Multicomponent Fokas–Lenells equations on Hermitian symmetric spaces

Gerdjikov, Vladimir S.; Ivanov, Rossen I.

doi:10.1088/1361-6544/abcc4b

Cited by 8 publications

(4 citation statements)

References 57 publications

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“…Typically, by h we denote the Coxeter number of the corresponding simple Lie algebra. Here, we include the families of KdV and MKdV eqs., as well as 2-dimensional Toda field theories [96][97][98][99][100][101][102][103][104][105] as well as some Camassa-Holm type equations [106][107][108][109][110][111][112].…”

Section: Discussionmentioning

confidence: 99%

Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions

Gerdjikov,

Stefanov

2023

Symmetry

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The standard approach to integrable nonlinear evolution equations (NLEE) usually uses the following steps: (1) Lax representation [L,M]=0; (2) construction of fundamental analytic solutions (FAS); (3) reducing the inverse scattering problem (ISP) to a Riemann-Hilbert problem (RHP) ξ+(x,t,λ)=ξ−(x,t,λ)G(x,tλ) on a contour Γ with sewing function G(x,t,λ); (4) soliton solutions and possible applications. Step 1 involves several assumptions: the choice of the Lie algebra g underlying L, as well as its dependence on the spectral parameter, typically linear or quadratic in λ. In the present paper, we propose another approach that substantially extends the classes of integrable NLEE. Its first advantage is that one can effectively use any polynomial dependence in both L and M. We use the following steps: (A) Start with canonically normalized RHP with predefined contour Γ; (B) Specify the x and t dependence of the sewing function defined on Γ; (C) Introduce convenient parametrization for the solutions ξ±(x,t,λ) of the RHP and formulate the Lax pair and the nonlinear evolution equations (NLEE); (D) use Zakharov–Shabat dressing method to derive their soliton solutions. This requires correctly taking into account the symmetries of the RHP. (E) Define the resolvent of the Lax operator and use it to analyze its spectral properties.

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Section: Discussionmentioning

confidence: 99%

Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions

Gerdjikov,

Stefanov

2023

Symmetry

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“…Here we include the families of KdV and MKdV eqs., as well as 2-dimensional Toda field theories [4,[49][50][51]55,79,80,97,102,103] as well as some Camassa-Holm type eqs. [11,44,66,68,69,75,76]. Kulish-Sklyanin system, or in other words the 3-component NLS related to the BD.I symmetric space has important applications to the spin-1 Bose-Einstein condensate, see [23,24,34,38,38,41,60,61,72,73,84,85,105,[113][114][115][116][117]124].…”

Section: Discussionmentioning

confidence: 99%

Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions

Gerdjikov,

Stefanov

2023

Preprint

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The standard approach to integrable nonlinear evolution equations (NLEE) usually uses the following steps: 1) Lax representation $[L,M]=0$; 2)~construction of fundamental analytic solutions (FAS); 3) reducing the inverse scattering problem (ISP) to a Riemann-Hilbert problem (RHP) $\xi^+(x,t,\lambda) =\xi^-(x,t,\lambda) G(x,t\lambda)$ on a contour $\Gamma$ with sewing function $G(x,t,\lambda)$; 4) soliton solutions and possible applications.Step 1) involves several assumptions: the choice of the Lie algebra $\mathfrak{g}$ underlying $L$, as well as its dependence on the spectral parameter, typically linear or quadratic in $\lambda$. In the present paper we propose another approach which substantially extends the classes of integrable NLEE.Its first advantage is that one can effectively use any polynomial dependence in both $L$ and $M$. We use the following steps: A) Start with canonically normalized RHP with predefined contour $\Gamma$; B) Specify the $x$ and $t$ dependence of the sewing function defined on $\Gamma$; C) Introduce convenient parametrization for the solutions $ \xi^\pm(x,t,\lambda)$ of the RHP and formulate the Lax pair and the nonlinear evolution equations (NLEE); D) use Zakharov-Shabat dressing method to derive their soliton solutions. This needs correctly taking into account the symmetries of the RHP. E) Define the resolvent of the Lax operator and use it analyze its spectral properties.

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“…Here matrix Λ depends on parameters ω i and describes a shift of the orbit obtained in the framework of the Adler-Kostant-Symes theorem [18]. All these results were reproduced in the various textbooks [17,19,20,21], where readers can find all the necessary definitions and details, see also [9] and references within. When all ω α = 0, Hamiltonian H (1.3) commutes with a family of the noncommutative linear integrals of motion associated with the various combinations of rotations.…”

Section: Introductionmentioning

confidence: 99%

“…15) and N ij(2.16) are associated with two realizations of so * (3) by using independent triple rotations in R9 . The Lie-Poisson brackets are {M 12 , M 13 } = M 23 , {M 13 , M 23 } = M 12 , {M 23 , M 12 } = M 13 .…”

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confidence: 99%

Second order Killing tensors related to symmetric spaces

Porubov¹,

V.²

2023

Preprint

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We discuss the pairs of quadratic integrals of motion belonging to the n-dimensional space of independent integrals of motion in involution, that provide integrability of the corresponding Hamiltonian equations of motion by quadratures. In contrast to the Eisenhart theory, additional integrals of motion are polynomials of the fourth, sixth and other orders in momenta. The main focus is on the second-order Killing tensors corresponding to quadratic integrals of motion and relating to the special combinations of rotations and translations in Euclidean space.

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Multicomponent Fokas–Lenells equations on Hermitian symmetric spaces

Cited by 8 publications

References 57 publications

Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions

Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions

Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions

Second order Killing tensors related to symmetric spaces

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