Quadratic bundle and nonlinear equations

Gerdjikov, Vladimir S.; Ivanov, M. I.; Kulish, P. P.

doi:10.1007/bf01029043

Cited by 89 publications

(94 citation statements)

References 6 publications

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“…while the first negative flow of the matrix Kaup-Newell hierarchy reads as [16,17] These are the matrix generalizations of massive Thirring-type models [18][19][20][21][22], where m denotes an arbitrary nonzero constant responsible for the mass terms. The massless limit m → 0 is not considered in this paper; hereafter, the value of m is set as 1, without loss of generality.…”

Section: Massive Thirring-like Model With Symplectic Invariance 231mentioning

confidence: 99%

“…For the reduced system to assume a concise form without the imaginary unit and fractions, the following vector reduction (l 1 = 1, l 2 = M) is considered: 21) which also implies the simple relation qr = 0. System (2.19) is then reduced to another system of coupled derivative mKdV equations:…”

Section: Reductionmentioning

confidence: 99%

“…Once we have identified systems (1.4) and (1.5) as the reductions of the matrix DNLStype systems, it is not difficult to further extend the class of Sp(m)-invariant integrable systems. First, we consider the integrable matrix generalizations [16,17] of massive Thirring-type models [18][19][20][21][22], which are the first negative flows of the matrix DNLS hierarchies. Then, via the same type of reduction, we can directly obtain a new integrable variant of the massive Thirring model that is a hyperbolic system with Sp(m)-invariance, 6) as well as an equivalent system up to the interchange of x and τ .…”

Section: Introductionmentioning

confidence: 99%

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New reductions of integrable matrix partial differential equations: Sp(m)-invariant systems

Tsuchida

2010

Journal of Mathematical Physics

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We propose a new type of reduction for integrable systems of coupled matrix PDEs; this reduction equates one matrix variable with the transposition of another multiplied by an antisymmetric constant matrix. Via this reduction, we obtain a new integrable system of coupled derivative mKdV equations and a new integrable variant of the massive Thirring model, in addition to the already known systems. We also discuss integrable semi-discretizations of the obtained systems and present new soliton solutions to both continuous and semi-discrete systems. As a by-product, a new integrable semi-discretization of the Manakov model (self-focusing vector NLS equation) is obtained.

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Section: Massive Thirring-like Model With Symplectic Invariance 231mentioning

confidence: 99%

Section: Reductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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New reductions of integrable matrix partial differential equations: Sp(m)-invariant systems

Tsuchida

2010

Journal of Mathematical Physics

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“…The study of DNSE was continued by Gerdjikov, Kulish and Ivanov [15] who developed the generalised Fourier interpretation for DNSE in terms of generating operators, squared solutions etc., found action-angle variables for it and thus proved its complete integrability. Later Gerdjikov and Ivanov [10,11] carried out an exhaustive study of the generic quadratic bundle…”

Section: Introductionmentioning

confidence: 99%

On the Quadratic Bundles Related to Hermitian Symmetric Spaces

Valchev¹

2013

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“…Gerzhikov et al also considered this case [6]. Morris and Dodd considered the two-component derivative nonlinear Schrödinger equation using a larger scattering problem (i.e., n = 3) [15].…”

Section: Examplesmentioning

confidence: 99%

On the Dissipative Evolution Equations Associated with the Zakharov-Shabat System with a Quadratic Spectral Parameter

Lee¹

1989

Transactions of the American Mathematical Society

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Abstract.In this paper we derive some results for the Zakharov-Shabat system of the form dmfdx = z2[J, m] + (zQ + P)m ; J is diagonal and skewHermitian [8,10, 12]. Following the idea of R. Beals and R. R. Coifman, we estimate the wedge products of the columns of m by L2-norm of the potential (Ö. P) [4]. By this result we have the global existence of the dissipative evolution equations associated with this spectral problem if the generic initial data (Q(x, 0), P(x, 0)) = (ßo. Po) is of Schwartz class. Direct and inverse scattering for the Z-S system WITH A QUADRATIC PARAMETERBy a potential here we mean a pair of functions (Q,P): R -► Mn(C) =set of « x « complex matrices; Q is off-diagonal and the diagonal part of P equals the diagonal part of Q(adJ)~xQ, and Q, Qx , P, Px E Lx . We

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Quadratic bundle and nonlinear equations

Cited by 89 publications

References 6 publications

New reductions of integrable matrix partial differential equations: Sp(m)-invariant systems

New reductions of integrable matrix partial differential equations: Sp(m)-invariant systems

On the Quadratic Bundles Related to Hermitian Symmetric Spaces

On the Dissipative Evolution Equations Associated with the Zakharov-Shabat System with a Quadratic Spectral Parameter

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