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Inversion of the Miura transformation. (English. Russian original) Zbl 0702.35222

Math. Notes 46, No. 4, 762-769 (1989); translation from Mat. Zametki 46, No. 4, 14-24 (1989).
It is proved that the global (\(\forall\, x\in\mathbb{R})\) inverse Miura transformation is possible, more precisely, it is proved that the Riccati equation \(v(x,t)=u_x(x,t)+u^2(x,t)\) is solvable for all \(x\in\mathbb{R}\), under the boundary conditions \(u(x,t)\sim C_1\) (resp. \(C_2)\) for \(x\to -\infty\) (resp. \(x\to \infty)\), \(v(x,t)\sim a^2\) (resp. \(b^2)\) for \(x\to -\infty\) (resp. \(x\to \infty)\) (where \(c_1=\pm a\), \(c_ 2\pm b\), \(a>0\), \(b>0)\) and the summability conditions \[ \int^{0}_{- \infty}(1+| x|)| v(x)-a^2| \,dx+\int^{\infty}_{0}(1+x)| v(x)-b^2| \,dx<\infty, \]
\[ \int^{\infty}_{-\infty}[| \phi'(x)|^ 2+v(x)| \phi(x)|^2] \,dx>0,\quad \forall \phi (x)\in C_0^{\infty}(\mathbb{R}). \]
Reviewer: Y. C. Yang

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A30 Geometric theory, characteristics, transformations in context of PDEs
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
34L25 Scattering theory, inverse scattering involving ordinary differential operators
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