Shallow-Water Wave Dynamics: Butterfly Waves, X-Waves, Multiple-Lump Waves, Rogue Waves, Stripe Soliton Interactions, Generalized Breathers, and Kuznetsov–Ma Breathers
A nonlinear <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics><...
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MDPI AG
2025-01-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/9/1/31 |
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| author | Sarfaraz Ahmed Ujala Rehman Jianbo Fei Muhammad Irslan Khalid Xiangsheng Chen |
| author_facet | Sarfaraz Ahmed Ujala Rehman Jianbo Fei Muhammad Irslan Khalid Xiangsheng Chen |
| author_sort | Sarfaraz Ahmed |
| collection | DOAJ |
| description | A nonlinear <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-dimensional nonlinear Geng equation that can be utilized to explain the dynamics of shallow-water waves in fluids is given special attention. Various wave solutions are produced with the aid of the Hirota bilinear and Cole–Hopf transformation techniques. By selecting the appropriate polynomial function and implementing the distinct transformations in bilinear form, bright lump waves, dark lump waves, and rogue waves (RWs) are generated. A positive quadratic transformation and cosine function are combined in Hirota bilinear form to evaluate the RW solutions. Typically, RWs have crests that are noticeably higher than those of surrounding waves. These waves are also known as killer, freak, or monster waves. The lump periodic solutions (LPSs) are obtained using a combination of the cosine and positive quadratic functions. The lump-one stripe solutions are computed by using a mix of positive quadratic and exponential transformations to the governing equation. The lump two-stripe solutions are obtained by using a mix of positive quadratic and exponential transformations to the governing equation. The interactional solutions of lump, kink, and periodic wave solutions are obtained. Additionally, mixed solutions with butterfly waves, X-waves and lump waves are computed. The Ma breather (MB), Kuznetsov–Ma breather (KMB), and generalized breathers GBs are generated. Furthermore, solitary wave solution is obtained and a relation for energy of the wave via ansatz function technique. |
| format | Article |
| id | doaj-art-32030202a11a448d9d34d2f476dc850c |
| institution | Kabale University |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-32030202a11a448d9d34d2f476dc850c2025-01-24T13:33:26ZengMDPI AGFractal and Fractional2504-31102025-01-01913110.3390/fractalfract9010031Shallow-Water Wave Dynamics: Butterfly Waves, X-Waves, Multiple-Lump Waves, Rogue Waves, Stripe Soliton Interactions, Generalized Breathers, and Kuznetsov–Ma BreathersSarfaraz Ahmed0Ujala Rehman1Jianbo Fei2Muhammad Irslan Khalid3Xiangsheng Chen4State Key Laboratory of Intelligent Geotechnics and Tunnelling, Shenzhen University, Shenzhen 518060, ChinaState Key Laboratory of Intelligent Geotechnics and Tunnelling, Shenzhen University, Shenzhen 518060, ChinaState Key Laboratory of Intelligent Geotechnics and Tunnelling, Shenzhen University, Shenzhen 518060, ChinaState Key Laboratory of Intelligent Geotechnics and Tunnelling, Shenzhen University, Shenzhen 518060, ChinaState Key Laboratory of Intelligent Geotechnics and Tunnelling, Shenzhen University, Shenzhen 518060, ChinaA nonlinear <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-dimensional nonlinear Geng equation that can be utilized to explain the dynamics of shallow-water waves in fluids is given special attention. Various wave solutions are produced with the aid of the Hirota bilinear and Cole–Hopf transformation techniques. By selecting the appropriate polynomial function and implementing the distinct transformations in bilinear form, bright lump waves, dark lump waves, and rogue waves (RWs) are generated. A positive quadratic transformation and cosine function are combined in Hirota bilinear form to evaluate the RW solutions. Typically, RWs have crests that are noticeably higher than those of surrounding waves. These waves are also known as killer, freak, or monster waves. The lump periodic solutions (LPSs) are obtained using a combination of the cosine and positive quadratic functions. The lump-one stripe solutions are computed by using a mix of positive quadratic and exponential transformations to the governing equation. The lump two-stripe solutions are obtained by using a mix of positive quadratic and exponential transformations to the governing equation. The interactional solutions of lump, kink, and periodic wave solutions are obtained. Additionally, mixed solutions with butterfly waves, X-waves and lump waves are computed. The Ma breather (MB), Kuznetsov–Ma breather (KMB), and generalized breathers GBs are generated. Furthermore, solitary wave solution is obtained and a relation for energy of the wave via ansatz function technique.https://www.mdpi.com/2504-3110/9/1/31rogue wavesbreatherslumprational solitonsthe (3 + 1)-dimensional Geng equationHirota bilinear method (HBM) |
| spellingShingle | Sarfaraz Ahmed Ujala Rehman Jianbo Fei Muhammad Irslan Khalid Xiangsheng Chen Shallow-Water Wave Dynamics: Butterfly Waves, X-Waves, Multiple-Lump Waves, Rogue Waves, Stripe Soliton Interactions, Generalized Breathers, and Kuznetsov–Ma Breathers Fractal and Fractional rogue waves breathers lump rational solitons the (3 + 1)-dimensional Geng equation Hirota bilinear method (HBM) |
| title | Shallow-Water Wave Dynamics: Butterfly Waves, X-Waves, Multiple-Lump Waves, Rogue Waves, Stripe Soliton Interactions, Generalized Breathers, and Kuznetsov–Ma Breathers |
| title_full | Shallow-Water Wave Dynamics: Butterfly Waves, X-Waves, Multiple-Lump Waves, Rogue Waves, Stripe Soliton Interactions, Generalized Breathers, and Kuznetsov–Ma Breathers |
| title_fullStr | Shallow-Water Wave Dynamics: Butterfly Waves, X-Waves, Multiple-Lump Waves, Rogue Waves, Stripe Soliton Interactions, Generalized Breathers, and Kuznetsov–Ma Breathers |
| title_full_unstemmed | Shallow-Water Wave Dynamics: Butterfly Waves, X-Waves, Multiple-Lump Waves, Rogue Waves, Stripe Soliton Interactions, Generalized Breathers, and Kuznetsov–Ma Breathers |
| title_short | Shallow-Water Wave Dynamics: Butterfly Waves, X-Waves, Multiple-Lump Waves, Rogue Waves, Stripe Soliton Interactions, Generalized Breathers, and Kuznetsov–Ma Breathers |
| title_sort | shallow water wave dynamics butterfly waves x waves multiple lump waves rogue waves stripe soliton interactions generalized breathers and kuznetsov ma breathers |
| topic | rogue waves breathers lump rational solitons the (3 + 1)-dimensional Geng equation Hirota bilinear method (HBM) |
| url | https://www.mdpi.com/2504-3110/9/1/31 |
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