The families of explicit solutions for the Hirota equation. (English) Zbl 1460.35311
The authors present some explicit solutions for the Hirota equation derived from the Ablowitz-Kaup-Newell-Segur (AKNS) shallow water wave equation: A dark one soliton solution is obtained by using homogeneous balance for the Hirota equation; multiple soliton solutions and multiple singular solutions are derived by using the so-called Hirota bilinear form of the equation. Finally, the authors obtain one- and two-periodic solutions for the AKNS equations in the form of Riemann theta functions by employing the Hirota bilinear form of the AKNS equation. The asymptotic behaviour of the two periodic solutions is then indicated.
Reviewer: Catalin Popa (Iaşi)
MSC:
| 35Q51 | Soliton equations |
| 35C08 | Soliton solutions |
| 68W30 | Symbolic computation and algebraic computation |
| 35B10 | Periodic solutions to PDEs |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 76B25 | Solitary waves for incompressible inviscid fluids |
| 14H42 | Theta functions and curves; Schottky problem |
Keywords:
Hirota equation; shallow water wave equation; dark soliton solution; multiple soliton solution; periodic solutionReferences:
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