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In this paper, we introduce a novel class of magnetic curves and call it pseudo‐solitonic magnetic curves by considering the connection between two significant types of integrable equations, that is, the nonlinear heat system/the nonlinear Schrödinger equation and moving space curves in the Minkowski 3‐space. Later, we construct a very special class of soliton surfaces and call it pseudo‐solitonic magnetic surfaces by considering the effects of the pseudo‐solitonic magnetic flows in the Minkowski 3‐space. These curves and surfaces are very useful since they not only contain geometric and physical features in their inner forms but also their constructions rely on a simple geometric procedure compared to other models. [ABSTRACT FROM AUTHOR]

In this article, some exact, atypical, topological soliton solutions of the integrable Kundu–Mukherjee–Naskar (KMN) equation have been derived. Those soliton solutions can have atypical shapes like U shaped soliton, W shaped soliton, three hump soliton, periodic soliton etc. on the x - y plane. This property is quite unusual for integrable systems, because solitons usually travel along a line with constant amplitude and velocity. The property of curvature of the solitons on x - y plane occur due to the presence of an arbitrary nonlinear function in the argument of the solution. Such arbitrary function in the soliton solution can exist because of the current like nonlinearity and Galilean covariance present in the KMN equation. These solutions are quite uncommon in the field of integrable systems; that may be used for better mathematical modelling of physical systems enriching the field of nonlinear optics. [ABSTRACT FROM AUTHOR]

We derive a new system of integrable derivative non-linear Schrödinger equations with an L operator, quadratic in the spectral parameter with coefficients belonging to the Kac–Moody algebra A 2 (1) . The construction of the fundamental analytic solutions of L is outlined and they are used to introduce the scattering data, thus formulating the scattering problem for the Lax pair L , M . [ABSTRACT FROM AUTHOR]

Nowadays, breather solutions are generally accepted models of rogue waves. However, breathers exist on a finite background and therefore are not localized, while wavefields in nature can generally be considered as localized due to the limited sizes of physical domain. Hence, the theory of rogue waves needs to be supplemented with localized solutions, which evolve locally as breathers. In this paper, we present a universal method for constructing such solutions from exact multisoliton solutions, which consists in replacing the plane wave in the dressing construction of the breathers with a specific exact N‐soliton solution converging asymptotically to the plane wave at large number of solitons N. On the example of the Peregrine, Akhmediev, Kuznetsov–Ma, and Tajiri–Watanabe breathers, we show that constructed with our method multisoliton solutions, being localized in space with characteristic width proportional to N, are practically indistinguishable from the breathers in a wide region of space and time at large N. Our method makes it possible to build solitonic models with the same dynamical properties for the higher order rational and super‐regular breathers, and can be applied to general multibreather solutions, breathers on a nontrivial background (e.g., cnoidal waves), and other integrable systems. The constructed multisoliton solutions can also be generalized to capture the spontaneous emergence of rogue waves through the spontaneous synchronization of soliton norming constants, though finding these synchronization conditions represents a challenging problem for future studies. [ABSTRACT FROM AUTHOR]

This paper primarily focuses on the development of a novel category of magnetic curves. While we explore conventional techniques, viewing these curves through a geometric lens and considering the electromagnetic principles of physics, our emphasis shifts towards investigating integrability conditions and solitonic behaviors. Consequently, we introduce diverse varieties of hybrid magnetic curves linked to the modified non-linear Schrödinger (mNLS) equation within the context of three-dimensional Minkowski space. We also scrutinize alterations in energy distribution and the pseudo-parallel transport of magnetic and electric vector lines associated with flows, enabling the construction of a new class of hybrid magnetic soliton surfaces. Finally, we employ the conformable fractional method to derive specific solution families for these integrable systems, presenting them visually. [ABSTRACT FROM AUTHOR]

We study theoretically the nonlinear interactions of vector breathers propagating on an unstable wavefield background. As a model, we use the two‐component extension of the one‐dimensional focusing nonlinear Schrödinger equation—the Manakov system. With the dressing method, we generate the multibreather solutions to the Manakov model. As shown previously in [D. Kraus, G. Biondini, and G. Kovačič, Nonlinearity 28(9), 3101, (2015)], the class of vector breathers is presented by three fundamental types I, II, and III. Their interactions produce a broad family of the two‐component (polarized) nonlinear wave patterns. First, we demonstrate that the type I and the types II and III correspond to two different branches of the dispersion law of the Manakov system in the presence of the unstable background. Then, we investigate the key interaction scenarios, including collisions of standing and moving breathers and resonance breather transformations. Analysis of the two‐breather solution allows us to derive general formulas describing phase and space shifts acquired by breathers in mutual collisions. The found expressions enable us to describe the asymptotic states of the breather interactions and interpret the resonance fusion and decay of breathers as a limiting case of infinite space shift in the case of merging breather eigenvalues. Finally, we demonstrate that only type I breathers participate in the development of modulation instability from small‐amplitude perturbations withing the superregular scenario, while the breathers of types II and III, belonging to the stable branch of the dispersion law, are not involved in this process. [ABSTRACT FROM AUTHOR]

The inverse scattering transform allows explicit construction of solutions to many physically significant nonlinear wave equations. Notably, this method can be extended to fractional nonlinear evolution equations characterized by anomalous dispersion using completeness of suitable eigenfunctions of the associated linear scattering problem. In anomalous diffusion, the mean squared displacement is proportional to t α , α > 0, while in anomalous dispersion, the speed of localized waves is proportional to A α , where A is the amplitude of the wave. Fractional extensions of the modified Kortewegâ€"deVries (mKdV), sine-Gordon (sineG) and sinh-Gordon (sinhG) and associated hierarchies are obtained. Using symmetries present in the linear scattering problem, these equations can be connected with a scalar family of nonlinear evolution equations of which fractional mKdV (fmKdV), fractional sineG (fsineG), and fractional sinhG (fsinhG) are special cases. Completeness of solutions to the scalar problem is obtained and, from this, the nonlinear evolution equation is characterized in terms of a spectral expansion. In particular, fmKdV, fsineG, and fsinhG are explicitly written. One-soliton solutions are derived for fmKdV and fsineG using the inverse scattering transform and these solitons are shown to exhibit anomalous dispersion. [ABSTRACT FROM AUTHOR]

We establish the explicit correspondence between the theory of soliton gases in classical integrable dispersive hydrodynamics, and generalized hydrodynamics (GHD), the hydrodynamic theory for many-body quantum and classical integrable systems. This is done by constructing the GHD description of the soliton gas for the Kortewegâ€"de Vries equation. We further predict the exact form of the free energy density and flux, and of the static correlation matrices of conserved charges and currents, for the soliton gas. For this purpose, we identify the solitons’ statistics with that of classical particles, and confirm the resulting GHD static correlation matrices by numerical simulations of the soliton gas. Finally, we express conjectured dynamical correlation functions for the soliton gas by simply borrowing the GHD results. In principle, other conjectures are also immediately available, such as diffusion and large-deviation functions for fluctuations of soliton transport. [ABSTRACT FROM AUTHOR]

We propose a systematic method for constructing integrable delay-difference and delay-differential analogues of known soliton equations such as the Lotkaâ€"Volterra, Toda lattice (TL), and sine-Gordon equations and their multi-soliton solutions. It is carried out by applying a reduction and delay-differential limit to the discrete KP or discrete two-dimensional TL equations. Each of the delay-difference and delay-differential equations has the N -soliton solution, which depends on the delay parameter and converges to an N -soliton solution of a known soliton equation as the delay parameter approaches 0. [ABSTRACT FROM AUTHOR]

Solitary waves are interesting phenomena arising in various fields of physics, chemistry, and biology. Nonlinear continuous and discrete models supporting wave solutions of solitary behaviour have received increasing attention in recent years. Some examples of such integrable systems include Korteweg de-Vries (KdV) equation, the nonlinear Schr¨odinger (NLS) equation, and the Manakov system (MS). In this paper, we propose a discrete nonlocal version of the nonlinear Manakov system which admits spatial and temporal PT-symmetry. PT-symmetry property gives relevance to various fields in physics and has received a lot of attention in the studies of integrable nonlinear equations. In this work, the time evolution of solitary and periodic wave solutions in the proposed system has been numerically investigated. Suitable initial conditions have been considered to construct bright and dark solitons. The variational iteration method (VIM) was used to simulate the solution of the system. The error measurement of the simulation demonstrates the efficiency of the numerical method in constructing the different types of wave solutions. [ABSTRACT FROM AUTHOR]

Simple visual approaches are used to examine the ultradiscrete KdV equation with arbitrary real values for the dependent variable. The number and sizes of the solitons in the system can be obtained straightforwardly using these visualizations. By identifying the initial soliton sizes and positions, the initial profile can be decomposed into its constituent parts, which can then be evolved separately, and then the evolved profile can be reconstructed. [ABSTRACT FROM AUTHOR]

We study the dynamics of explicit solutions of the -dimensional (2D) sine-Gordon equation. The Darboux transformation is applied to the associated linear eigenvalue problem to construct nontrivial solutions of the D sine-Gordon equation in terms of a ratio of determinants. We obtain a generalized expression for an -fold transformed dynamical variable, which enables us to calculate explicit expressions of nontrivial solutions. To explore the dynamics of kink soliton solutions, explicit expressions for one- and two-soliton solutions are derived for particular column solutions. Different profiles of kink–kink and kink–anti-kink interactions are illustrated for different parameters and arbitrary functions. We also present a first-order bound state solution. [ABSTRACT FROM AUTHOR]

In this brief report we study numerically the spontaneous emergence of rogue waves in 1) modulationally unstable plane wave at its long-time statistically stationary state and 2) bound-state multi-soliton solutions representing the solitonic model of this state. Focusing our analysis on the cohort of the largest rogue waves, we find their practically identical dynamical and statistical properties for both systems, that strongly suggests that the main mechanism of rogue wave formation for the modulational instability case is multi-soliton interaction. Additionally, we demonstrate that most of the largest rogue waves are very well approximated–simultaneously in space and in time–by the amplitude-scaled rational breather solution of the second order.

The basic κ -color box-ball (BBS) system is an integrable cellular automaton on one dimensional lattice whose local states take { 0 , 1 , ... , κ } with 0 regarded as an empty box. The time evolution is defined by a combinatorial rule of quantum group theoretical origin, and the complete set of conserved quantities is given by a κ -tuple of Young diagrams. In the randomized BBS, a probability distribution on { 0 , 1 , ... , κ } to independently fill the consecutive n sites in the initial state induces a highly nontrivial probability measure on the κ -tuple of those invariant Young diagrams. In a recent work Kuniba et al. (Nucl Phys B 937:240–271, 2018), their large n 'equilibrium shape' has been determined in terms of Schur polynomials by a Markov chain method and also by a very different approach of thermodynamic Bethe ansatz (TBA). In this paper, we establish a large deviations principle for the row lengths of the invariant Young diagrams. As a corollary, they are shown to converge almost surely to the equilibrium shape at an exponential rate. We also refine the TBA analysis and obtain the exact scaling form of the vacancy, the row length and the column multiplicity, which exhibit nontrivial factorization in a one-parameter specialization. [ABSTRACT FROM AUTHOR]

We formulate an isoperimetric deformation of curves on the Minkowski plane, which is governed by the defocusing modified Korteweg-de Vries (mKdV) equation. Two classes of exact solutions to the defocusing mKdV equation are also presented in terms of the τ functions. By using one of these classes, we construct an explicit formula for the corresponding motion of curves on the Minkowski plane even though those solutions have singular points. Another class gives regular solutions to the defocusing mKdV equation. Some pictures illustrating the typical dynamics of the curves are presented. [ABSTRACT FROM AUTHOR]

Abstract We present the results of study of a nonlinear evolutionary PDE (more precisely, a one-parameter family of PDEs) associated with the chain of pre-stressed granules. The PDE in question supports solitary waves of compression and rarefaction (bright and dark compactons) and can be written in Hamiltonian form. We investigate inter alia integrability properties of this PDE and its generalized symmetries and conservation laws. For the compacton solutions we perform a stability test followed by the numerical study. In particular, we simulate the temporal evolution of a single compacton, and the interactions of compacton pairs. The results of numerical simulations performed for our model are compared with the numerical evolution of corresponding Cauchy data for the discrete model of chain of pre-stressed elastic granules. [ABSTRACT FROM AUTHOR]

We review some of the fundamental notions associated with the theory of solitons. More precisely, we focus on the issue of conservation laws via the existence of the Lax pair and also on methods that provide solutions to partial or ordinary differential equations that are associated to discrete or continuous integrable systems. The Riccati equation associated to a given continuous integrable system is also solved and hence suitable conserved quantities are derived. The notion of the Darboux–Bäcklund transformation is introduced and employed in order to obtain soliton solutions for specific examples of integrable equations. The Zakharov–Shabat dressing scheme and the Gelfand–Levitan–Marchenko equation are also introduced. Via this method, generic solutions are produced and integrable hierarchies are explicitly derived. Various discrete and continuous integrable models are employed as examples such as the Toda chain, the discrete nonlinear Schrödinger model, the Korteweg–de Vries and nonlinear Schrödinger equations as well as the sine-Gordon and Liouville models. [ABSTRACT FROM AUTHOR]

A discrete analog of the dressing method is presented and used to derive integrable nonlinear evolution equations, including two infinite families of novel continuous and discrete coupled integrable systems of equations of nonlinear Schrödinger type. First, a demonstration is given of how discrete nonlinear integrable equations can be derived starting from their linear counterparts. Then, starting from two uncoupled, discrete one‐directional linear wave equations, an appropriate matrix Riemann‐Hilbert problem is constructed, and a discrete matrix nonlinear Schrödinger system of equations is derived, together with its Lax pair. The corresponding compatible vector reductions admitted by these systems are also discussed, as well as their continuum limits. Finally, by increasing the size of the problem, three‐component discrete and continuous integrable discrete systems are derived, as well as their generalizations to systems with an arbitrary number of components. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for the focusing nonlinear Schrödinger (fNLS) equation. Using the ∂ ‾ generalization of the nonlinear steepest descent method we compute the long-time asymptotic expansion of the solution ψ ( x , t ) in any fixed space-time cone C ( x 1 , x 2 , v 1 , v 2 ) = { ( x , t ) ∈ R 2 : x = x 0 + v t with x 0 ∈ [ x 1 , x 2 ] , v ∈ [ v 1 , v 2 ] } up to an (optimal) residual error of order O ( t − 3 / 4 ) . In each cone C the leading order term in this expansion is a multi-soliton whose parameters are modulated by soliton–soliton and soliton–radiation interactions as one moves through the cone. Our results require that the initial data possess one L 2 ( R ) moment and (weak) derivative and that it not generate any spectral singularities. [ABSTRACT FROM AUTHOR]

In this paper, we study integrable multilayer spin systems, namely, the multilayer M-LIII equation. We investigate their relation with the geometric flows of interacting curves and surfaces in some space . Then we present their Lakshmanan equivalent counterparts. We show that these equivalent counterparts are, in fact, the vector nonlinear Schrödinger equation (NLSE). It is well known that the vector NLSE is equivalent to the -spin system. Also, we have presented the transformations which give the relation between solutions of the -spin system and the multilayer M-LIII equation. It is interesting to note that the integrable multilayer M-LIII equation contains constant magnetic field . It seems that this constant magnetic vector plays an important role in the theory of 'integrable multilayer spin system' and in nonlinear dynamics of magnetic systems. Finally, we present some classes of integrable models of interacting vortices. [ABSTRACT FROM AUTHOR]

Integrable spin systems possess interesting geometrical and gauge invariance properties and have important applications in applied magnetism and nanophysics. They are also intimately connected to the nonlinear Schrödinger family of equations. In this paper, we identify three different integrable spin systems in (2 + 1) dimensions by introducing the interaction of the spin field with more than one scalar potential, or vector potential, or both. We also obtain the associated Lax pairs. We discuss various interesting reductions in (2 + 1) and (1 + 1) dimensions. We also deduce the equivalent nonlinear Schrödinger family of equations, including the (2 + 1)-dimensional version of nonlinear Schrödinger-Hirota-Maxwell-Bloch equations, along with their Lax pairs. [ABSTRACT FROM AUTHOR]

We formulate and discuss integrable analogue of the sine-Gordon equation on arbitrary time scales. This unification contains the sine-Gordon equation, discrete sine-Gordon equation and the Hirota equation (doubly discrete sine-Gordon equation) as special cases. We present the Lax pair, check compatibility conditions and construct the Darboux–Bäcklund transformation. Finally, we obtain a soliton solution on arbitrary time scale. The solution is expressed by the so-called Cayley exponential function. [ABSTRACT FROM AUTHOR]

We study the symplectic geometry of the Jaynes–Cummings–Gaudin model with n = 2 m − 1 spins. We show that there are focus–focus singularities of maximal Williamson type ( 0 , 0 , m ) . We construct the linearized normal flows in the vicinity of such a point and show that soliton type solutions extend them globally on the critical torus. This allows us to compute the leading term in the Taylor expansion of the symplectic invariants and the monodromy associated to this singularity. [ABSTRACT FROM AUTHOR]

The inverse scattering transform for the defocusing Manakov system with nonzero boundary conditions at infinity is rigorously studied. Several new results are obtained: (i) The analyticity of the Jost eigenfunctions is investigated, and precise conditions on the potential that guarantee such analyticity are provided. (ii) The analyticity of the scattering coefficients is established. (iii) The behavior of the eigenfunctions and scattering coefficients at the branch points is discussed. (iv) New symmetries are derived for the analytic eigenfunctions (which differ from those in the scalar case). (v) These symmetries are used to obtain a rigorous characterization of the discrete spectrum and to rigorously derive the symmetries of the associated norming constants. (vi) The asymptotic behavior of the Jost eigenfunctions is derived systematically. (vii) A general formulation of the inverse scattering problem as a Riemann-Hilbert problem is presented. (viii) Precise results guaranteeing the existence and uniqueness of solutions of the Riemann-Hilbert problem are provided. (ix) Explicit relations among all reflection coefficients are given, and all entries of the scattering matrix are determined in the case of reflectionless solutions. (x) A compact, closed-form expression is presented for general soliton solutions, including any combination of dark-dark and dark-bright solitons. (xi) A consistent framework is formulated for obtaining solutions corresponding to double zeros of the analytic scattering coefficients, leading to double poles in the Riemann-Hilbert problem, and such solutions are constructed explicitly. [ABSTRACT FROM AUTHOR]

Using a variational formulation for partial differential equations combined with numerical simulations on ordinary differential equations (ODEs), we find two categories (pulses and snakes) of dissipative solitons, and analyze the dependence of both their shape and stability on the physical parameters of the cubic-quintic Ginzburg-Landau equation (CGLE). In contrast to the regular solitary waves investigated in numerous integrable and non-integrable systems over the last three decades, these dissipative solitons are not stationary in time. Rather, they are spatially confined pulse-type structures whose envelopes exhibit complicated temporal dynamics. Numerical simulations reveal very interesting bifurcations sequences as the parameters of the CGLE are varied. Our predictions on the variation of the soliton amplitude, width, position, speed and phase of the solutions using the variational formulation agree with simulation results. Firstly, we develop a variational formalism which explores the various classes of dissipative solitons. Given the complex dynamics, the trial functions have been generalized considerably over conventional ones to keep the shape relatively simple, and the trial function integrable while allowing arbitrary temporal variation of the amplitude, width, position, speed and phase of the pulses and snakes. In addition, the resulting Euler-Lagrange (EL) equations from the variational formulation are treated in a completely novel way. Rather than considering the stable fixed points which correspond to the well-known stationary solitons, we use dynamical systems theory to focus on more complex attractors, viz. periodic (pulses) and quasiperiodic (snakes). Periodic evolution of the trial function parameters on stable periodic attractors yields solitons whose amplitudes and widths are non-stationary or time dependent. Secondly, we investigate the dissipative solitons of the CGLE and analyze its qualitative behavior by using numerical methods for ODEs. To solve numerically the nonlinear systems of ODEs that represent EL equations obtained from variational technique, we use an explicit Runge-Kutta fourth-order method. Finally, we elucidate the Hopf bifurcation mechanism responsible for the various pulsating solitary waves, as well as its absence in Hamiltonian and integrable systems where such structures are absent due to the lack of dissipation. [ABSTRACT FROM AUTHOR]

The construction of axially symmetric soliton solutions with non-zero Hopf topological charges according to a theory known as the extended Skyrme-Faddeev model, was performed in [1]. In this paper we show how masses of glueballs are predicted within this model.

In this paper, we discuss some specific features of symmetries of integrable systems which can be used to construct the Fokas-Gel'fand formula for the immersion of 2D-soliton surfaces, associated with such systems, in Lie algebras. We establish a sufficient condition for the applicability of this formula. This condition requires the existence of two vector fields which generate a common symmetry of the initial system and its corresponding linear spectral problem. This means that these two fields have to be group-related and we determine an explicit form of this relation. It provides a criterion for the selection of symmetries suitable for use in the Fokas-Gel'fand formula. We include some examples illustrating its application. [ABSTRACT FROM AUTHOR]

The dressing method of Zakharov and Shabat [Fund. Anal. Appl. 8, 226 (1974) and ibid. 13, 166 (1980)] has been employed to the generalized coupled dispersionless integrable system in two dimensions. The dressed solutions to the Lax pair and to the nonlinear matrix equation have been obtained in terms of Hermitian projectors. The dressing method has been related with the quasi-determinant solutions obtained by using the standard matrix Darboux transformation. The iteration of dressing procedure has been shown to give N-soliton solutions of the system. At the end, the explicit soliton solution has been obtained for the system based on Lie group SU(2). [ABSTRACT FROM AUTHOR]

Optical signal propagating through a non-linear fibre medium when coupled to an Erbium-doped resonant medium is known to produce a cleaner solitonic pulse, described by the self-induced transparency (SIT) coupled to the non-linear Schrödinger (NLS) equation. Extending this idea to ultra short pulses when description through derivative NLS (DNLS) becomes relevant, we propose and investigate a new model of a coupled DNLS-SIT system for greater efficiency. It is shown that, the broadening of the optical pulse, a well-known problem in fibre transmission, can be controlled by regulating a certain initial profile related to the population inversion. This effect can be enhanced by using the constrained integrable hierarchy of the DNLS-SIT system. [ABSTRACT FROM PUBLISHER]

Soliton solutions of the KP equation have been studied since 1970. when Kadomtsev and Petviashvili [Kadomtsev BB. Petviashvili VI (1970) Sov Phys Dokl 15:539-541] proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that the Wronskian approach to the KP equation provides a method to construct soliton solutions. The regular soliton solutions that one obtains in this way come from points of the totally nonnegative part of the Grassmannian. In this paper we explain how the theory of total positivity and cluster algebras provides a framework for understanding these soliton solutions to the KP equation. We then use this framework to give an explicit construction of certain soliton contour graphs and solve the inverse problem for soliton solutions coming from the totally positive part of the Grassmannian. [ABSTRACT FROM AUTHOR]

We study Darboux transformations for the two boson (TB) hierarchy both in the scalar as well as in the matrix descriptions of the linear equation. While Darboux transformations have been extensively studied for integrable models based on SL(2, R) within the AKNS framework, this model is based on SL(2, R)⊗U(1). The connection between the scalar and the matrix descriptions in this case implies that the generic Darboux matrix for the TB hierarchy has a different structure from that in the models based on SL(2, R) studied thus far. The conventional Darboux transformation is shown to be quite restricted in this model. We construct a modified Darboux transformation which has a much richer structure and which also allows for multi-soliton solutions to be written in terms of Wronskians. Using the modified Darboux transformations, we explicitly construct one-soliton/kink solutions for the model. [ABSTRACT FROM AUTHOR]

In this paper we present Darboux transformation for the principal chiral and WZW models in two dimensions and construct multi-soliton solutions in terms of quasideterminants. We also establish the Darboux transformation on the holomorphic conserved currents of the WZW model and expressed them in terms of the quasideterminant. We discuss the model based on the Lie group SU(n) and obtain explicit soliton solutions for the SU(2) model. [ABSTRACT FROM AUTHOR]

This paper studies optical solitons with linear attenuation and bandpass filters. The Kerr and power laws of nonlinearity are considered. The governing equations are integrated by the aid of He's semi-inverse variational principle. The parameter domains are identified. [ABSTRACT FROM AUTHOR]

We solve the initial-boundary value problem (IBVP) for the Ablowitz-Ladik system on the natural numbers with certain linearizable boundary conditions. We do so by employing a nonlinear method of images, namely, by extending the scattering potential to all integers in such a way that the extended potential satisfies certain symmetry relations. Using these extensions and the solution of the initial value problem (IVP), we then characterize the symmetries of the discrete spectrum of the scattering problem, and we show that discrete eigenvalues in the IBVP appear in octets, as opposed to quartets in the IVP. Furthermore, we derive explicit relations between the norming constants associated with symmetric eigenvalues, and we identify a new kind of linearizable IBVP. Finally, we characterize the soliton solutions of these IBVPs, which describe the soliton reflection at the boundary of the lattice. [ABSTRACT FROM AUTHOR]

Abstract: In this work, we study two systems of two-component integrable equations. The Cole–Hopf transformation and Hirota’s bilinear method are applied to emphasize the integrability of each system. Multiple soliton solutions and multiple singular soliton solutions are formally derived. [Copyright &y& Elsevier]

Abstract: We prove a global Birkhoff decomposition for almost split real forms of loop groups, when an underlying finite dimensional Lie group is compact. Among applications, this shows that the dressing action–by the whole subgroup of loops which extend holomorphically to the exterior disc–on the -hierarchy of the ZS-AKNS systems, on curved flats and on various other integrable systems, is global for compact cases. It also implies a global infinite dimensional Weierstrass-type representation for Lorentzian harmonic maps ( wave maps) from surfaces into compact symmetric spaces. An “Iwasawa-type” decomposition of the same type of real form, with respect to a fixed point subgroup of an involution of the second kind, is also proved, and an application given. [Copyright &y& Elsevier]