Nonlinear scattering and analyticity properties of solitons
The author considers the scattering of a soliton or solitary wave by a linear potential and shows that the amount of mass and energy lost by the solitary wave during a scattering event is exponentially small for a strong nonlinearity. In the case of a delta function potential or a meromorphic potential and the cubic nonlinear Schrödinger equation (NLS), for large distance \(Z\) into the medium, he found that the velocity of the soliton decays as \((\log (Z))^{-1}\). The author shows that the decay is more slow than \((\log (Z))^{-1}\) for a potential which is an entire function of a complex variable. Specifically, after making a Galilean change of variables, the NLS equation becomes \[ i \psi_t = - \psi_{xx} - 2|\psi|^2 \psi + \varepsilon \delta(x + 2bt) \psi. \] With an initial condition in the past, the solution should reduce to the exact solitary wave solution \[ \psi(x,t) \rightarrow a \text{sech}(ax) \exp(ia^2t), t \rightarrow -\infty. \] The author assumes the ansatz \[ \psi(x,t) \approx \exp(ia^2t)[(a+\varepsilon a^{(1)}(t)(x-\varepsilon a^{(1)}(t)) \times \exp(i\varepsilon \psi^{(1)}(t)+i\varepsilon b^{(1)}(t)x+\varepsilon \psi^R+O(\varepsilon^2))]. \] He introduces the eigenfunction decomposition \[ \psi^R = \int A(k,\alpha,t)f(k,x) \exp(-i\lambda(k)t)+\overline{A}(k,x,\alpha)\overline{g}(k,x) \exp(i\lambda(k)t)dk \] and shows \[ A(x,\alpha)=A(x,\alpha,+\infty) \propto \frac{(k-1)^2+\alpha^2}{\text{cosh}(\frac{\pi}{4\alpha}(k^2+\alpha^2-1))}. \] From this estimate he shows the velocity \(b(z)\) to satisfy \(b(z) \approx c'\ln^{-1}(Z)\). The paper is generally well written and easy to read and contains comments on relations with previous results in these fields.
scientific article; zbMATH DE number 1138724