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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References:

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