New Inverse Spectral Problem and Its Application

Abstract

The origin of inverse spectral problems lies in natural science, but the problems themselves are purely mathematical. At the beginning these problems attracted attention of mathematicians by their nonstandard physical contents. But we think that today their place in mathematical physics is determined rather by the unexpected connection between inverse problems and nonlinear evolution equations which was discovered in 1967. This discovery was made in a famous paper by Gardner, Greene, Kruskal and Miura (1967). They found that the scattering data of a family H(t) (−∞ < t < ∞) (i.e. the reflection coefficients r(k, t) and normalizing coefficients m(ik _l , t)) of Schrödinger operators

EquationSource<math display=&#x2019;block&#x2019;> <mrow> <mi>H</mi><mrow><mo>(</mo> <mi>t</mi> <mo>)</mo></mrow><mo>=</mo><mo>&#x2212;</mo><mfrac> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> </mrow> <mrow> <mi>d</mi><msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo><mi>u</mi><mrow><mo>(</mo> <mrow> <mi>x</mi><mo></mo><mi>t</mi></mrow> <mo>)</mo></mrow></mrow> </math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$H\left( t \right) = - \frac{{{d^2}}}{{d{x^2}}} + u\left( {x,t} \right)$$

satisfy linear differential equations

EquationSource<math display=&#x2019;block&#x2019;> <mrow> <msub> <mover accent=&#x2019;true&#x2019;> <mi>r</mi> <mo>&#x02D9;</mo> </mover> <mi>t</mi> </msub> <mo>=</mo><mn>8</mn><mi>i</mi><msup> <mi>k</mi> <mn>3</mn> </msup> <mi>r</mi><mo></mo><mtext></mtext><msub> <mover accent=&#x2019;true&#x2019;> <mi>m</mi> <mo>&#x02D9;</mo> </mover> <mi>t</mi> </msub> <mo>=</mo><mn>8</mn><mi>i</mi><msup> <mrow> <mrow><mo>(</mo> <mrow> <mi>i</mi><msub> <mi>k</mi> <mi>l</mi> </msub> </mrow> <mo>)</mo></mrow></mrow> <mn>3</mn> </msup> <mi>m</mi></mrow> </math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\dot r_t} = 8i{k^3}r,{\text{ }}{\dot m_t} = 8i{\left( {i{k_l}} \right)^3}m$$

if the potentials u(x, t) are rapidly decreasing solutions of the KdV equation

EquationSource<math display=&#x2019;block&#x2019;> <mrow> <msub> <mover accent=&#x2019;true&#x2019;> <mi>u</mi> <mo>&#x02D9;</mo> </mover> <mi>t</mi> </msub> <mo>=</mo><mn>6</mn><mi>u</mi><msub> <mi>u</mi> <mi>x</mi> </msub> <mo>=</mo><msub> <mi>u</mi> <mrow> <mi>x</mi><mi>x</mi><mi>x</mi></mrow> </msub> </mrow> </math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\dot u_t} = 6u{u_x} = {u_{xxx}}$$

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