By Gu Ch., Hг H., Zhou Z.
ISBN-10: 0792310691
ISBN-13: 9780792310693
The Darboux transformation procedure is without doubt one of the most well known equipment for developing particular ideas of partial differential equations that are referred to as integrable platforms and play very important roles in mechanics, physics and differential geometry.This ebook offers the Darboux alterations in matrix shape and offers simply algebraic algorithms for developing the specific strategies. A foundation for utilizing symbolic computations to procure the categorical specified strategies for plenty of integrable structures is confirmed. additionally, the habit of easy and multi-solutions, even in multi-dimensional circumstances, should be elucidated essentially. the tactic covers a chain of vital equations akin to several types of AKNS structures in R1+n, harmonic maps from 2-dimensional manifolds, self-dual Yang-Mills fields and the generalizations to better dimensional case, idea of line congruences in 3 dimensions or better dimensional area and so on. these kind of situations are defined intimately. This publication comprises many effects that have been acquired by means of the authors long ago few years.
Read Online or Download Darboux Transformations in Integrable Systems: Theory and their Applications to Geometry PDF
Best geometry and topology books
Geometric Aspects of Functional Analysis by Joram Lindenstrauss, Vitali D. Milman PDF
This can be the 3rd released quantity of the complaints of the Israel Seminar on Geometric features of practical research. the big majority of the papers during this quantity are unique learn papers. there has been final 12 months a powerful emphasis on classical finite-dimensional convexity concept and its reference to Banach area idea.
- Problemas Matematicos - Geometria
- Lectures on Homotopy Theory
- Introduction géométrique à quelques théories physiques
- Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications
- Encyclopedie des sciences mathematiques. Geometrie descriptive et elementaire
- Geometrical and Topological Methods in Gauge Theories
Extra resources for Darboux Transformations in Integrable Systems: Theory and their Applications to Geometry
Sample text
93) has a solution satisfying S(x0 , t0 ) = S0 . That is, the system is integrable. Proof. First suppose the Jordan form of S0 is a diagonal matrix. Suppose its eigenvalues are λ1 , · · ·, λN and the corresponding eigenvectors are h0i . 95) satisfying hi (x0 , t0 ) = h0i . Then these hi are linearly independent in a neighborhood of (x0 , t0 ). 93) has a solution. If the Jordan form of S0 is not diagonal, then there is a series of matrices (k) (k) (k) {S S0 } such that the Jordan form of S0 is diagonal and S0 → S0 as (k) k → ∞.
The proof here is given by [52] (2 × 2 case) and [33] (N × N case). This proof does not depend on any boundary conditions and the permutation of the parameters is expressed definitely. From the solution (P, Φ(λ)), we can construct the Darboux transfor(1) (1) (1) (1) mation with parameters λ1 , · · ·, λN and the solutions hi = Φ(λi )li (1) of the Lax pair. Then the solution (P (1) , Φ(1) (λ)) is obtained. Here li ’s are N constant vectors. Next, construct a Darboux matrix for (P (1) , (2) (2) (2) Φ(1) (λ)) with parameters λ1 , · · ·, λN and li to get (P (1,2) , Φ(1,2) (λ)).
They are the important special cases of the 2 × 2 AKNS system: (1) KdV hierarchy: p is real and q = −1; (2) MKdV-SG hierarchy: q = −p is real; (3) Nonlinear Schr¨ o¨dinger hierarchy: q = −¯ p. g. [82, 88, 91, 105, 117, 118]). Here we use a unified method to deal with the whole hierarchy, and the coefficients may depend on t [32, 45]. At the end of this section, we discuss the general AKNS system with u(N ) reduction. This is a generalization of the nonlinear Schr¨ odinger hierarchy and has many applications to other problems.
Darboux Transformations in Integrable Systems: Theory and their Applications to Geometry by Gu Ch., Hг H., Zhou Z.
by Robert
4.4