Newton’s method for periodic travelling wave solutions of the KdV equation. (Chinese. English summary) Zbl 0604.35067
B. P. Liu and C. V. Pao [Appl. Anal. 14, 293-301 (1983; Zbl 0515.35077)] showed that the problem to find the periodic travelling wave solutions \(u(x,t)=v(x-bt),v(y)=v(y+2T)\) of the equation \(u_ t+f(u)_ x+u_{xxx}=0\) can be reduced to one of solving the equation
\[
F(v)\equiv v(y)-\int^{2T}_{0}K(y,z)f(v(z))dz=0,
\]
where \(K(y,z)=(2\lambda sh \lambda T)^{-1}ch \lambda (T-| y-z|)-(2T \lambda^ 2)^{-1}.\)
The author proposes a Newton’s iteration procedure to solve the equation \(F(v)=0\) and establishes the convergence of the iteration under some technical suppositions.
The author proposes a Newton’s iteration procedure to solve the equation \(F(v)=0\) and establishes the convergence of the iteration under some technical suppositions.
Reviewer: G.Tu
MSC:
35Q99 | Partial differential equations of mathematical physics and other areas of application |
35B10 | Periodic solutions to PDEs |
35B40 | Asymptotic behavior of solutions to PDEs |
35G20 | Nonlinear higher-order PDEs |