Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions †
Abstract
:1. Introduction
- 1.
- Scattering data The minimal sets of scattering data are determined by the asymptotics of
- 2.
- Resolvent The FAS determine the kernel of the resolvent of L. Applying the contour integration method on , one can derive the spectral expansions for L, i.e., the completeness relation of the FAS.
- 3.
- Dressing method Zakharov–Shabat dressing method is a very effective and convenient method to construct the class of reflectionless potentials of L and to derive the soliton solutions of the NLEE. The simplest dressing factor has pole singularities at , which determine the new discrete eigenvalues that are added to the spectrum of the initial Lax operator.
- 4.
- Generalized Fourier transforms (GFTs) Here, we start with a GZS system related to a simple Lie algebra with Cartan-Weyl basis , [21] and construct the so-called ‘squared solutions’
- 5.
- Hierarchies of Hamiltonian structures The GFTs described above allow one to prove that each of the NLEE related to L allows a hierarchy of Hamiltonian structures. More precisely, each NLEE allows a hierarchy of Hamiltonians and a hierarchy of symplectic forms (or a hierarchy of Poisson brackets) such that for any n they produce the relevant NLEE [22,35,36].
- 6.
- Complete integrability and action-angle variables Starting from the famous paper by Zakharov and Faddeev [1] it is known that some of the NLEE allow action-angle variables. The difficulty here is that these NLEE are Hamiltonian systems with infinitely many degrees of freedom. Therefore, the strict derivation of the proof must be based on the completeness relation for the ‘squared solutions’. In fact, V. Gerdjikov and E. Khristov [27,28] (see also [30]) proposed the so-called ‘symplectic basis’ of squared solutions, which maps the variation of the potential of the AKNS system to the variation of the action-variables. Unfortunately, for many multi-component systems such bases are not yet known.
2. From the Lax Representation to the RHP
2.1. N-Waves According to Manakov and Zakharov
- C.1
- By we mean that possesses smooth derivatives of all orders and falls off to zero for faster than any power of x:
- C.2
- is such that the corresponding operator L has only a finite number of simple discrete eigenvalues.
2.2. MNLS Equations According to Manakov, Fordy and Kulish
2.3. Generic Lax Representation
3. Jost Solutions and FAS of
3.1. Jost Solutions and Scattering Matrix
3.2. Construction of the FAS
3.3. The Time-Dependence of
4. RHP and Integrable NLEE
4.1. Uniqueness of the Regular Solution of RHP
4.2. Zakharov–Shabat Theorem
5. Mikhailov’s Reduction Groups and the Contours of RHP
5.1. General Theory
5.2. Involutive Reductions
5.3. Reduction Groups
5.4. Reduction Groups
6. Parametrizing the RHP with Canonical Normalization
6.1. Generic Parametrization of the RHP with Canonical Normalization
6.2. The Family of N-Wave Equations with Cubic Non-linearities
6.3. The Main Idea of the Dressing Method
6.4. Dressing of N-Wave Equations: Two Involutions
6.5. Dressing of N-Wave Equations: Three Involutions
7. MNLS Family and Symmetric Spaces
7.1. Lax Pairs on Symmetric Spaces. Generic Case
7.2. NLEE on Symmetric Spaces: A.III
7.3. MNLS Equations Related to D.III and C.I Symmetric Spaces
7.4. MNLS Related to BD.I-Type Symmetric Spaces
8. Soliton Solutions of the MNLS Equations
8.1. Dressing for NLEE on Symmetric Spaces: A.III Case
8.2. Dressing for NLEE on Symmetric Spaces: C.I and D.III Cases
8.3. Dressing for NLEE on Symmetric Spaces: BD.I Cases
9. Multi-Soliton Solutions
10. The Resolvent and Spectral Properties of Lax Operators
- C.1
- possesses smooth derivatives of all orders with respect to x and falls off to zero for faster than any power of x:
- C.2
- is such that the corresponding RHP has finite index. For the class of RHP that we have been dealing with this means that the solution of the RHP must have finite number of simple zeroes and pole singularities.
- 1.
- if is a zero or pole of , then there must exist , which is also a zero or pole of ;
- 2.
- if is a zero or pole of , then there must exist , which is also zero or pole of .
- 3.
- if is a zero or pole of , then there must exist which is a zero or pole of .
- (i) the continuous spectrum of consists of all points for which is an unbounded integral operator;
- (ii) the discrete spectrum of consists of all points for which develops pole singularities.
- 1.
- is obvious from the fact that are the FAS of (86). It is also easy to see that if satisfies conditions (C.1) and (C.2) then and will also satisfy condition C1. In addition obviously will satisfy condition C2 and will have poles and zeroes at the points , see Remark 10.
- 2.
- Assume that and consider the asymptotic behavior of for . From Equation (86) we find that
- 3.
- For the arguments of (2) can not be applied because the exponentials in the right-hand side of (276) only oscillate. Thus, we conclude that for is only a bounded function of x and thus the corresponding operator is an unbounded integral operator.
- 4.
- The proof of Equation (275) follows from the fact that and
11. Natural Generalizations
11.1. Cubic Pencils as Lax Operators
11.2. Cubic Pencils with Dihedral Reduction Group
12. Discussion and Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Root Systems of Simple Lie Algebras
- The Cartan subalgebras of all simple Lie algebras can be represented by diagonal matrices;
- There is a one-to-one mapping between the elements of and the vectors in r-dimensional Euclidean space ;
- The Weil generators are defined as eigenvectors of all the elements of , i.e.,
- Root systems
- Dynkin diagrams
- Cartan-Weyl basis
- Automorphisms of finite order
- If then is also a root, .
- Each root system is split into positive and negative roots:
- In each root system, one can introduce a basis, known as system of simple roots. By definition , are simple roots if: (i) they are linearly independent and form a basis in ; (ii) they are positive roots such that ;
- Each positive root can be expressed as sum of simple roots where all are integers;
- There is a maximal (resp. minimal) root (resp ) such that (resp. ) is not a root;
- Symmetry properties of and Weyl group. Introduce the Weyl reflections by:The Weyl reflections form a finite group, which preserves , i.e., .
Appendix A.1. The Root System of Algebras
Appendix A.2. The Root System of Algebras
Appendix A.3. The Root System of Algebras
Appendix A.4. The Root System of Algebras
Appendix B. Gauss Decompositions
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Gerdjikov, V.S.; Stefanov, A.A. Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions. Symmetry 2023, 15, 1933. https://doi.org/10.3390/sym15101933
Gerdjikov VS, Stefanov AA. Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions. Symmetry. 2023; 15(10):1933. https://doi.org/10.3390/sym15101933
Chicago/Turabian StyleGerdjikov, Vladimir Stefanov, and Aleksander Aleksiev Stefanov. 2023. "Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions" Symmetry 15, no. 10: 1933. https://doi.org/10.3390/sym15101933
APA StyleGerdjikov, V. S., & Stefanov, A. A. (2023). Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions. Symmetry, 15(10), 1933. https://doi.org/10.3390/sym15101933