Periodic-background solutions of Kadomtsev-Petviashvili I equation. (English) Zbl 1509.35267
Summary: A systematic method to obtain exact solutions of the Zakharov-Shabat spectral problems with Jacobi elliptic function potentials is constructed by using the Baker-Akhiezer functions which appear in the study quasi-periodic solutions. As a result, periodic-background lump (PB-lump for short) solutions of the Kadomtsev-Petviashvili I (KPI) equation are constructed. Like lump solutions, PB-lump solutions are localized in the xy-plane and propagate at constant velocities. Different from lump solutions, which are rational solutions and tend to a constant amplitude when \(x\) and \(y\) are large, PB-lump solutions are composed of polynomials, elliptic functions and other special functions, and tend to the periodic background when \(x\) and \(y\) are large. What’s more, the amplitude of the crest for the PB-lump solution changes periodically as it propagates on the xy-plane. The trajectories of the PB-lumps for the KPI equation are presented on the basis of the formal series expansion technique. As an illustration, both first- and second-order PB-lumps and their exact formulas are obtained.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K15 | Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems |
| 35K35 | Initial-boundary value problems for higher-order parabolic equations |
| 35C08 | Soliton solutions |
Keywords:
periodic-background lumps; Darboux transformations; Baker-Akhiezer functions; formal series expansionReferences:
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