Oceanic long-gravity-water-wave investigations on a variable-coefficient nonlinear dispersive-wave system

Abstract

Oceanic water waves are actively investigated. One example is a variable-coefficient nonlinear dispersive-wave system modelling the long gravity water waves in a shallow oceanic environment. The system describes the surface velocity of the water wave and the wave elevation. Making use of symbolic computation, we perform the Painlevé analysis and work out two sets of the bilinear forms, two sets of the N-soliton solutions and one set of the similarity reductions for the aforementioned system, with N being a positive integer. We also graphically discuss those soliton solutions. What we accomplish should rely on the variable coefficients. This paper could be of some use for the future oceanic studies.

Keywords:

• Oceans
• water waves
• dispersive-wave system
• variable coefficients
• Painlevé analysis
• bilinear forms
• N-soliton solutions
• similarity reductions
• symbolic computation

Acknowledgements

We express our sincere thanks to the Editors and Reviewers for their valuable comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 For example, presently interesting, with respect to the shallow water waves, Ref. [31] has constructed some bilinear auto-Bäcklund transformations and soliton solutions of a (3+1)-dimensional generalized nonlinear evolution equation.

2 It can be noted that the hetero-Bäcklund transformations have also been named the non-auto-Bäcklund transformations, which are usually interpreted as the relations between the solutions of different partial differential equations [44,46–48].

3 More symbolic-computation investigations could be seen, e.g., in Refs. [59–79].

4 Note that the scaling transformation in Ref. [44] is correct. However, a part of the scaling transformation in Ref. [45] is wrong, which we have corrected here.

5 similar to those in Refs.  [88–95]

Additional information

Funding

This work has been supported by the National Natural Science Foundation of China [grant number 11871116] and Fundamental Research Funds for the Central Universities of China [grant number 2019XD-A11]. XYG also thanks the National Scholarship for Doctoral Students of China and BUPT Innovation and Entrepreneurship Support Program, Beijing University of Posts and Telecommunications.