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Perturbation of a solitary wave of the nonlinear Klein-Gordon equation. (English. Russian original) Zbl 0946.35083
Sib. Math. J. 41, No. 2, 281-293 (2000); translation from Sib. Mat. Zh. 41, No. 2, 345-358 (2000).
The author considers the Goursat problem for the nonlinear Klein-Gordon equation \[ \partial_t\partial_x\Phi - \Phi/2 +\Phi^3/2=\varepsilon f(\Phi),\quad \varepsilon>0. \eqno{(1)} \] The boundary conditions are written as \[ \Phi|_{t=0}=\phi_0(z)|_{t=0,\varepsilon=0},\quad\Phi|_{x\to -\infty}=1, \eqno{(2)} \] with \(\phi_0(z)|_{\varepsilon=0}\) a solution to equation (1) for \(\varepsilon=0\) which is a soliton (a kink). Here the phase variable \(z\) depends on \(x,t,\varepsilon\). An asymptotic solution to problem (1), (2) is constructed in the following form: \[ \phi(x,t,\varepsilon)=\phi_0(z)+\varepsilon u(x,t,\varepsilon)+ \varepsilon^2 v(x,t,\varepsilon). \eqno{(3)} \] It is shown that a solution is representable in the form (3) and this representation is valid for \(x\in \mathbb R\) and \(0\leq t\leq T_0\varepsilon^{-1}\) with some \(T_0>0\). The asymptotics of the solution (3) is of the form \(\phi=\phi_0+\varepsilon u+ O(\varepsilon^{1+\alpha})\) (\(\alpha>0\)) uniformly in \(x\in \mathbb R\) and \(0\leq t\leq T_0\varepsilon^{-1}\).

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35C20 Asymptotic expansions of solutions to PDEs
35L70 Second-order nonlinear hyperbolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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References:
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