Gauge-covariant formulation and complete integrable structure of the nonlocal PT-symmetric multicomponent nonlinear Schrödinger hierarchies on symmetric spaces

The inverse scattering transform for nonlocal PT-symmetric reductions of the multicomponent nonlinear Schrödinger (NLS) equations on symmetric spaces — the Manakov-type models on the Hermitian symmetric space A.III and the Kulish–Sklyanin models on BD.I, each with two nonlocal involutions — was constructed by Grahovski, Mustafa and Susanto up to the direct scattering problem, fundamental analytic solutions and low soliton solutions. Their conclusion poses the extension program: build the gauge-covariant formulation of these hierarchies, namely the Wronskian relations, the squared eigenfunctions and their completeness relations, the description of the associated class of nonlinear evolution equations, the recursion operator and its spectral decomposition, the infinite set of integrals of motion, the hierarchy of Hamiltonian structures and the r-matrix formulation, together with the companion extensions to gauge-equivalent multicomponent ferromagnets, further nonlocal reductions, generalized Darboux transformations with multi-soliton and rational solutions, nonzero boundary conditions, and the quadratic/rational-bundle (derivative NLS and Fokas–Lenells type) hierarchies on symmetric spaces. Partial templates exist (scalar nonlocal NLS, a Kulish–Sklyanin-type sketch, ferromagnet nonlocal reductions, Fokas–Lenells formulations, scalar nonzero-boundary results), but the full program for the nonlocal multicomponent symmetric-space hierarchies — above all their Hamiltonian and r-matrix structure — remains open.

Background

The nonlinear Schrödinger (NLS) equation $iq_t+q_{xx}+2|q|^2q=0$ and its multicomponent generalizations describe vector solitons in birefringent fibres, spinor Bose–Einstein condensates and multichannel wave propagation. Two multicomponent families are organized by symmetric spaces $G/K$ (coset spaces whose Lie algebra splits into even and odd parts, the odd part constraining the potential of the Lax pair): the Manakov-type vector NLS associated with the Hermitian symmetric space A.III $SU(p+q)/S(U(p)\times U(q))$, and the Kulish–Sklyanin matrix NLS associated with BD.I $SO(2r+1)/S(O(2r-1)\times O(2))$, which in the local case models spin-1 condensates. All are solvable by the inverse scattering transform (IST): the nonlinear equation is the compatibility condition of a Lax pair of linear problems, one of which — the Zakharov–Shabat scattering problem — is solved by Jost solutions whose asymptotic coefficients form a scattering matrix; fundamental analytic solutions (FAS) are combinations of Jost solutions analytic in sectors of the spectral plane and convert the IST into a Riemann–Hilbert problem.

In 2013 Ablowitz and Musslimani discovered that the nonlocal reduction $q(x,t)\mapsto q^*(-x,t)$ (a parity–time, or PT, symmetric involution coupling the field at $x$ to the field at $-x$) yields an integrable equation with the self-induced potential $V(x,t)=2q(x,t)q^*(-x,t)$, launching the field of nonlocal integrable systems motivated by PT-symmetric optics with balanced gain and loss. Grahovski, Mustafa and Susanto extended this to symmetric spaces: for the Manakov-type (A.III) and Kulish–Sklyanin (BD.I) multicomponent NLS they introduced two nonlocal involutions each — one, (B), acting essentially componentwise through $q(x,t)\mapsto q^*(-x,t)$ up to constant matrices, the other, (C), mixing components via a Weyl reflection — and constructed the direct scattering problem (Jost solutions, scattering matrix, minimal scattering data, FAS), obtaining 1- and 2-soliton solutions by a modified Zakharov–Shabat dressing method with both regular and singular regimes.

For the local multicomponent NLS hierarchies the deeper structure is known under the name of the gauge-covariant formulation, or generalized Fourier transform. Its ingredients: Wronskian relations, bilinear identities between Jost solutions that map the potential and its variations to the scattering data; the squared solutions (squared eigenfunctions), bilinear combinations of right and adjoint Jost solutions that behave as generalized exponentials and satisfy completeness relations; the recursion (generating) operator, whose eigenfunctions are the squared solutions, which generates the whole class of nonlinear evolution equations (NLEEs) sharing the scattering problem and whose spectral decomposition yields the infinite set of integrals of motion; the Hamiltonian formulation, recasting the IST as a canonical transformation to action-angle variables organized into a hierarchy of compatible Hamiltonian structures; and the classical r-matrix, encoding the fundamental Poisson brackets of the monodromy (transfer) matrix from which the Hamiltonian structures follow. For the local symmetric-space multicomponent NLS this program was completed by Gerdjikov (Inverse Problems 1986) and Gerdjikov–Grahovski–Mikhailov–Valchev (SIGMA 2010, 2011). Neighbouring structures include gauge equivalence to Heisenberg-ferromagnet-type spin systems (the multicomponent NLS and the corresponding ferromagnet equations are related by a gauge transformation of the Lax pair); Darboux transformations, dressing-like operations generating exact solutions whose iteration and degeneration produce multi-soliton and rational solutions; formulations with nonzero boundary conditions, where a nontrivial background at spatial infinity moves the continuous spectrum onto Riemann surfaces; and generalized spectral problems — quadratic bundles (the derivative NLS family of Kaup–Newell, Chen–Lee–Liu and Gerdjikov–Ivanov) and rational bundles (Fokas–Lenells). The 2018 paper's conclusion poses the transfer of this entire apparatus to the nonlocal PT-symmetric multicomponent hierarchies, listing six linked directions: the gauge-covariant formulation; the gauge-equivalent multicomponent ferromagnets; further reduction types (nonlocal analogues of the $\mathbb{Z}_4$ and $\mathbb{Z}_6$ reductions); generalized Darboux transformations with multi-soliton and rational solutions; nonzero boundary conditions; and the quadratic/rational-bundle hierarchies on symmetric spaces.

Problem Statement

Extend the inverse-scattering theory constructed by Grahovski, Mustafa and Susanto for the nonlocal PT-symmetric reductions of the multicomponent nonlinear Schrödinger (NLS) equations on symmetric spaces — the Manakov-type vector NLS related to the Hermitian symmetric space A.III $SU(p+q)/S(U(p)\times U(q))$ and the Kulish–Sklyanin NLS related to BD.I $SO(2r+1)/S(O(2r-1)\times O(2))$, each with the two nonlocal involutions denoted (B) and (C) in that paper — to the full gauge-covariant formulation and complete integrable structure of the corresponding hierarchies. Concretely, for these nonlocal hierarchies construct: (i) the Wronskian relations linking the potential and its variations to the scattering data; (ii) the squared eigenfunctions and their completeness relations; (iii) the description of the class of nonlinear evolution equations associated with the given scattering problem; (iv) the generating (recursion) operator and its spectral decomposition; (v) the infinite set of integrals of motion; (vi) the hierarchy of Hamiltonian structures; and (vii) the r-matrix formulation. The source's conclusion further identifies, as companion extensions of the same theory, the following directions: the gauge-equivalent multicomponent Heisenberg-ferromagnet-type systems on the same symmetric spaces under the nonlocal reductions; the nonlocal analogues of the $\mathbb{Z}_4$ and $\mathbb{Z}_6$ reductions (the order-4 and order-6 reductions of the multicomponent NLS on symmetric spaces generated by the reduction groups $\mathbb{Z}_4$ and $\mathbb{Z}_6$ in Grahovski, Gerdjikov, Kostov and Atanasov, arXiv:nlin/0603066 (2006)); generalized Darboux transformations for the nonlocal reductions on Hermitian symmetric spaces, with multi-soliton and rational solutions obtained from them; the nonzero-boundary-condition (nontrivial background) regime; and the nonlocal reductions of the quadratic- and rational-bundle hierarchies on symmetric spaces (derivative NLS of Kaup–Newell and Gerdjikov–Ivanov type, and Fokas–Lenells type). The common objective uniting the core items (i)–(vii) is to bring the nonlocal PT-symmetric multicomponent NLS hierarchies on symmetric spaces to the same level of structural completeness as their local counterparts, for which all items (i)–(vii) are established. A resolution must establish these core items (i)–(vii) for both families — the Manakov-type A.III models and the Kulish–Sklyanin BD.I models — and for each of the two nonlocal involutions (B) and (C) in each family; a resolution is judged on the core items alone, and the companion extensions listed above are further directions of the same programme that a resolution is not required to cover.

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