| Volume 14, Number 4, 2024, Pages - DOI：10.11948/JAAC-2023-0320 |
| B\ |
| Yadong Shang,Huafei Di |
| Keywords:a variant Boussinesq equations, nonlinear transformation, exact linearization, explicit exact solution, singular wave solution. |
| Abstract: |
| This paper deals with a variant Boussinesq equations which describes the propagation of
shallow water waves in a lake or near an ocean beach. We derive out two hetero-B\"{a}cklund transformations between the variant Boussinesq equations and two linear parabolic equations by using the extended homogeneous balance method. We also obtain two hetero-B\"{a}cklund transformations between the variant Boussinesq equations and Burgers equations. Furthermore, we obtain two hetero-B\"{a}cklund transformation between the variant Boussinesq equations and heat equations. By using these B\"{a}cklund transformations and so-called "seed solution", we obtain a large number of explicit exact solutions of the variant Boussinesq equations. Especially, The infinite explicit exact singular wave solutions of variant Boussinesq equations are obtained for the first time. It is worth noting that these singular wave solutions of variant Boussinesq equations will blow up on some lines or curves in the $(x,t)$ plane. These facts reflect the complexity of the structure of the solution of variant Boussinesq equations. It also reflects the complexity of shallow water wave propagation from one aspect. |
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