| Citation: |
Chaudry Masood Khalique, Oke Davies Adeyemo. APPLICATIONS OF SOLITARY WAVE SOLUTIONS AND CONSERVED FLOWS OF A GENERALIZED GEOPHYSICAL KDV MODEL WITH NONLINEAR DUAL POWER LAW IN GEOPHYSICS AND MARINE SCIENCE[J]. Journal of Applied Analysis & Computation, 2026, 16(5): 2540-2568. doi: 10.11948/20250156
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APPLICATIONS OF SOLITARY WAVE SOLUTIONS AND CONSERVED FLOWS OF A GENERALIZED GEOPHYSICAL KDV MODEL WITH NONLINEAR DUAL POWER LAW IN GEOPHYSICS AND MARINE SCIENCE
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Abstract
This paper analytically examines a generalized geophysical Korteweg-de Vries model with a nonlinear power-law in ocean science. To begin with, we apply Lie symmetry analysis in generating the point symmetries of the equation. This further leads to the model being simplified to a nonlinear ordinary differential equation. Thus, for the very first time as far as we know, we attain diverse solitary wave solutions for the model. Initially, a direct integration technique is adopted to obtain solutions to the equation. Additionally, we obtain more broadly defined exact solutions for the generalized geophysical Korteweg-de Vries model. This is achieved via an extended Jacobi expansion approach with elliptic functions. This is a widely recognized technique for achieving closed-form solutions to evolution equations. As a result, one obtains different cnoidal, snoidal, and dnoidal wave solutions to the less-explored model. The tabulated copolar trio illustrates that these solutions can revert to different hyperbolic and trigonometric functions given specific conditions. Furthermore, various graphical representations of the dynamic characteristics of the obtained results are shown. This is done to achieve a clear comprehension of the physical phenomena of the fundamental model. In the latter section, conserved vectors related to the aforementioned model are obtained using the standard multiplier method as well as Noether's theorem. We adopt Lie symmetry analysis in studying the dual power version of the KdV-type equation for the first time. Besides, the inclusion of Noether theorem as well as multiplier technique makes it novel compared to some other forms of work earlier done.
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References
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Chaudry Masood Khalique, Oke Davies Adeyemo. APPLICATIONS OF SOLITARY WAVE SOLUTIONS AND CONSERVED FLOWS OF A GENERALIZED GEOPHYSICAL KDV MODEL WITH NONLINEAR DUAL POWER LAW IN GEOPHYSICS AND MARINE SCIENCE[J]. Journal of Applied Analysis & Computation, 2026, 16(5): 2540-2568. doi: 10.11948/20250156
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Figure 1.
Wave depiction of cnoidal wave solution (2.38) using the assigned values
$ c = 1 $ $ b = 10 $ $ \nu = 4 $ $ d = 2 $ $ \omega = 0.5 $ $ -10 \leq t, x, \leq 10 $ -
Figure 2.
Wave representation of cnoidal wave solution (2.38) using the assigned values
$ c = 1 $ $ b = 10 $ $ \nu = 4 $ $ d = 2 $ $ \omega = 0.7 $ $ -5 \leq t, x, \leq 5 $ -
Figure 3.
Wave depiction of cnoidal wave solution (2.39) using the assigned values
$ c = 1 $ $ b = 3 $ $ \nu = 0.2 $ $ d = 2 $ $ \omega = 1 $ $ -10 \leq t, x, \leq 10 $ -
Figure 4.
Wave representation of cnoidal wave solution (2.39) using the assigned values
$ c = 0.1 $ $ b = 3 $ $ \nu = 0.2 $ $ d = 2 $ $ \omega = 1 $ $ -5 \leq t, x, \leq 5 $ -
Figure 5.
Singular Periodic wave depiction of snoidal wave solution (2.45) using the assigned values
$ c = 2 $ $ b = 10 $ $ \nu = 4 $ $ d = -1.2 $ $ \omega = 0.9 $ $ -5 \leq t, x, \leq 5 $ $ -4 \leq t, x, \leq 4 $ -
Figure 6.
Singular periodic wave depiction of snoidal wave solution (2.46) using the allocated parameter values
$ c = 1.2 $ $ b = 3 $ $ \nu = 0.2 $ $ d = -2 $ $ \omega = 0.8 $ $ -10 \leq t, x, \leq 10 $ -
Figure 7.
Periodic wave depiction of snoidal wave solution (2.47) using the assigned values
$ c = 2 $ $ b = 8 $ $ \nu = 3 $ $ d = -1.2 $ $ \omega = 0.9 $ $ -4 \leq t, x, \leq 4 $ -
Figure 8.
Smooth periodic wave depiction of snoidal wave solution (2.47) using the assigned values
$ c = 2 $ $ b = 8 $ $ \nu = 3 $ $ d = -1.2 $ $ \omega = 0.9 $ $ -8 \leq t, x, \leq 8 $ -
Figure 9.
Periodic wave portrayal of dnoidal wave solution (2.52) obtained by invoking the assigned values
$ c = 2.1 $ $ b = 10 $ $ \nu = 4.02 $ $ d = 1.2 $ $ \omega = 0.88 $ $ -5 \leq t, x, \leq 5 $ -
Figure 10.
Periodic wave exhibition of dnoidal wave solution (2.53) achieved by engaging the assigned values
$ c = 1.2 $ $ b = 3.2 $ $ \nu = 0.2 $ $ d = 1.2 $ $ \omega = 0.98 $ $ -6 \leq t, x, \leq 6 $ -
Figure 11.
Wave depiction of periodic solution (2.54) using the values of constant parameters as
$ c = 2.4 $ $ b = 10.03 $ $ \nu = 4.02 $ $ d = 60.4 $ $ \omega = 0.89 $ $ -4 \leq t, x \leq 4 $ -
Figure 12.
Diagrammatic representation of Earth's magnetic field.
https://scitechdaily.com/earths-magnetosphere-protecting-our-planet-from-harmful-space-radiation . -
Figure 13.
Schematic illustration of phytoplankton diatoms ocean food chains foundations for fishes.
https://www.imbrsea.eu/product/74 . -
Figure 14.
Carbon dioxide is present in the atmosphere and is also dissolved in water. Photosynthesis transforms carbon dioxide gas into organic carbon, while respiration returns the organic carbon to carbon dioxide gas. The enduring storage of organic carbon happens when material from living organisms is buried deep within the earth and transforms into fossils. Volcanic eruptions and, more recently, emissions from humans reintroduce this stored carbon into the carbon cycle.
https://openoregon.pressbooks.pub/envirobiology/chapter/3-2-biogeochemical-cycles -
Figure 15.
a Components of storm surge (tide), terminology and illustration of interactions. b Key interactions among mean sea level, waves, storm surges caused by the atmosphere, tides, wave setup, and flooding [29].