zbMATH — the first resource for mathematics

The infinite-dimensional dressing dynamical system. (English) Zbl 0762.35098
The infinite chain of ordinary differential equations \[ (f_ j+f_{j+1})_ x=f^ 2_ j-f^ 2_{j+1}+\lambda_ j- \lambda_{j+1},\quad j=0,\pm 1,\pm 2,\dots \] is considered, that is closely related to the spectral theory of the linear Schrödinger equation \[ \psi_{xx}+(q(x)+\lambda)\psi=0. \] This chain is named the dressing dynamical system because the following theorem describing dressing, i.e. constructing a potential with additional eigenvalues, is valid.
Theorem. If the functions \(\varphi_ j\), \(j=1,\dots,n\) are solutions of the Schrödinger equation with eigenvalues \(\lambda_ j\), respectively, then the functions \[ f_ n=(d/dx)\log\varphi_ n,\quad f_{n- 1}=(d/dx)\log{\langle\varphi_ n,\varphi_{n-1}\rangle\over\varphi_ n},\dots, \quad f_ 1=(d/dx)\log{\langle\varphi_ n,\dots,\varphi_ 1\rangle\over\langle\varphi_ n,\dots ,\varphi_ 2\rangle}, \] \(\langle,\rangle\) denoting Wronskians, satisfy the chain of dressing equations \[ (d/dx)(f_ j+f_{j-1})=f_ j^ 2-f^ 2_{j- 1}+\lambda_ j-\lambda_{j-1},\quad j=2,\dots,n. \] The result of \(n\)- times dressing is therefore \[ q_ 0=2(d^ 2/dx^ 2)\log\langle\varphi_ n,\dots,\varphi_ 1\rangle+q_ n. \] The appropriate choice of \(\varphi_ n\), \(\varphi_{n-1},\dots\) leads to \(n\)-soliton solutions of the Korteweg-de Vries equation. The main point of this paper is the investigation of the asymptotic behaviour of \(n\)- eigenvalue potential as \(n\to\infty\).

35Q51 Soliton equations
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
34L25 Scattering theory, inverse scattering involving ordinary differential operators
Full Text: DOI