Dynamical Visualization and Qualitative Analysis of the (4+1)-Dimensional KdV-CBS Equation Using Lie Symmetry Analysis
This study investigates novel optical solitons within the intriguing (4+1)-dimensional Korteweg–de Vries–Calogero–Bogoyavlenskii–Schiff (KdV-CBS) equation, which integrates features from both the Korteweg–de Vries and the Calogero–Bogoyavlenskii–Schiff equations. Firstly, all possible symmetry gener...
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2024-12-01
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| author | Maria Luz Gandarias Nauman Raza Muhammad Umair Yahya Almalki |
| author_facet | Maria Luz Gandarias Nauman Raza Muhammad Umair Yahya Almalki |
| author_sort | Maria Luz Gandarias |
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| description | This study investigates novel optical solitons within the intriguing (4+1)-dimensional Korteweg–de Vries–Calogero–Bogoyavlenskii–Schiff (KdV-CBS) equation, which integrates features from both the Korteweg–de Vries and the Calogero–Bogoyavlenskii–Schiff equations. Firstly, all possible symmetry generators are found by applying Lie symmetry analysis. By using these generators, the given model is converted into an ordinary differential equation. An adaptive approach, the generalized exp(-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">S</mi></mrow></semantics></math></inline-formula>(<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>χ</mi></semantics></math></inline-formula>)) expansion technique has been utilized to uncover closed-form solitary wave solutions. The findings reveal a range of soliton types, including exponential, rational, hyperbolic, and trigonometric functions, represented as bright, singular, rational, periodic, and new solitary waves. These results are illustrated numerically and accompanied by insightful physical interpretations, enriching the comprehension of the complex dynamics modeled by these equations. Our approach’s novelty lies in applying a new methodology to this problem, yielding a variety of novel optical soliton solutions. Additionally, we employ bifurcation and chaos techniques for a qualitative analysis of the model, extracting a planar system from the original equation and mapping all possible phase portraits. A thorough sensitivity analysis of the governing equation is also presented. These results highlight the effectiveness of our methodology in tackling nonlinear problems in both mathematics and engineering, surpassing previous research efforts. |
| format | Article |
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| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-b4a8ae8d59094e50a419fb42e96959af2025-01-10T13:18:13ZengMDPI AGMathematics2227-73902024-12-011318910.3390/math13010089Dynamical Visualization and Qualitative Analysis of the (4+1)-Dimensional KdV-CBS Equation Using Lie Symmetry AnalysisMaria Luz Gandarias0Nauman Raza1Muhammad Umair2Yahya Almalki3Department of Mathematics, University of Cadiz, 11510 Puerto Real, SpainDepartment of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, PakistanDepartment of Mathematics, University of Engineering and Technology, Lahore 54890, PakistanDepartment of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi ArabiaThis study investigates novel optical solitons within the intriguing (4+1)-dimensional Korteweg–de Vries–Calogero–Bogoyavlenskii–Schiff (KdV-CBS) equation, which integrates features from both the Korteweg–de Vries and the Calogero–Bogoyavlenskii–Schiff equations. Firstly, all possible symmetry generators are found by applying Lie symmetry analysis. By using these generators, the given model is converted into an ordinary differential equation. An adaptive approach, the generalized exp(-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="fraktur">S</mi></mrow></semantics></math></inline-formula>(<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>χ</mi></semantics></math></inline-formula>)) expansion technique has been utilized to uncover closed-form solitary wave solutions. The findings reveal a range of soliton types, including exponential, rational, hyperbolic, and trigonometric functions, represented as bright, singular, rational, periodic, and new solitary waves. These results are illustrated numerically and accompanied by insightful physical interpretations, enriching the comprehension of the complex dynamics modeled by these equations. Our approach’s novelty lies in applying a new methodology to this problem, yielding a variety of novel optical soliton solutions. Additionally, we employ bifurcation and chaos techniques for a qualitative analysis of the model, extracting a planar system from the original equation and mapping all possible phase portraits. A thorough sensitivity analysis of the governing equation is also presented. These results highlight the effectiveness of our methodology in tackling nonlinear problems in both mathematics and engineering, surpassing previous research efforts.https://www.mdpi.com/2227-7390/13/1/89(4+1)-D KdV–CBS equationgeneralized exp(-?(χ)) expansion methodnovel solitonsdynamic study of bifurcation and chaotic behaviorsensitivity analysis |
| spellingShingle | Maria Luz Gandarias Nauman Raza Muhammad Umair Yahya Almalki Dynamical Visualization and Qualitative Analysis of the (4+1)-Dimensional KdV-CBS Equation Using Lie Symmetry Analysis Mathematics (4+1)-D KdV–CBS equation generalized exp(-?(χ)) expansion method novel solitons dynamic study of bifurcation and chaotic behavior sensitivity analysis |
| title | Dynamical Visualization and Qualitative Analysis of the (4+1)-Dimensional KdV-CBS Equation Using Lie Symmetry Analysis |
| title_full | Dynamical Visualization and Qualitative Analysis of the (4+1)-Dimensional KdV-CBS Equation Using Lie Symmetry Analysis |
| title_fullStr | Dynamical Visualization and Qualitative Analysis of the (4+1)-Dimensional KdV-CBS Equation Using Lie Symmetry Analysis |
| title_full_unstemmed | Dynamical Visualization and Qualitative Analysis of the (4+1)-Dimensional KdV-CBS Equation Using Lie Symmetry Analysis |
| title_short | Dynamical Visualization and Qualitative Analysis of the (4+1)-Dimensional KdV-CBS Equation Using Lie Symmetry Analysis |
| title_sort | dynamical visualization and qualitative analysis of the 4 1 dimensional kdv cbs equation using lie symmetry analysis |
| topic | (4+1)-D KdV–CBS equation generalized exp(-?(χ)) expansion method novel solitons dynamic study of bifurcation and chaotic behavior sensitivity analysis |
| url | https://www.mdpi.com/2227-7390/13/1/89 |
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