A Dressing Method in Mathematical Physics

This monograph systematically develops and considers the so-called "dressing method" for solving differential equations (both linear and nonlinear), a means to generate new non-trivial solutions for a given equation from the (perhaps trivial) solution of the same or related equation. Throughout, the text exploits the "linear experience" of presentation, with special attention given to the algebraic aspects of the main mathematical constructions and to practical rules of obtaining new solutions.

The monograph is devoted to the systematic presentation of the so called "dressing method" for solving differential equations (both linear and nonlinear) of mathematical physics. The essence of the dressing method consists in a generation of new non-trivial solutions of a given equation from (maybe trivial) solution of the same or related equation. The Moutard and Darboux transformations discovered in XIX century as applied to linear equations, the Bäcklund transformation in differential geometry of surfaces, the factorization method, the Riemann-Hilbert problem in the form proposed by Shabat and Zakharov for soliton equations and its extension in terms of the d-bar formalism comprise the main objects of the book. Throughout the text, a generally sufficient "linear experience" of readers is exploited, with a special attention to the algebraic aspects of the main mathematical constructions and to practical rules of obtaining new solutions. Various linear equations of classical and quantum mechanics are solved by the Darboux and factorization methods. An extension of the classical Darboux transformations to nonlinear equations in 1+1 and 2+1 dimensions, as well as its factorization are discussed in detail. The applicability of the local and non-local Riemann-Hilbert problem-based approach and its generalization in terms of the d-bar method are illustrated on various nonlinear equations.

INTRODUCTION 1 Mathematical preliminaries





1.1 Intertwine relations. Ladder operators

1.2 Factorization of matrices.





1.3 Factorization of l -matrix.





2 Factorization and dressing



2.1 Left and right division of ordinary differential operators. Bell polynomials.

2.2 Generalized Bell polynomials.

2.3 Division and factorization of differential operators. Generalized Riccati equations.

2.4 Darboux transformation. Generalized Burgers equations.

2.5 Darboux transformations in associative ring with automorphism. Quasideterminants.

2.6 Joint covariance of equations and nonlinear problems.

2.7 Example. Nonabelian Hirota system.

2.8 Second example. Nahm equation.

2.9 On symmetry and supersymmetry.



3 Darboux transformations



3.1 Gauge transformations and general definition of DT.

3.2 Basic notations: algebraic objects.

3.3 Zakharov - Shabat equations for two projectors. Elementary DT.

3.4 Elementary and binary Darboux Transformations for ZS equations with three projectors.

3.5 General case. Elementary and binary Darboux transformations.

3.6 The limit case - analog of Schlesinger transformation.

3.7 N-wave equations.

3.8 Higher combinations. Hirota-Satsuma (integrable CKdV) and KdV-MKdV equations.

3.9 Infinitesimal transforms for iterated DTs.

3.10 Geometric aspect.



4 Applications in linear problems



4.1 Integrable potentials in quantum mechanics.

4.2 Darboux transformations in continuous spectrum. Scattering problem.

4.3 Radial Schrödinger equation.

4.4 Darboux transformations and potentials in multidimensions.

4.5 Zero-range potentials, dressing and electron-molecule scattering.

4.5 Linear Darboux auto-transformation.

4.6 One-dimensional Dirac equation.

4.7 Non-stationary problems.

4.8 Dressing in classical mechanics. One-dimensional problems.

4.9 Poisson form of dynamics equations. Darboux theorem for classical evolution.



5 Dressing chain equations



5.1 Scalar case. The Boussinesq equation.

5.2 Chain equations for noncommutative field formulation.

5.3 The example of Zakharov-Shabat problem.

5.4 General operator. Stationary equations as eigenvalue problems and chains.

5.5 Finite closures of the chain equations. Finite-gap solutions.



6 The dressing in 2+1



6.1 Laplace transformations.

6.2 Combined Darboux-Laplace transforms.

6.3 Moutard transformation.

6.4 Goursat transformation.

6.5 The addition of the lower level to spectra of matrix and scalar components of d=2 SUSY Hamiltonian.



7 Important links



7.1 Bilinear formalism. The Hirota method.

7.2 Bäcklund transformations.

7.3 Singular manifold method.

7.4 General ideas for integral equations.

7.5 The Zakharov-Shabat theory.

7.6 Reduction conditions.



8 Dressing via the local Riemann-Hilbert problem



8.1 The RH problem and generation of new solutions.

8.2 The nonlinear Shchrödinger equation.

8.3 The modified NLS equation.

8.4 The Ablowitz-Ladik equation.

8.5 Three wave resonant interaction equations.

8.6 Homoclinic orbits by the dressing method.



9 Dressing via the non-local RH problem



9.1 The Benjamin-Ono equation.

9.2 The Kadomtsev-Petviashvili I equation.

9.3 The Davey-Stewartson I equation.

9.4 The modified Kadomtsev-Petviashvili I equation.



10 Generating new solutions via the d-bar problem



10.1 Elements of the d-bar formalism.

10.2 The KdV equation with self-consistent source.

10.3 The NLS, modified NLS, and modified Manakov equations with self-consistent source.

10.4 The Kadomtsev-Petviashvili II equation.

10.5 The Davey-Stewartson II equation.



CONCLUSION



REFERENCES



AUTHOR INDEX



SUBJECT INDEX

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