On a class of polynomials connected with the Korteweg-deVries equation

A new and simpler construction of the family of rational solutions of the Korteweg-deVries equation is given. This construction is related to a factorization of the Sturm-Liouville operators into first order operators and a new deformation problem for the latter. In the final section the spectral representation for the corresponding complex potentials is discussed.

[1]  Robert M. Miura,et al.  Korteweg-de Vries Equation and Generalizations. I. A Remarkable Explicit Nonlinear Transformation , 1968 .

[2]  C. S. Gardner,et al.  Korteweg-devries equation and generalizations. VI. methods for exact solution , 1974 .

[3]  H. McKean,et al.  The spectrum of Hill's equation , 1975 .

[4]  Peter D. Lax,et al.  Almost Periodic Solutions of the KdV Equation , 1976 .

[5]  H. McKean,et al.  Rational and elliptic solutions of the korteweg‐de vries equation and a related many‐body problem , 1977 .

[6]  S. Tuan,et al.  Potentials with zero scattering phase , 2018 .

[7]  Soliton Solutions of the Korteweg-de Vries Equation , 1974 .

[8]  F. Calogero.  Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations and related «solvable» many-body problems , 1978 .

[9]  D. V. Choodnovsky,et al.  Pole expansions of nonlinear partial differential equations , 1977 .

[10]  Peter J. Olver,et al.  Evolution equations possessing infinitely many symmetries , 1977 .

[11]  M. Wadati,et al.  Relationships among Inverse Method, Bäcklund Transformation and an Infinite Number of Conservation Laws , 1975 .

[12]  H. Moses.  A solution of the Korteweg–de Vries equation in a half‐space bounded by a wall , 1976 .

[13]  V. Kulikov.  Degenerate elliptic curves and resolution of uni- and bimodal singularities , 1975 .

[14]  J. L. Burchnall,et al.  Commutative Ordinary Differential Operators , 1928 .

[15]  P. Lax.  INTEGRALS OF NONLINEAR EQUATIONS OF EVOLUTION AND SOLITARY WAVES. , 1968 .

[16]  J. Satsuma.  Solitons and Rational Solutions of Nonlinear Evolution Equations (Theory of Nonlinear Waves) , 1978 .

[17]  J. L. Burchnall,et al.  Commutative Ordinary Differential Operators , 1923 .

[18]  V. Matveev,et al.  Hill's operator with finitely many gaps , 1975 .

[19]  M. Crum.  ASSOCIATED STURM-LIOUVILLE SYSTEMS , 1999, physics/9908019.

[20]  P. Deift.  Applications of a commutation formula , 1978 .