, 31 (2023), 7, 2350063
- He, C. H., A Variational Principle for a Fractal Nano/Microelectromechanical (N/MEMS) System, International Journal of Numerical Methods for Heat & Fluid Flow, 33 (2023), 1, 351-359
- He, J.-H., et al., Geometrical Explanation of the Fractional Complex Transform and derivative Chain Rule for Fractional Calculus, Physics Letters A, 376 (2012), 4, pp. 257-259
- He, J.-H., et al., A Variational Principle for a Non-Linear Oscillator Arising in the Microelectromechanical System, Journal of Applied and Computational Mechanics, 7 (2021), 1, pp. 78-83
- He, C. H., Chao, L., Variational Principle for Singular Waves, Chaos Solitons & Fractals, 172 (2023), 113566
- He, J.-H., Variational Principles for Some Non-Linear Partial Differential Equations with Variable Coefficients, Chaos Solitons & Fractals, 19 (2004), 4, pp. 847-851
- Sun J. S., An Insight on the (2+1)-Dimensional Fractal Non-Linear Boiti-Leon- Manna-Pempinelli Equations, Fractals, 30 (2022), 9, 2250188
- He, J.-H., Chang S., A Variational Principle for a Thin Film Equation, Journal of Mathematical Chemistry, 57 (2019), Aug., pp. 2075-2081
- He, J.-H. Variational Principles for Some Non-Linear Partial Differential Equations with Variable Coefficients, Chaos Solitons & Fractals, 19 (2004), 4, pp. 847-851
- He, J.-H., Generalized Variational Principles for Buckling Analysis of Circular Cylinders, Acta Mechanica, 231 (2020), 3, pp. 899-906
- Sun, J. S., Traveling Wave Solution of Fractal KDV-Burgers-Kuramoto Equation Within Local Fractional Differential Operator, Fractals, 29 (2021), 7, 2150231
- He, J.-H., Lagrange Crisis and Generalized Variational Principle for 3D Unsteady Flow, International Journal of Numerical Methods for Heat & Fluid Flow, 30 (2019), 3, pp. 1189-1196
- He, J.-H., Semi-Inverse Method of Establishing Generalized Variational Principles for Fluid Mechanics with Emphasis on Turbomachinery Aerodynamics, International Journal of Turbo & Jet-Engines, 14 (1997), 1, pp. 23-28
- He, J.-H., et al., On a Strong Minimum Condition of a Fractal Variational Principle, Applied Mathematics Letters, 119 (2021), 107199
- He, J.-H., et al., Solitary Waves Travelling Along an Unsmooth Boundary, Results in Physics, 24 (2021), 104104
- Sun, J. S., Variational Principle and Solitary Wave of the Fractal Fourth-Order Non-Linear Ablowitz-Kaup-Newell-Segur Water Wave Model, Fractals, 31 (2023), 5, 2350036
- He, J.-H., Asymptotic Methods for Solitary Solutions and Compactons, Abstract and Applied Analysis, 2012 (2012), 1, 916793