On the Cauchy problem for the linearized GPKdV and gauge transformations for a quadratic pencil and AKNS system

The present work in the area of soliton theory studies two problems which arise when seeking analytic solutions to certain nonlinear partial differential equations. In the first one, Lax pairs associated with prolonged eigenfunctions and prolonged squared eigenfunctions (prolonged squares) are derived for a Schrödinger equation with a potential depending polynomially on the spectral parameter (of degree N) and its respective hierarchy of nonlinear evolution equations (here named generalized polynomial Korteweg-de Vries equations or GPKdV). It is shown that the prolonged squares satisfy the linearized GPKdV equations. On that basis, the Cauchy problem for the linearized GPKdV has been solved by finding a complete set of such prolonged squares and applying an expansion formula derived in another work by the author. The results are a generalization of the ones by Sachs (SIAM J. Math. Anal. 14, 1983, 674-683). Moreover, a condition on the so-called recursion operator A is derived which generates the whole hierarchy of Lax pairs associated with the prolonged squares. As for the second part of the work, it developed a scheme for deriving gauge transformations between different linear spectral problems. Then the scheme is applied to obtain all known Darboux transformations for a quadratic pencil (the spectral problem considered in the first part at N = 2), Schrödinger equation (N = 1), Ablowitz-Kaup-Newell-Segur (AKNS) system and also derive the Jaulent-Miodek transformation. Moreover, the scheme yields a large family of new transformations of the above types. It also gives some insight on the structure of the transformations and emphasizes the symmetry with respect to inversion that they possess.

Name thesis:degree_name

Ph. D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

1992

Author dc:creator

• Yordanov, Russi Georgiev

Chair dc:contributor.committeechair

• Shaw, John K.

Committee members dc:contributor.committeemember

• Klaus, Martin
• Greenberg, William
• Zweifel, Paul F.
• Hagedorn, George

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en

Dc Identifier Other

etd-06062008-171329

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/38467