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An explicit finite-difference scheme for the solution of the Kadomtsev-Petviashvili equation. (English) Zbl 0904.65094
The nonlinear equation \[ (u_t+ 6uu_x+ u_{xxx})_x +3u_{yy}=0 \] is considered for \(t>0\), \((x,y) \in[L_0,L_1]^2 \equiv \Omega\); the boundary conditions (on \(\partial \Omega)\) are of the form: \(u|_{\partial \Omega} =b\); \({\partial u\over \partial u} |_{\partial \Omega} =0\); the initial conditions are prescribed for \(u\) and \({\partial u\over \partial t}\) (consistency of them is not analyzed).
An explicit three-level difference scheme is used on the basis of a rectangular grid and central difference approximations. The analysis of stability (for linearized equations) is heuristic since it is based on periodicity conditions. Numerical experiments are presented.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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References:
[1] Bratsos A. G., Numerical Solutions of Nonlinear Partial Differential Equations, Ph. D. (1993)
[2] DOI: 10.1093/imamat/32.1-3.125 · Zbl 0542.35079
[3] Hirota R., Direct methods in soliton theory (1980)
[4] Kadomtsev B. B., Sov. Phys. Dokl. 15 pp 539– (1970)
[5] Korteweg D. J., Phil. Mag. Ser. 5 pp 422– (1985)
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