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The author proceed to construct a dual pair for the AdS3×S3$AdS_3 \times S^3$ background by applying non‐Abelian T‐duality (here as Poisson–Lie [PL] T‐duality on a semi‐Abelian double). By using a certain parametrization of the 4‐dimensional Lie group A2⊗2A1${A}_2 \otimes 2{A}_1$ and by a suitable choice of spectator‐dependent matrices the original σ$\sigma$‐model including the AdS3×S3$AdS_3 \times S^3$ metric and a non‐trivial B$B$‐field are constructed. The dual background constructed by means of the PL T‐duality with the spectators is an asymptotically flat one with a potential black hole interpretation supported by a non‐trivial H$H$‐flux whose metric contains the true singularity with a single horizon. The question of classical integrability of the non‐Abelian T‐dual σ$\sigma$‐models under consideration is addressed, and their corresponding Lax pairs are found, depending on some spectral parameters. Finally, the conformal invariance conditions of the models are checked up to two‐loop order, and it has been concluded that the resulting model is indeed a solution of supergravity. [ABSTRACT FROM AUTHOR]

The current paper conducts comprehensive research on a 3D generalized Korteweg-de Vries (3D-gKdV) equation describing long-wave behavior in shallow water. First, the Bell polynomial method (BPM) is utilized to construct the Hirota bilinear (HB) form, bilinear Bäcklund (BB) transformation, and Lax pair of the governing equation. Based on the Painlevè test, the integrability of the governing equation is then analyzed, and as a result, multi-soliton waves to the 3D-gKdV equation are retrieved using the simplified Hirota method. In the end, through applying different ansatzes, lump, lump-soliton, and breather waves of the 3D-gKdV equation are extracted with the use of symbolic computations. Some graphical illustrations have been represented in two- and three-dimensional postures to clarify the physical characteristics of the nonlinear waves. The paper's achievements will undoubtedly enrich studies concerning nonlinear wave structures arising from KdV-type equations. [ABSTRACT FROM AUTHOR]

Through the modified deformation algorithm related to conservation laws, the (1+1)-dimensional AKNS(-1) equations are extended to a (4+1)-dimensional AKNS(-1) system. When one, two, or three of the independent variables are removed, the (4+1)-dimensional AKNS(-1) system degenerates to some novel (3+1)-dimensional, (2+1)-dimensional, and (1+1)-dimensional AKNS(-1) systems, respectively. Under a simple dependent transformation, the (1+1)-dimensional AKNS(-1) equations turn into the classical sine-Gordon equation. Then using the same deformation procedure, the (1+1)-dimensional sine-Gordon equation is generalized to a (3+1)-dimensional version. By introducing the deformation operators to the Lax pairs of the original (1+1)-dimensional models, the Lax integrability of both the (4+1)-dimensional AKNS(-1) system and the (3+1)-dimensional sine-Gordon equation is proven. Finally, the traveling wave solutions of the (4+1)-dimensional AKNS(-1) system and the (3+1)-dimensional sine-Gordon equation are implicitly given and expressed by tanh function and incomplete elliptic integral, respectively. These results may enhance our understanding of the complex physical phenomena described by the nonlinear system discussed in this paper. [ABSTRACT FROM AUTHOR]

We investigate the standard binary Darboux transformation (SBDT) for an -component sdC integrable system. For this, we construct the Darboux matrix using specific eigenvector solutions associated to the Lax pair, not only in the direct space but also in the adjoint space, resulting in the binary Darboux matrix. By the iterative application of the SBDT, we derive quasi-Grammian soliton solutions of the -component sdC integrable system. We also examine the Darboux transformation (DT) applied to matrix solutions of the sdC integrable system, expressing solutions using quasideterminants. Additionally, we thoroughly discuss the DT applied to scalar solutions of the system, expressing solutions as ratios of determinants. Furthermore, we investigate the SBDT and its application to obtaining quasi-Grammian multikink and multisoliton solutions for the -component sdC integrable system. Additionally, we demonstrate that quasi-Grammian solutions can be simplified to elementary solutions by reducing spectral parameters. [ABSTRACT FROM AUTHOR]

This study investigates the dynamics of boomerang solitons and their interactions in three-mode fiber under the influence of inhomogeneous effects governed by three-coupled nonlinear Schrödinger (3-CNLS) equations in GVD and nonlinearity coefficients. Also, 3-CNLS equation can be used to investigate the optical soliton propagation in three-core fibers, which contain some potential applications in the optical fiber communication system. Using the Darboux transformation method with computer simulation, the two-bright boomerang soliton solutions are attained analytically for the dispersion and nonlinearity functions. The propagation and interaction properties of boomerang solitons are investigated, while the variable coefficients are adopted as linear functions. Various interaction scenarios are illustrated by plotted figures to reveal the importance of obtained soliton solutions. The physical significance of boomerang solitons can be understood from the displayed figures. The obtained results will be benefit soliton path control and soliton phase management in an inhomogeneous three-core optical fiber. [ABSTRACT FROM AUTHOR]

Non-Kerr media, twin-core nonlinear optical fibers and waveguides are widely applied in optical fiber communications. In order to model the effects of quintic nonlinearity for the ultrashort optical pulse propagation in a non-Kerr medium, twin-core nonlinear optical fiber or waveguide, a coupled variable-coefficient cubic-quintic nonlinear Schrödinger system is investigated in this paper. For the two components of the electromagnetic fields, on the basis of the existing Lax pair, we take a gauge transformation to convert a Lax pair into an Ablowitz-Kaup-Newell-Segur system. Then we acquire a system, which is equal to the original system. The N-fold Darboux transformation is derived based on our Lax pair, where N is a positive integer. Applying Taylor series, we derive the N-fold generalized Darboux transformation. The second- and third-order vector breather solutions are obtained via the generalized Darboux transformation method. All our results depend on the variable coefficients of the original system. Our results might be of some use in a non-Kerr medium, twin-core nonlinear optical fiber or waveguide system. [ABSTRACT FROM AUTHOR]

In this paper, using a scaling symmetry, it is shown how to compute polynomial conservation laws, generalized symmetries, recursion operators, Lax pairs, and bilinear forms of polynomial nonlinear partial differential equations, thereby establishing their complete integrability. The Gardner equation is chosen as the key example, as it comprises both the Korteweg–de Vries and modified Korteweg–de Vries equations. The Gardner and Miura transformations, which connect these equations, are also computed using the concept of scaling homogeneity. Exact solitary wave solutions and solitons of the Gardner equation are derived using Hirota's method and other direct methods. The nature of these solutions depends on the sign of the cubic term in the Gardner equation and the underlying mKdV equation. It is shown that flat (table-top) waves of large amplitude only occur when the sign of the cubic nonlinearity is negative (defocusing case), whereas the focusing Gardner equation has standard elastically colliding solitons. This paper's aim is to provide a review of the integrability properties and solutions of the Gardner equation and to illustrate the applicability of the scaling symmetry approach. The methods and algorithms used in this paper have been implemented in Mathematica, but can be adapted for major computer algebra systems. [ABSTRACT FROM AUTHOR]

A (3 + 1) dimensional Kudryashov–Sinelshchikov equation is investigated in this paper, which describes bubbles in the liquid fluctuations. By virtue of the binary Bell polynomials, the bilinear representation, bilinear B a ̈ cklund transformation with associated Lax pair are obtained, respectively. Moreover, utilizing Hirota's bilinear representation, four new lump solutions are constructed and the interaction phenomenon between lump and periodic solution is thoroughly examined. The work also illustrates the intriguing dynamical behavior with the aid of Maple software, which plots the three-dimensional surface, two-dimensional density, and contour profiles of the solutions constructed in this work in various planes. [ABSTRACT FROM AUTHOR]

This article describes the bilinear form, biliear Bäcklund transformation and Lax pair of the geophysical Boussinesq equation using the Bell polynomial approach. The integrability of the said equation is asserted in the sense of the Lax pair. The Darboux transformation is employed to produce periodic solutions as well as single and N-complexiton-type solutions. Several steps of the Hirota bilinear technique are used to illustrate the flow characteristics of multi-solitons, higher-order breathers and rogue waves. Every sort of wave dynamics response to the Coriolis force is carefully studied. [ABSTRACT FROM AUTHOR]

Using Whitham modulation theory, this paper examined periodic solutions and the problem of discontinuous initial values for the higher-order good Jaulent-Miodek (JM) equation. The physical significance of the JM equations was discussed by considering the reduction of Euler's equation. Next, the zero- and one-phase periodic solutions of the JM equation, along with the associated Whitham equations, were derived. The analysis included the degeneration of the one-phase periodic solution and the genus-one Whitham equation by examining the limits of the modulus m of the Jacobi elliptic functions. Additionally, analytical and graphical representations of rarefaction wave solutions and periodic wave patterns were provided, and a solution for discontinuous initial values in the JM equation was presented. The results of this study offer a theoretical foundation for analyzing discontinuous initial values in nonlinear dispersion equations.

In this work, the stability of exact smooth multi-solitons for the Dullin-Gottwald-Holm (DGH) equation is investigated for the first time. Firstly, through the inverse scattering approach, we derive the conservation laws in terms of scattering data and multi-solitons related to the discrete spectrum based on the Lax pair of the DGH equation. Notably, a new series Hamiltonian derived from bi-Hamilton structure are equivalent to the conservation laws, and can also be expressed in terms of scattering data. Thus we successfully connect scattering data (multi-solitons) to the recursion operator between Hamiltonian, and to the Lyapunov functional constructed from Hamiltonian. With this Lyapunov functional, it is identified that these smooth multi-solitons serve as non-isolated constrained minimizers, adhering to a suitable variational nonlocal elliptic equation. Moreover, the integrable properties of the recursion operator are presented, which is the crucial for spectral analysis in this work. Consequently, the investigation of dynamical stability transforms into a problem on the spectrum of explicit linearized systems. It is worth noting that, compared to previous works on Camassa-Holm equation and KdV-type equations, recursion operator derived from existing Hamiltonian for the DGH equation do not possess good integrable properties. We construct a new series of Hamiltonians with both analysable recursion operator and simple initial variation derivative (δ H 1 / δ u = m) to overcome this problem. Furthermore, orbital stability of the smooth double solitons is proved at last of the work. [ABSTRACT FROM AUTHOR]

Using the Darboux transformation method, the general Lax equation is solved and a collection of new exact solutions together with one-soliton solutions, singular one-soliton solutions, periodic solutions, singular periodic solution, two-soliton solutions, singular two-soliton solutions, two-periodic solutions and singular two-periodic solutions is obtained. Using traveling wave transformation, the Lax equation is transfigured to a conservative dynamical system (CDS) of dimension four with three equilibrium points involving two parameters γ and v. The CDS has various quasi-periodic motions for fixed values of the parameters γ and v at different initial conditions. Furthermore, effects of the parameters γ and v are shown on the quasiperiodic motions of the CDS by means of phase sections and time series plots. This approach can be applied to a heterogeneity of nonlinear model equations or partial differential equations for describing their inherent nonlinear phenomena. [ABSTRACT FROM AUTHOR]

A new generalised non-local nonlinear Schrödinger (NLS) equation is introduced which possesses a Lax pair and is parity–time (PT)-symmetric. Thus, it is confirmed that the generalised non-local NLS equation is integrable. The inverse scattering transform for the generalised non-local NLS equation is developed using a Riemann–Hilbert problem for rapidly decaying initial data and an approach for finding pure soliton solutions is described. The analytical characteristics of the eigenfunctions, scattering data and their symmetries are discussed. Finally, using Mathematica some important two-dimensional plots of the wave solutions are shown to illustrate the dynamics of the model. [ABSTRACT FROM AUTHOR]

A higher-order spectral problem is introduced, based on a special Lie subalgebra of the general linear algebra. An associated matrix Liouville integrable hierarchy, each of which consists of four submatrix equations, is generated by means of the zero curvature formulation. The corresponding Hamiltonian structure is established by the trace variational identity and two integrable reductions, of which one is real and the other is complex, are constructed under similarity transformations. [ABSTRACT FROM AUTHOR]

This study introduces a 4 × 4 matrix eigenvalue problem and develops an integrable hierarchy with a bi-Hamiltonian structure. Integrability is ensured by the zero-curvature condition, while the Hamiltonian structure is supported by the trace identity. Explicit derivations yield second-order and third-order integrable equations, illustrating the integrable hierarchy. [ABSTRACT FROM AUTHOR]

In a recent paper Li et al. (Z Angew Math Phys 73:52, 2022. https://doi.org/10.1007/s00033-022-01681-4) have considered a generalized nonlinear Schrödinger equation which has extensive applications in various fields of physics and engineering. After proving Liouville integrability of this equation, they investigated the phenomenon of the modulational instability for the possible reason of the formation of the rogue waves. By means of the generalized ( 2 , N - 2 )-fold Darboux transformation, authors presented several mixed localized wave solutions, such as breathers, rogue waves and semi-rational solitons for their model equation, and accurately analyzed a number of important physical quantities. It is the aim of this Comment to point out that (i) the baseband modulation instability was developed in a wrong way and (ii) one of the two different types of Taylor series expansions for solution of Lax pair used in that article for building analytical solutions, especially the one obtained with ξ j = Z does not correspond to any solution of the spectral problem (2.1) when using u 0 x , t as the seed solution. Consequently, all mixed localized solutions that involve the mentioned Taylor series are invalid. [ABSTRACT FROM AUTHOR]

The paper aims to demonstrate that a linear expansion in a unital two-dimensional algebra can generate integrable couplings, proposing a novel approach for their construction. The integrable couplings presented encompass a range of perturbation equations and nonlinear integrable couplings. Their corresponding Lax pairs and hereditary recursion operators are explicitly detailed. Concrete applications to the KdV equation and the AKNS system of nonlinear Schrödinger equations are extensively explored. [ABSTRACT FROM AUTHOR]

The generalized mixed nonlinear Schrödinger (GMNLS) equation is a completely integrable model under certain parameter conditions, which has important applications in nonlinear optics, weak nonlinear dispersive water wave and quantum field theory. In this work, we use the unified transformation method to construct a Riemann-Hilbert problem and discuss the initial-boundary value problems of the GMNLS equation on the half-line. Furthermore, we also obtain that some spectral functions are not independent of each other, but meet a paramount global relation. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the modified Korteweg-de Vries equation with time-dependent coefficients and a self-consistent source in the class of rapidly decreasing functions. To solve the problem, we find Lax pairs and employ the inverse scattering method. Note that in the case under study the Dirac operator is not selfadjoint, and so the eigenvalues can be multiples. We find the equations describing the dynamics in time of the scattering data of a nonselfadjoint Dirac operator whose potential is a solution to the modified Korteweg-de Vries equation with time-dependent coefficients and a self-consistent source in the class of rapidly decreasing functions. As a special case, we examine a loaded modified Korteweg-de Vries equation with a self-consistent source. The equations describe the dynamics in time of the scattering data of a nonselfadjoint Dirac operator whose potential is a solution to the loaded modified Korteweg-de Vries equation with variable coefficients in the class of rapidly decreasing functions. We provide some examples that illustrate applications of the results. [ABSTRACT FROM AUTHOR]

The optical solitons represent a fascinating intersection of nonlinear physics and optical technology. In this study, we examine the three-component Dirac–Manakov system through the application of a unified Darboux transformation. We elaborate on the construction of the n-fold Darboux transformation and elucidate the correlation between optical solitons and seed solutions. Our research successfully yields both single-soliton and double-soliton solutions derived from zero-seed solutions. Additionally, we derive periodic solutions from non-zero seed solutions. To visualize these solutions, we employ Maple to generate three-dimensional and density diagrams, thereby facilitating a comprehensive understanding of the solution structures within the three-component Dirac–Manakov system. [ABSTRACT FROM AUTHOR]

Analytical methods provide crucial mathematical insights into the stable solitary waves hidden in nonlinear phenomena. The nonlinear Schrödinger (NLS) equation is one of the most important typical integrable soliton models. From a mathematical perspective, the essence of the celebrated derivative NLS (DNLS) equation’s difference from the classical NLS equation lies in its cubic potential being differentiated once by the spatial variable and multiplied by the imaginary unit, which leads to the former having some characteristics that the latter cannot have. This paper extends the DNLS equation to the fractional integrable case with conformable derivative operators, and uses Darboux transformations (DTs) and generalized DT (GDT) to solve it exactly. Specifically, Lax pairs generating the fractional DNLS equation are first given. Based on the given Lax pairs, then the n-fold DTs and GDT for the fractional DNLS equation are derived. Some special exact solutions of the fractional DNLS equation are further obtained by employing the derived n-fold DTs and GDT. Finally, several novel space–time structures and dynamical evolutions of the obtained exact solutions are analyzed. This paper reveals through the DT and GDT methods that the double power-law fractional orders in the exact solutions of the fractional DNLS equation can be used to dominate the variable velocity propagation and anomalous diffusion in fractional dimensional media at different geometric scales. [ABSTRACT FROM AUTHOR]

We develop lattice eigenfunction equations of the lattice KdV equation, which are equations obeyed by auxiliary functions, or eigenfunctions, of the Lax pair of the lattice KdV equation. These equations are three-dimensionally consistent quad-equations, that are closely related to lattice equations in the Adler-Bobenko-Suris (ABS) classification. The connection between the H3(δ), Q1(δ), Q2 and Q3(δ) equations in the ABS classification and the lattice eigenfunction equations is explicitly showed. In particular, we provide a natural interpretation of the δ term in those equations. This can be understood as 'interactions' between the eigenfunctions. Other integrable properties of the eigenfunction equations, such as exact solutions, discrete zero curvature conditions are also provided. We believe that the approach presented in this paper can be used as a means to search for integrable lattice equations.> [ABSTRACT FROM AUTHOR]

We study the asymptotic stability of peaked solitons under H1 × H1-perturbations of the two-component Novikov equation involving interaction between two components. This system, as a two-component generalization of the Novikov equation, is a completely integrable system which has Lax pair and bi-Hamiltonian structure. Interestingly, it admits the two-component peaked solitons with different phases, which are the weak solutions in the sense of distribution and lie in the energy space H1 × H1. It is shown that the peakons are asymptotically stable in the energy space H1 × H1 with non-negative momentum density by establishing a rigidity theorem for H1 × H1-almost localized solutions. Our proof generalizes the arguments for studying the Camassa-Holm and Novikov equations. There are three new ingredients in our proof. One is a new characteristic describing interaction of the two-components; the second is new additional conserved densities for establishing the main inequalities; while the third one is a new Lyapunov functional used to overcome the difficulty caused by the loss of momentum. [ABSTRACT FROM AUTHOR]

In the paper, we describe a method for deriving generalized symmetries for a generic discrete quadrilateral equation that allows a Lax pair. Its symmetry can be interpreted as a flow along the tangent direction of its solution evolving with a Lie group parameter t. Starting from the spectral problem of the quadrilateral equation and assuming the eigenfunction evolves with the parameter t, one can obtain a differential-difference equation hierarchy, of which the flows are proved to be commuting symmetries of the quadrilateral equation. We prove this result by using the zero-curvature representations of these flows. As an example, we apply this method to derive symmetries for the lattice potential Korteweg–de Vries equation. [ABSTRACT FROM AUTHOR]

The Benjamin–Ono equation is shown to be well-posed, both on the line and on the circle, in the Sobolev spaces H s for s > − 1 2 . The proof rests on a new gauge transformation and benefits from our introduction of a modified Lax pair representation of the full hierarchy. As we will show, these developments yield important additional dividends beyond well-posedness, including (i) the unification of the diverse approaches to polynomial conservation laws; (ii) a generalization of Gérard's explicit formula to the full hierarchy; and (iii) new virial-type identities covering all equations in the hierarchy. [ABSTRACT FROM AUTHOR]

We consider chiral-type systems admitting a Lax representation with values in a real or complex semisimple Lie algebra such that an additional regularity condition is satisfied (one of the matrices is a regular element of the Lie algebra). We prove that for a chiral-type system with vanishing torsion and a nonvanishing curvature, the existence of at least one pointwise cosymmetry is a necessary condition for the regular Lax representation. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is the inverse scattering transform for a nonlinear lattice equation, which can be used to study the fluctuation of nonlinear optics and dynamics of anharmonic lattices. Symmetries, analyticities and asymptotic behaviors of eigenfunctions will be obtained in the direct scattering analysis to establish a suitable Riemann-Hilbert problem. The Riemann-Hilbert problem of the scattering data with simple poles will be constructed. In particular, by using the Laurent expansion and the generalized residue condition to solve the Riemann-Hilbert problem, the determinant representation of N-soliton solution for the equation will be presented. One-dark-soliton under non-vanishing boundary conditions will be displayed through some representative reflectionless potentials. [ABSTRACT FROM AUTHOR]

This paper is dedicated to extending the focusing/defocusing Calogero–Moser–Sutherland cubic derivative Schrödinger equations (CMSdNLS) i ∂ t u + ∂ x 2 u = ± u D + | D | | u | 2 , D = - i ∂ x , x ∈ R or x ∈ T : = R / 2 π Z , which were initially introduced in Matsuno (Phys Lett A 278(1–2):53–58, 2000; Inverse Probl 18:1101–1125, 2002; J Phys Soc Jpn 71(6):1415–1418, 2002; Inverse Prob 20(2):437–445, 2004), Abanov et al. (J Phys A 42(13): 135201, 2009), Gérard and Lenzmann (The Calogero–Moser derivative nonlinear Schrödinger equation, Communications on Pure and Applied Mathematics. ) and Badreddine (On the global well-posedness of the Calogero–Sutherland derivative nonlinear Schrödinger equation, Pure and Applied Analysis. ; Traveling waves and finite gap potentials for the Calogero–Sutherland derivative nonlinear Schrödinger equation, Annales de l’Institut Henri Poincaré, Analyse Non Linéaire. ), to a system of two matrix-valued variables, leading to the following intertwined system, i ∂ t U + ∂ x 2 U = - 1 2 U D + | D | V ∗ U - 1 2 V D + | D | U ∗ U , i ∂ t V + ∂ x 2 V = - 1 2 V D + | D | U ∗ V - 1 2 U D + | D | V ∗ V. This system enjoys a Lax pair structure, enabling the establishment of an explicit formula for general solutions on both the 1-dimensional torus and the real line. Consequently, this system can be regarded as an integrable perturbation and extension of both the linear Schrödinger equation and the CMSdNLS equations. [ABSTRACT FROM AUTHOR]

In this article, a time-varying coefficient modified nonlinear Schrödinger equation (tvcmNLSE) with non-zero boundary conditions (NZBCs) at infinity, phase term of which contains time-varying coefficients, is proposed to describe the nonlinear propagation behavior of optical pulses in non-uniform optical fibre. Meanwhile, the Riemann–Hilbert method (RHM) for the tvcmNLSE with NZBCs is presented for the first time. Specifically, a special transformation is constructed to convert the tvcmNLSE with NZBCs into the case with constant NZBCs, and the associated asymptotic spectral problem (ASP) is obtained. By introducing a suitable two-piece Riemann surface to map the original spectral parameter into a single-valued one, and analyzing the ASP, we construct the corresponding Jost solutions and spectral matrix, and sequentially analyze their analytical, symmetric, and asymptotic properties. Then, a generalized Riemann–Hilbert problem (RHP) is established that relates to the converted tvcmNLSE with constant NZBCs. Using the reconstruction formula, we further derive N-soliton solution with NZBCs in the absence of reflection potential, and analyze the dynamic characteristics of single and double solitons. Most notably, by modulating discrete spectral points (DSPs) and time-varying coefficients, we discover some previously unreported new waveforms, such as parabolic and periodic solitons with NZBCs. These waveforms have important implications in both theory and practice. In addition, the influences of different time-varying coefficients, NZBCs, and DSPs on soliton solutions are also discussed and analyzed. This study shows that the selection of time-varying coefficients, distribution of DSPs, and constraints from NZBCs all have significant impacts on the properties of soliton solutions.

The modified Helmholtz equation qxx+qyy−4β2q=0, is one of the basic equations of classical mathematical physics. In this paper we have obtained the solution of the boundary-value problems for the modified Helmholtz equation in an equilateral triangle. An additional mixed boundary condition related to the symmetry of the solution is taken into consideration. We have analysed the Global relation and only used the algebraic techniques to obtain the explicit solution of modified Helmholtz equation bypassing the Riemann Hilbert approach. This solution is applied to the problem of diffusion-limited coalescence, A+A⇌A.

A new (2 + 1)-dimensional generalized KP-type equation and its Lax pair are established. Based on gauge transformation between spectral problems, a Darboux transformation for the (2 + 1)-dimensional generalized KP-type equation is obtained. Some exact solutions are derived by utilizing the formula of Darboux transformation, including soliton solutions, trigonometric function solution, lump solutions, lump-soliton solution. In order to analyze the dynamical behavior of the solutions, the plots of solutions are depicted by Mathematical software. [ABSTRACT FROM AUTHOR]

In this paper, a study of the coupled Schrödinger Maxwell-Bloch equations, which described the propagation of two optical pulses in an optical medium with coherent three-level atoms, is presented via the Riemann-Hilbert approach. First, we performed a spectral analysis on the basis of Lax pair with the negative flow, and then the Jost function, scattering matrix, and their analytic and symmetric properties are given. Second, the Riemann-Hilbert problem is established successfully through a standard dressing procedure via the Riemann-Hilbert approach, and then the potential function related to the solution of the Riemann-Hilbert problem is reconstructed. By introducing the special matrix functions, we can transform the irregular Riemann-Hilbert problem into the regular one, which can be solved by the Plemelj formula. Finally, some applications are given to solve the Riemann-Hilbert problem without reflection for the coupled Schrödinger Maxwell-Bloch equation, and the multi-soliton solutions are obtained explicitly. Moreover, the dynamic behaviors of some soliton solutions are discussed graphically by choosing appropriate parameters. [ABSTRACT FROM AUTHOR]

In this paper, a compound equation which is a mix of the Wadati--Konno--Ichikawa (WKI) equation and the short-pulse (SP) equation is first studied. By transforming both the independent and dependent variables in the equation, we introduce a novel hodograph transformation to convert the compoundWKI--SP equation into the mKdV--SG (modified Korteweg--de Vries and sine- Gordon) equation. The multiloop soliton solutions in the form of the parametric representation are found. It is shown that the N-loop soliton solution may be decomposed exactly into N separate soliton elements by using a Moloney--Hodnett-type decomposition. By virtue of the decomposed soliton solutions, the asymptotic behaviors of N = 2 and N = 3 are investigated in detail. The corresponding phase shifts of each loop or antiloop soliton caused by its interaction with the other ones are calculated. Furthermore, a new hierarchy of WKI--SP-type equations possessing multiloop soliton solutions is constructed. These deduced equations are all with time-varying coefficients and the corresponding dispersion relation will have a time-dependent velocity. The whole hierarchy of equations which include the WKI-type equations, the SP-type equations, and the compound generalized WKI--SP equations, are illustrated Lax integrable. The specific equation in the hierarchy is labeled as WKI--SP(n,m) equation so that its Lax pairs can be directly written out with the help of n and m. A unified hodograph transformation is established to relate the compound WKI--SP hierarchy with the mKdV--SG hierarchy. [ABSTRACT FROM AUTHOR]

For the discrete PT-symmetric nonlinear Schrödinger (dPTNLS) equation, this paper gives a rigorous proof of the N-fold Darboux transformation (DT) and especially verifies the PT-symmetric relation between transformed potentials in the Lax pair. Meanwhile, some determinant identities are developed in completing the proof. When the tanh-function solution is directly selected as a seed for the focusing case, the onefold DT yields a three-soliton solution that exhibits the solitonic behavior with a wide range of parameter regimes. Moreover, it is shown that the solution contains three pairs of asymptotic solitons, and that each asymptotic soliton can display both the dark and antidark soliton profiles or vanish as t → ± ∞. It indicates that the focusing dPTNLS equation admits a rich variety of soliton interactions over the nonzero background, behaving like those in the continuous counterpart. [ABSTRACT FROM AUTHOR]

We prove that the topological recursion formalism can be used to quantize any generic classical spectral curve with smooth ramification points and simply ramified away from poles. For this purpose, we build both the associated quantum curve, i.e. the differential operator quantizing the algebraic equation defining the classical spectral curve considered, and a basis of wave functions, that is to say a basis of solutions of the corresponding differential equation. We further build a Lax pair representing the resulting quantum curve and thus present it as a point in an associated space of meromorphic connections on the Riemann sphere, a first step towards isomonodromic deformations. We finally propose two examples: the derivation of a 2-parameter family of formal trans-series solutions to Painlevé 2 equation and the quantization of a degree three spectral curve with pole only at infinity. [ABSTRACT FROM AUTHOR]

It has been unknown whether Hirota's discrete Korteweg-de Vries equation and the lattice sine-Gordon equation have the consistency around a cube (CAC) property. In this paper, we show that they have the CAC property. Moreover, we also show that they can be extended to systems on the 3-dimensional integer lattice. [ABSTRACT FROM AUTHOR]

By using vertex operator representation of polynomial Lie algebra b ∞ (n) , a theoretical interpretation of coupled BKP hierarchy is given, where corresponding Hirota bilinear equations are derived. Then based upon this, a wave function matrix is introduced, so that we can construct the corresponding dressing operator matrix and obtain the coupled BKP constraints and Sato equation, which allows us to define the Lax operator matrix and obtain the Lax equation. It is found that the coupled BKP hierarchy is a special case of matrix KP theory. Lastly, we present the Bäcklund–Darboux transformations for the coupled BKP equations from both fermionic and bosonic pictures. [ABSTRACT FROM AUTHOR]

We show that the same algebraic data that permit to construct the Lax pair and the r-matrix of an integrable non-linear σ-model in 1 + 1 dimensions can be also used for the construction of Lax pairs and of r-matrices of several other non-trivial integrable theories in 1 + 0 dimension. We call those new integrable theories the point particle E -models, we describe their structure and give their physical interpretation. We work out in detail the point particle E -modelsassociated to the bi-Yang–Baxter deformation of the SU(N) principal chiral model. In particular, for each complex flag manifold we thus obtain a two-parameter family of integrable models living on it. [ABSTRACT FROM AUTHOR]

This paper conducts an in-depth study on the self-similar transformation, Darboux transformation, and the excitation and propagation characteristics of high-order bright–dark rogue wave solutions in the (2+1)-dimensional variable-coefficient Zakharov equation. The Zakharov equation is instrumental for studying complex nonlinear interactions in these areas, with specific implications for energy transfer processes in plasma and nonlinear wave propagation systems. By analyzing bright–dark rogue wave solutions—phenomena that are critical in understanding high-energy events in optical and fluid environments—this research elucidates the intricate dynamics of energy concentration and dissipation. Using the self-similar transformation method, we map the (2+1)-dimensional equation to a more tractable (1+1)-dimensional nonlinear Schrödinger equation form. Through the Lax pair and Darboux transformation, we successfully construct high-order solutions that reveal how variable coefficients influence rogue wave features, such as shape, amplitude, and dynamics. Numerical simulations demonstrate the evolution of these rogue waves, offering novel perspectives for predicting and mitigating extreme wave events in engineering applications.This paper crucially advances the practical understanding and manipulation of nonlinear wave phenomena in variable environments, providing significant insights for applications in optical fibers, atmospheric physics, and marine engineering. [ABSTRACT FROM AUTHOR]

With the inhomogeneity of optical fiber media taken into account, under investigation in this paper is the variable coefficient Kundu-nonlinear Schrödinger equation, which describes the pulses propagation in optical fibers. Based on Lax pair, the Nth-order Darboux transformation is constructed. Depending on plane wave solution, the first- and second-order breather solutions are derived and the interactions between breathers are graphically analyzed. The Kuznetsov–Ma breather, Akhmediev breather, and spatial-temporal breather have been obtained. Moreover, the first-, second-, and third-order rogue wave solutions have been constructed. The usual rogue waves and first- and second-order line rogue waves are observed. The weak and strong interactions between the first-, second-order rogue waves, and spatial-temporal period breather are studied. Furthermore, variable coefficient δ (t) causes rogue waves to produce some interesting evolutionary phenomena, which have been systematically analyzed. In addition, the influences of parameters for the properties of solutions are discussed. [ABSTRACT FROM AUTHOR]

In geophysical hydrodynamics, baroclinic instability denotes the process in which the perturbations draw the energy from the mean flow potential power. Researchers focus their attention on the baroclinic instability in the Earth's atmosphere and oceans for the meteorological diagnosis and prediction. Under investigation in this paper is a repulsive nonlinear system modeling the marginally unstable baroclinic wave packets in a baroclinic flow. With respect to the amplitude of the baroclinic wave packet and correction to the mean flow resulting from the self-rectification of the baroclinic wave, we present a Lax pair with the changeable parameters and then derive the N-dark-dark soliton solutions, where N is a positive integer. Asymptotic analysis on the N-dark-dark solitons is processed to obtain the algebraic expressions of the N-dark-dark soliton components. We find that the obtained phase shift of each dark-dark soliton component is relevant with the N − 1 spectral parameters. Furthermore, we take N = 3 as an example and graphically illustrate the 3-dark-dark solitons, which are consistent with our asymptotic-analysis results. Our analysis may provide the explanations of the complex and variable natural mechanisms of the baroclinic instability. [ABSTRACT FROM AUTHOR]

This study makes consideration of the recently devised Lax-integrable three-dimensional sixth-order Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation owing to the great relevance of higher-dimensional nonlinear evolution equation models in guiding various nonlinear phenomena in scientific domains. Thus, as only a little is known about the new model, the present study is therefore, obliged to unravel more about the admissibility of supplementary exact solutions by the model through the use of the promising Kudryashov method. In fact, all the special models arising from the governing equation have been reduced with regard to both the equations as well as the corresponding exact solutions. In addition, the study graphically analyses the state of the art with regard to the temporal variation as well as the influence of the Kudryashov-index on the evolution of waves associated with the model. Lastly, the model of concern is further extended to the n-dimensional frame and equally analyzed using the adopted methodology. [ABSTRACT FROM AUTHOR]

The goal of this article is to characterize the lax pullback complement of a given partial morphism along a total morphism, in an -partial morphism category, where is an exponentiable stable system. To achieve this, we first show that if a lax pullback complement of a partial morphism exists in the partial morphism category, then a lax pullback complement of its total part exists in the base category. For the converse, we consider two cases. In the first case the base category is assumed to be adhesive and the morphism along which we take the lax pullback complement is admissible (i.e., its pushout along a monomorphism forms a pullback complement square), while in the second case the base category is -cohesive, however the morphism along which we take the lax pullback complement is an arbitrary total morphism. Finally as a byproduct we provide the connection between exponentials in the partial morphism category and its base category. [ABSTRACT FROM AUTHOR]

This paper provides some information on the eigenvalues and eigenfunctions of some left‐definite system of first‐order differential equations subject to eigenparameter‐dependent boundary conditions. Namely, we show that the pair of solutions of the system of equations satisfying some initial conditions exists and is unique, and this pair is analytic in the spectral parameter of order 1/2. We also introduce Lagrange's formula for the left‐definite equation. Using some Prüfer angels, we investigate oscillation of zeros of eigenfunctions and asymptotics equations for the eigenvalues of the problem. Moreover, we share some ordinary and Fréchet derivatives of eigenvalues and eigenfunctions with respect to some elements of data. [ABSTRACT FROM AUTHOR]

In this paper, we study a three‐component modified nonlinear Schrödinger equation and main results include: (1) Lax pair and infinitely‐many conservation laws, (2) modulation instability of continuous waves, (3) semi‐rational solutions obtained through the Darboux transformation, and (4) four new types of mixed semi‐rational solutions provided for the three components q1$$ {q}_1 $$, q2$$ {q}_2 $$ and q3$$ {q}_3 $$. [ABSTRACT FROM AUTHOR]

In this paper, we study parameter deformations of matrix valued orthogonal polynomials. These deformations are built on the use of certain matrix valued operators which are symmetric with respect to the matrix valued inner product defined by the orthogonality weight. We show that the recurrence coefficients associated with these operators satisfy generalizations of the non-Abelian lattice equations. We provide a Lax pair formulation for these equations, and an example of deformed Hermite-type matrix valued polynomials is discussed in detail. [ABSTRACT FROM AUTHOR]

In this paper, we study the breathers in the framework of the sixth-order nonlinear Schrödinger equation by using the Darboux transformation. The primary objective of this research is twofold. First, we consider the nonlinear superposition of breathers on the plane wave background. Based on the concept that rogue waves are formed from colliding Akhmediev breathers, we obtain rogue wave sequences and a first-order Akhmediev breather with a central second-order rogue wave peak. Second, we consider the formation of breathers on the periodic wave background. The difficulty of solving the Lax pair is overcome, and we successfully construct the breathers on the cn- and dn-periodic wave background. [ABSTRACT FROM AUTHOR]

The Cauchy problem of the coupled modified complex short-pulse equation with decaying boundary conditions $ (x\rightarrow \pm\infty) $ are studied by utilizing the Riemann-Hilbert approach and nonlinear steepest descent method. On the basis of the spectral analysis of $ 4\times 4 $ matrix Lax pair and the inverse scattering transform, the solution to the Cauchy problem is converted to solving a basic Riemann-Hilbert problem. As a special example, the explicit formulas for the one-soliton solutions and breather solutions are given. A model Riemann-Hilbert problem, which can be solved by the parabolic cylindrical functions, is obtained by successive deformation of various corresponding Riemann-Hilbert problems. Finally, the leading order asymptotic behavior of the solution of the Cauchy problem for the coupled modified complex short-pulse equation is derived. [ABSTRACT FROM AUTHOR]