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Integrable quantum field models are known to exist mostly in one space-dimension. Exploiting the concept of multi-time in integrable systems and a Lax matrix of higher scaling order, we construct a novel quantum field model in quasi-two dimensions involving interacting fields. The Yang–Baxter integrability is proved for the model by finding a new kind of commutation rule for its basic fields, representing nonstandard scalar fields along the transverse direction. In spite of a close link with the quantum Landau–Lifshitz equation, the present model differs widely from it, in its content and the result obtained. Using further the algebraic Bethe ansatz we solve exactly the eigenvalue problem of this quantum field model for all its higher conserved operators. The idea presented here should instigate the construction of a novel class of integrable field and lattice models and exploration of a new type of underlying algebras.

A new, completely integrable, two dimensional evolution equation is derived for an ion acoustic wave propagating in a magnetized, collisionless plasma. The equation is a multidimensional generalization of a modulated wavepacket with weak transverse propagation, which has resemblance to nonlinear Schrodinger (NLS) equation and has a connection to Kadomtsev-Petviashvili equation through a constraint relation. Higher soliton solutions of the equation are derived through Hirota bili- nearization procedure, and an exact lump solution is calculated exhibiting 2D structure. Some mathe- matical properties demonstrating the completely integrable nature of this equation are described. Modulational instability using nonlinear frequency correction is derived, and the corresponding growth rate is calculated, which shows the directional asymmetry of the system. The discovery of this novel (2{\th}1) dimensional integrable NLS type equation for a magnetized plasma should pave a new direction of research in the field., Comment: 11 pages, 8 figures

Bending of a shape-invariant optical beam is achieved so far along parabolic or circular curves. Borrowing ideas used in nonlinear optical communication, we propose such a bending along any preassigned curve or surface, controlled by the boundary population inversion of atoms in an Erbium doped medium. The optical beam generated in a nonlinear Kerr medium and transmitted through a doped resonant medium as an accelerating soliton predicted here, should be realizable experimentally and applicable to nonlinear events in other areas like plasma or ocean wave., 10 pages

We define and illustrate the novel notion of dual integrable hierarchies, on the example of the nonlinear Schr\"odinger (NLS) hierarchy. For each integrable nonlinear evolution equation (NLEE) in the hierarchy, dual integrable structures are characterized by the fact that the zero-curvature representation of the NLEE can be realized by two Hamiltonian formulations stemming from two distinct choices of the configuration space, yielding two inequivalent Poisson structures on the corresponding phase space and two distinct Hamiltonians. This is fundamentally different from the standard bi-Hamiltonian or generally multitime structure. The first formulation chooses purely space-dependent fields as configuration space; it yields the standard Poisson structure for NLS. The other one is new: it chooses purely time-dependent fields as configuration space and yields a different Poisson structure at each level of the hierarchy. The corresponding NLEE becomes a {\it space} evolution equation. We emphasize the role of the Lagrangian formulation as a unifying framework for deriving both Poisson structures, using ideas from covariant field theory. One of our main results is to show that the two matrices of the Lax pair satisfy the same form of ultralocal Poisson algebra (up to a sign) characterized by an $r$-matrix structure, whereas traditionally only one of them is involved in the classical $r$-matrix method. We construct explicit dual hierarchies of Hamiltonians, and Lax representations of the triggered dynamics, from the monodromy matrices of either Lax matrix. An appealing procedure to build a multi-dimensional lattice of Lax pair, through successive uses of the dual Poisson structures, is briefly introduced., Comment: 28 pages. Two references added and improved explanations of the novelty of our results. Final version matching published version in Nucl. Phys. B as Frontiers Article. Also available in open access at http://www.sciencedirect.com/science/article/pii/S0550321315004083

We construct well-known integrable equations with their Lax pairs from the corresponding linear equations using our nonlinearization scheme. Using negative powers in the spectral flow to deform the time Lax operator, we find a class of perturbations that unlike the usual perturbations, which spoil the system integrability, exhibit a twofold integrable hierarchy, including those for the KdV, modified KdV, sine-Gordon, nonlinear Schrodinger (NLS), and derivative NLS equations. We discover hidden possibilities of using the perturbed hierarchy of the NLS equations to amplify and control optical solitons propagating through a fiber in a doped nonlinear resonant medium.

Topological Skyrmions as intricate spin textures were observed experimentally in helimagnets on 2d plane. Theoretical foundation of such solitonic states to appear in pure ferromagnetic model, as exact solutions expressed through any analytic function, was made long ago by Belavin and Polyakov (BP). We propose an innovative generalization of the BP solution for an anisotropic ferromagnet, based on a physically motivated geometric (in-)equality, which takes the exact Skyrmion to a new class of functions beyond analyticity. The possibility of stabilizing such metastable states in helimagnets is discussed with the construction of individual Skyrmion and Skyrmion crystal with asymmetry, likely to be detected in precision experiments., 12 pages, latex, 3 figures, published in Nucl Phys B (As Frontiers article)

Interacting matter–radiation system described by multi-atom Jaynes–Cummings models without rotating wave approximation and solvable by Bethe ansatz is proposed. The core integrable system with explicit matter–matter interaction is constructed from the elliptic Gaudin model at high spin limit, where the radiative excitation term is included perturbatively.

Integrable inhomogeneous versions of the models like NLS, Toda chain, Ablowitz-Ladik model etc., though well known at the classical level, have never been investigated for their possible quantum extensions. We propose a unifying scheme for constructing and solving such quantum integrable inhomogeneous models including a novel inhomogeneous sine-Gordon model, which avoids the difficulty related to the customary non-isospectral flow by introducing the inhomogeneities through some central elements of the underlying algebra., 12 pages, no figure, latex. Two new chapters on general inhom. trigonometric models and inhom. SG model included. Accepted in Phys.Lett.B

New integrable multi-atom matter-radiation models with and without rotating wave approximation (RWA) are constructed and exactly solved through algebraic Bethe ansatz. The models with RWA are generated through ancestor model approach in an unified way. The rational case yields the standard type of matter-radiaton models, while the trigonometric case corresponds to their q-deformations. The models without RWA are obtained from the elliptic case at the Gaudin and high spin limit., 9 pages, no figure, talk presented in int. conf. NEEDS04 (Gallipoli, Italy, July 2004)

A challenge in the theory of integrable systems is to show for every nonultralocal quantum integrable model, a possible connection to an ultralocal model. Some of such gauge connections were discovered earlier. We complete the task by identifying the same for the remaining ones along with two new models. We also unveil the underlying algebraic structure for these nonultralocal models., Comment: Latex, 10 pages, no figure, final version (with more technical details) to be published in Phys. Lett. B

Applying a unifying Lax operator approach to statistical systems a new class of integrable vertex models based on quantum algebra is proposed, which exhibits a rich variety for generic q, q roots of unity and q -> 1. Exact solutions are formulated through algebraic Bethe ansatz and a novel possibility of hybrid vertex models is introduced., Latex, 9 pages, 1 figure; version to be published in J. Phys.A vol.35 (2002) L

We propose an alternative way for managing near shore surging waves, including extreme waves like tsunamis, going beyond the conventional passive measures like the warning system. We study theoretically the possibility of influencing the nonlinear surface waves through a leakage boundary effect at the bottom. It has been found through analytic result, that the controlled leakage at the bottom might regulate the amplitude of the surface solitary waves. This could lead to a possible decay of the surging waves to reduce its hazardous effects near the shore. Our theoretical results are estimated by applying it to a real coastal bathymetry of the Bay of Bengal in India.

Exact quantum integrability is established for a class of multi-chain electron models with correlated hopping and spin models with interchain interactions, by constructing the related Lax operators and R-matrices through twisting and gauge transformations. Exact solution of the eigenvalue problem for commuting conserved quantities of such systems is achieved through Algebraic Bethe ansatz, on the examples of Hubbard and t-J models with correlated hopping. Our systematic construction identifies the integrable subclass of such known solvable models and also generates new systems including the generalized t-J models. At the same time it makes proper correction to a well known model and resolves recent controversies regarding the equivalence and solvability of some known models., Comment: Latex, 25 pages, no figure, v2: 5 refs. added, version to be published in Nucl. Phys. B

A class of quantum integrable quasi-two- and higher-dimensional quantum spin as well as strongly correlated electron systems with localized interactions are proposed. The basic idea of construction is to introduce interchain interactions in an array of spin chains or one-dimensional Hubbard models through twisting transformation. The models allow explicit quantum R-matrix, Lax operator, and exact eigenvalue solution.

We study a system of one-dimensional t-J models coupled to a ladder system. A special choice of the interaction between neighbouring rungs leads to an integrable model with supersymmetry, which is broken by the presence of rung interactions. We analyze the spectrum of low-lying excitations and ground state phase diagram at zero temperature.

A 1D Bose gas interacting through $\ensuremath{\delta},{\ensuremath{\delta}}^{\ensuremath{'}}$ and double $\ensuremath{\delta}$ function potentials is shown to be equivalent to a $\ensuremath{\delta}$ anyon gas, allowing an exact Bethe ansatz solution. In the noninteracting limit, it describes an ideal gas with generalized exclusion statistics and solves some recent controversies.

Integrable inhomogenous or impurity models are usually constructed by either shifting the spectral parameter in the Lax operator or using another representation of the spin algebra. We propose a more involved general method for such construction in which the Lax operator contains generators of a novel quadratic algebra, a generalization of the known quantum algebra. In forming the monodromy matrix, we can replace any number of the local Lax operators with different realizations of the underlying algebra, which can result in spin chains with nonspin impurities causing changed coupling across the impurity sites, as well as with impurities in the form of bosonic operators. Following the same idea, we can also generate integrable inhomogeneous versions of the generalized lattice sine-Gordon model, nonlinear Schrodinger equation, Liouville model, relativistic and nonrelativistic Toda chains, etc.

It is well known through a recent work of Bernard, Gaudin, Haldane and Pasquier (BGHP) that the usual spin Calogero-Sutherland (CS) model, containing particles with $M$ internal degrees of freedom, respects the $Y(gl_M)$ Yangian symmetry. By following and suitably modifying the approach of BGHP, in this article we construct a novel class of spin CS models which exhibit multi-parameter deformed or `nonstandard' variants of $Y(gl_M)$ Yangian symmetry. An interesting feature of such CS Hamiltonians is that they contain many-body spin dependent interactions, which can be calculated directly from the associated rational solutions of Yang-Baxter equation. Moreover, these spin dependent interactions often lead to `anyon like' representations of permutation algebra on the combined internal space of all particles. We also find out the general forms of conserved quantities as well as Lax pairs for the above mentioned class of spin CS models, and describe the method of constructing their exact wave functions., Plain Latex file, 27 pages, Latex page size corrected, text completely unchanged

A scheme suitable for describing quantum nonultralocal models including supersymmetric ones is proposed. Braided algebras are generalised to be used through Baxterisation for constructing braided quantum Yang--Baxter equations. Supersymmetric and some known nonultralocal models are derived in the framework of the present approach. As further applications of this scheme construction of new quantum integrable nonultralocal models like mKdV and anyonic supersymmetric models including deformed anyonic super algebra are outlined., Comment: 23 pages (Latex), replaced by extended published version

The basic ideas behind the construction of integrable statistical as well as quantum models are reviewed, showing the connection between the related Yang-Baxter equations and the quantum group algebra structures. Such quantized algebra is shown to play an important role in integrable theories for generating different classes of quantum integrable systems in a rather systematic way.

It is shown that the matrix KP hierarchy can yield new integrable equations in $(2+1)$-dimensions along with the corresponding Lax pair. For particular gauge choice this may result derivative and also a higher order nonlinear extension of the Devay-Stewartson equation (DSE),the higher order DSE being a higher dimensional generalisation of the Kundu- Eckhaus equation. Such gauge transformation is shown also to produce significant extensions to the constrained matrix KP system.

Integrable systems in 1+1 dimensions arise from the KP hierarchy as symmetry reductions involving square eigenfunctions. Exploiting the residual gauge freedom in these constraints new integrable systems are derived. They include generalizations of the hierarchy of the Kundu–Eckhaus equation and higher‐order extensions of the Yajima–Oikawa and Melnikov hierarchies. Constrained modified KP flows yield further integrable equations such as the hierarchies of the derivative NLS equation, the Gerdjikov–Ivanov equation, and the Chen–Lee–Liu equation.

Rogue waves are extraordinarily high and steep isolated waves, which appear suddenly in a calm sea and disappear equally fast. However, though the Rogue waves are localized surface waves, their theoretical models and experimental observations are available mostly in one dimension(1D) with the majority of them admitting only limited and fixed amplitude and modular inclination of the wave. We propose a two-dimensional(2D), exactly solvable Nonlinear Schr\"odinger equation(NLS), derivable from the basic hydrodynamic equations and endowed with integrable structures. The proposed 2D equation exhibits modulation instability and frequency correction induced by the nonlinear effect, with a directional preference, all of which can be determined through precise analytic result. The 2D NLS equation allows also an exact lump solution which can model a full grown surface Rogue wave with adjustable height and modular inclination. The lump soliton under the influence of an ocean current appear and disappear preceded by a hole state, with its dynamics controlled by the current term.These desirable properties make our exact model promising for describing ocean rogue waves., Comment: 17 pages, 5 figures, Latex

A lattice version of quantum nonlinear Schrodinger (NLS) equation is considered, which has significantly simple form and fullfils most of the criteria desirable for such lattice variants of field models. Unlike most of the known lattice NLS, the present model belongs to a class which does not exhibit the usual symmetry properties. However this lack of symmetry itself seems to be responsible for the remarkable simplification of the relevant objects in the theory, such as the Lax operator, the Hamiltonian and other commuting conserved quantities as well as their spectrum. The model allows exact quantum solution through algebraic Bethe ansatz and also a straightforward and natural generalisation to the vector case, giving thus a new exact lattice version of the vector NLS model. A deformation representing a new quantum integrable system involving Tamm-Dancoff like $q$-boson operators is constructed., Comment: 14 pages (plain TEX)

We translate effectively our earlier quantum constructions to the classical language and using Yang-Baxterisation of the Faddeev-Reshetikhin-Takhtajan algebra are able to construct Lax operators and associated $r$-matrices of classical integrable models. Thus new as well as known lattice systems of different classes are generated including new types of collective integrable models and canonical models with nonstandard $r$ matrices., Comment: 7 pages; Talk presented at NEEDS'93 (Gallipoli,Italy)

A 'colour' representation of Faddeev-Reshetikhin-Takhtajan (FRT) algebra, which, in contrast to the standard case, is related to the coloured braid group representation with generic values of q is presented. Explicit realizations of L(+or-)-matrices, occurring in this coloured variant of FRT algebra, are also obtained for the Uq(gl(2)) quantized algebra. Though these realizations are found to depend manifestly on the colour parameters, the underlying quantum group structure and associated co-product are interestingly free from such dependence. This allows us to perform the Yang-Baxterization of the coloured FRT algebra successfully, which leads to the construction of an ancestor Law operator associated with a new non-additive-type quantum R-matrix. Through different realizations of this Lax operator, a new class of quantum integrable models representing 'colour' generalizations of the well known models, such as the lattice sine-Gordon model, the Ablowitz-Ladik model lattice and the derivative nonlinear Schrodinger model etc, is generated.

An ultralocal structure discovered in a derivative nonlinear Schrodinger model enables us to investigate the complete integrability of the classical system through action‐angle variables and solve exactly the quantum model by quantum inverse scattering transformation. Exploiting the underlying quantum group related Sklyanin‐like algebra an exact lattice version of the model is constructed, and also the deep‐rooted reason for the existence of such an ultralocal model is revealed.

Fiber guided optical signal propagating in a Erbium doped nonlinear resonant medium is known to produce cleaner solitonic pulse, described by the self induced transparency (SIT) coupled to nonlinear Schroedinger equation. We discover two new possibilities hidden in its integrable structure, for amplification and control of the optical pulse. Using the variable soliton width permitted by the integrability of this model, the broadening pulse can be regulated by adjusting the initial population inversion of the opant atoms. The effect can be enhanced by another innovative application of its constrained integrable hierarchy, proposing a system of multiple SIT media. These theoretical predictions are workable analytically in details, correcting the limitation of a well known result., Comment: 10 pages, 3 figures, latex

R-matrix solutions for the nonadditive Yang-Baxter equation are obtained in arbitrary dimension using a factorization ansatz in spectral parameters. A symmetry transformation determined for the 'particle conserving' R-matrix is used to formulate a re-Yang-Baxterization scheme with multicomponent spectral parameters, extending the notion of conventional Yang-Baxterization. The possible scope for such a transformation, which has a striking resemblance to Reshetikhin's deformation of R-matrices but extended now to include spectral parameters, is discussed.

Yang-Baxterization of Faddeev-Reshetikhin-Takhtajan algebra (1989) leading to quantum YBE is considered for a class of R-matrices which satisfy Hecke relations. The authors apply such construction of the gl(N) case together with its multiparameter deformations revealing the connection between FRT relations and the extended trigonometric Sklyanin algebra. New realizations of FRT algebra through q-oscillator modes are presented, and their potential importance in the context of quantum-integrable models is discussed. A multiparametrized R-matrix with the inclusion of spectral parameter is constructed as a by-product.

A relativistic quantum field model, recently solved exactly [Y.K. Zhou, Phys. Lett. B 276 (1992) 135], is shown to have an underlying quantum group structure. The baxterisation of the FRT algebra enables us to construct through q-oscillators an exact lattice regularised version of the model. Moreover this model is found to be hybridisation of a new quantum integrable bosonic model.

A systematic approach for generation of integrable quantum lattice models exploiting the underlying Uq(2) quantum group structure as well as its multiparameter generalization is presented. We find an extension of trigonometric Sklyanin algebra and also its deformation through "symmetry breaking transformation," which after consistent bosonization (or q-bosonization) construct a series of integrable lattice models. A novel quantum solvable derivative NLS, a relativistic Toda chain and a lattice model involving q-oscillators warrant special mention. As an added advantage, along with the integrable models the corresponding quantum R-matrices are also specified.

Different q -bosonisation schemes of the quantum group including a q -deformed Holstein-Primakoff transformation as well as a realisation of the trigonometric Sklyanin algebra through a single q -oscillator mode are presented. A deforming map for q - oscillators is introduced, which in turn may generate deforming functionals for various other algebras like superalgebra, Virasoro algebra, etc.

Canonical action-angle type bosonizations of SU(2)q and q-oscillators found by us lead to a novel single q-oscillator mode realization of quantum group, representing a q-deformed Holstein-Primakoff transformation. The same canonical bosonizations help us to find a deforming map for q-oscillators, which in turn induces deforming functionals for centreless Virasoro algebra. Such bosonizations are also shown to play an important role in generating integrable field theories like sine-Gordon model.