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Sasa–Satsuma (SS)-type integrable matrix modified Korteweg–de Vries (mKdV) equations are derived from two group constraints, involving the replacement of the spectral matrix in the Ablowitz–Kaup–Newell–Segur matrix eigenproblems with its matrix transpose and its Hermitian transpose. Using the Lax pairs and dual Lax pairs of matrix eigenproblems as a foundation, binary Darboux transformations are constructed. These transformations, initiated with a zero seed solution, facilitate the generation of soliton solutions for the SS-type integrable matrix mKdV equations presented. [ABSTRACT FROM AUTHOR]

This paper aims to generate a Liouville integrable Hamiltonian hierarchy by introducing a specific matrix eigenvalue problem with four components. The adopted approach is the zero curvature formulation. A bi-Hamiltonian formulation is furnished through applying the trace identity, which shows the Liouville integrability of the resulting hierarchy. Two examples of generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations are presented. [ABSTRACT FROM AUTHOR]

For vector nonlocal reverse-space nonlinear Schrödinger equations, a binary Darboux transformation is formulated by using two sets of eigenfunctions and adjoint eigenfunctions. The resulting binary Darboux transformation has been decomposed into an N -fold product of single binary Darboux transformations. An application starting from zero seed potentials generates a class of soliton solutions. [ABSTRACT FROM AUTHOR]

The main work of this paper is to construct the Wronskian solution and investigate the integrability characteristics of the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation. Firstly, the Wronskian technique is used to acquire a sufficient condition of the Wronskian solution. According to the Wronskian form, the soliton solution is obtained by selecting the elements in the determinant that satisfy the linear partial differential systems. Secondly, the bilinear Bäcklund transformation and Bell-polynomial-typed Bäcklund transformation are derived directly via the Hirota bilinear method and the Bell polynomial theory, respectively. Finally, Painlevé analysis proves that this equation possesses the Painlevé property, and a Painlevé-typed Bäcklund transformation is constructed to solve a family of exact solutions by selecting appropriate seed solution. It shows that the Wronskian technique, Bäcklund transformation, Bell polynomial and Painlevé analysis are applicable to obtain the exact solutions of the nonlinear evolution equations, e.g., soliton solution, single-wave solution and two-wave solution. [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]

This investigation delves into the analytic optical exploration of the Biswas–Milovic equation featuring both parabolic and power law nonlinearity. It provides valuable insights into the distinctive characteristics of optical soliton solutions, enhancing our comprehension of soliton dynamics in diverse optical systems. Employing the potent Sardar-subequation analytical method, we construct a variety of wave structures, revealing a spectrum of optical soliton solutions that span trigonometric and hyperbolic functions. Thorough validation with Mathematica software ensures precision and dynamic visual representations depict soliton solutions that exhibit diverse patterns. These patterns include bright solitons, dark solitons, singular solitons, king solutions, bell-shaped patterns, as well as parabolic-shaped and hyperbolic-shaped patterns. These solutions hold significance in the realms of optical fiber and wave dynamics within various optical systems. Our approach above demonstrates versatility and applicability in solving a wide range of mathematical and physical problems, indicating its usefulness in generating such optical solutions. [ABSTRACT FROM AUTHOR]

The goal of this paper is to find exact solutions to the improved modified Korteweg-de Vries (mKdV) equation with a conformable derivative using the Jacobi elliptic function expansion method. The improved mKdV equation is a prominent mathematical model in the realm of nonlinear partial differential equations, with widespread applicability in diverse scientific and engineering domains. This study leverages the well-known effectiveness of the Jacobi elliptic function expansion method in solving nonlinear differential equations, specifically focusing on the intricacies of the improved mKdV problem. The investigation reveals innovative and explicit solutions, providing insight into the dynamics of the related physical processes. This paper provides a comprehensive examination of these solutions, emphasizing their distinct features and depictions using Jacobi elliptic functions. These findings are especially advantageous for specialists in the fields of nonlinear science and mathematical physics, providing significant insights into the behavior and development of nonlinear waves in various physical situations. This work also contributes to our knowledge of the improved mKdV equation and shows that the Jacobi elliptic function expansion method is a useful tool for solving complex nonlinear situations. The study is enhanced with graphical illustrations of various solutions, which further enhance its analytical complexity. [ABSTRACT FROM AUTHOR]

The aim of this paper is to analyze a specific fourth-order matrix spectral problem involving four potentials and two free nonzero parameters and construct an associated integrable hierarchy of bi-Hamiltonian equations within the zero curvature formulation. A hereditary recursion operator is explicitly computed, and the corresponding bi-Hamiltonian formulation is established by the so-called trace identity, showing the Liouville integrability of the obtained hierarchy. Two illustrative examples are novel generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations with four components and two adjustable parameters. [ABSTRACT FROM AUTHOR]

The main topic of the paper is to investigate the generalized (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) and Korteweg–de Vries (KdV) equations, which are widely used in many physical areas, especially in fluids. A new Wronskian formulation is presented for these two equations associated with the bilinear Bäcklund transformation. Based on Wronskian identities of the bilinear Kadomtsev–Petviashvili (KP) hierarchy, the Wronskian determinant solution is verified by a direct and concise calculation. The newly introduced Wronskian formulation provides a comprehensive way for building rational solutions. A few rational Wronskian solutions of lower order are computed for the generalized (2 + 1)-dimensional DJKM equation. Our work can show that the extended (2 + 1)-dimensional KdV equation possesses the similar rational Wronskian solutions through the corresponding logarithmic transformation. [ABSTRACT FROM AUTHOR]

A class of higher-order matrix spectral problems is formulated and the associated integrable hierarchies are generated via the zero-curvature formulation. The trace identity is used to furnish Hamiltonian structures and thus explore the Liouville integrability of the obtained hierarchies. Illuminating examples are given in terms of coupled nonlinear Schrödinger equations and coupled modified Korteweg–de Vries equations with four components. [ABSTRACT FROM AUTHOR]

This paper aims to present two hierarchies of nonlocal reduced integrable mKdV-type equations from a vector integrable hierarchy associated with a matrix Lie algebra, not being A type. The key point is to make similarity transformations for the spectral matrix, which keep the associated zero curvature equations invariant and then there follow reduced nonlocal integrable mKdV-type equations. The success lies in determining a Laurent series solution to the corresponding reduced stationary zero curvature equation, which generates temporal matrix spectral problems in the zero curvature formation. [ABSTRACT FROM AUTHOR]

The inverse scattering transform for the defocusing–defocusing coupled Hirota equations with non-zero boundary conditions at infinity is thoroughly discussed. We delve into the analytical properties of the Jost eigenfunctions and scrutinize the characteristics of the scattering coefficients. To enhance our investigation of the fundamental eigenfunctions, we have derived additional auxiliary eigenfunctions with the help of the adjoint problem. Two symmetry conditions are studied to constrain the behavior of the eigenfunctions and scattering coefficients. Utilizing these symmetries, we precisely delineate the discrete spectrum and establish the associated symmetries of the scattering data. By framing the inverse problem within the context of the Riemann–Hilbert problem, we develop suitable jump conditions to express the eigenfunctions. Consequently, we have not only derived the pure soliton solutions from the defocusing–defocusing coupled Hirota equations but also provided the multiple double-pole solutions for the first time. • The inverse scattering transform for the defocusing–defocusing coupled Hirota equations with non-zero boundary conditions is investigated using the Riemann-Hilbert approach. • The analyticity properties of the Jost eigenfunctions and scattering coefficients are investigated. • The asymptotic behavior of the Jost eigenfunctions is derived systematically. • We have provided the multiple double-pole solutions for the defocusing–defocusing coupled Hirota equations for the first time. [ABSTRACT FROM AUTHOR]

In this paper, a (3 + 1) -dimensional generalized KP-Boussinesq equation is introduced and its associate Hirota bilinear form is also given. Based on finding the positive quadratic function solutions of the associate Hirota bilinear equation, the lump solutions of the proposed (3 + 1) -dimensional generalized KP-Boussinesq equation and its corresponding reduced equations in (2 + 1) dimensions are obtained. Furthermore, the sufficient and necessary conditions for guaranteeing the analyticity and rational localization of lump solutions are derived and expressed in the form of free parameters, which are involved in lump solutions and play a key role in controlling the dynamic properties of lump solutions. The localized properties are also analyzed and shown graphically. [ABSTRACT FROM AUTHOR]

By the Lagrangian multiplier and constraint variational derivative, a relationship between conserved quantities and multi‐Hamiltonian structures is built. Using the relation, a method is founded to prove the infinite‐dimensional Liouville integrability of evolution equations with continuous variables. As the application, the conservation laws of the Kundu equation are first obtained. Its conserved quantities are deduced for comparing by Fokas' method different from the method used in the existed literature. The integrability of the equation is proved through taking the conservation laws as a starting point. [ABSTRACT FROM AUTHOR]

This study aims to explore the connection between integrable couplings and three-dimensional unital algebras. Expansions in these algebras generate integrable couplings, including nonlinear integrable couplings, and their corresponding Lax pairs and hereditary recursion operators possess specific structures, associated with non-semisimple Lie algebras. Applications to the KdV equation are explicitly presented. • Integrable couplings. • Three dimensional unital algebras. • Recursion operators associated with non-semi-simple matrix Lie algebras. [ABSTRACT FROM AUTHOR]

In this paper, an effective algorithm for constructing nonlinear evolution equations (NLEEs) has been proposed. Particularly, the existence of resonant multi-soliton solutions in the newly generated NLEEs is verified and demonstrated, and the accuracy of the extracted resonant multi-soliton solutions has been proved at the same time. Firstly, via the linear superposition principle along with reverse engineering two new NLEEs arising from the B-type Kadomtsev–Petviashvili (BKP) equation are established and investigated as well. The first new NLEE is constructed with three time derivative terms, and the second one is constructed with three space dissipative terms, respectively. Besides, the infinite resonant multi-soliton solutions are extracted which enjoy a variety of inelastic interactions due to the fact that they are constructed with variable parameters. Then, the reliable judgments to the multi-soliton solutions are carried out. Finally, the Painlevé test is applied to examine the new equations and none of them passed the test. It is important to highlight that the presented method and NLEEs could be extended to diversify the problem of physical nature. [ABSTRACT FROM AUTHOR]

We analyze N-soliton solutions and explore the Hirota N-soliton conditions for scalar (1 + 1)-dimensional equations, within the Hirota bilinear formulation. An algorithm to verify the Hirota conditions is proposed by factoring out common factors out of the Hirota function in N wave vectors and comparing degrees of the involved polynomials containing the common factors. Applications to a class of generalized KdV equations and a class of generalized higher-order KdV equations are made, together with all proofs of the existence of N-soliton solutions to all equations in two classes. [ABSTRACT FROM AUTHOR]

A universal property in terms of Hirota differential operators is presented, with an aim to construct Hirota bilinear equations possessing Pfaffian formulations in (2+1)‐dimensions. It is then shown that there exist linear combination solutions of exponential waves to various extended Hirota bilinear equations. Applications are made for the (2+1)‐dimensional generalized Hirota–Satsuma–Ito (HSI) equation, the special fourth‐order (2+1)‐dimensional nonlinear equation, and the fifth‐order Korteweg–de Vries (KdV) equation, thereby presenting their particular Pfaffian and multiple wave solutions. [ABSTRACT FROM AUTHOR]

Upon introducing a specific 4 × 4 matrix eigenvalue problem with four components, we would like to construct a Liouville integrable Hamiltonian hierarchy, within the zero curvature formulation. Bi-Hamiltonian formulations are furnished via the trace identity, through which the Liouville integrability of the resulting hierarchy is explored. The first two nonlinear examples are novel generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations. [ABSTRACT FROM AUTHOR]

A (2+1)-dimensional generalized Kadomtsev–Petviashvili–Ito equation is introduced. Upon adding some second-order derivative terms, its various lump solutions are explicitly constructed by utilizing the Hirota bilinear method and calculated through the symbolic computation system Maple. Furthermore, two specific lump solutions are obtained with particular choices of the parameters and their dynamical behaviors are analyzed through three-dimensional plots and contour plots. [ABSTRACT FROM AUTHOR]

We investigate and concentrate on new infinitesimal generators of Lie symmetries for an extended (2 + 1)-dimensional Calogero-Bogoyavlenskii-Schif (eCBS) equation using the commutator table which results in a system of nonlinear ordinary differential equations (ODEs) which can be manually solved. Through two stages of Lie symmetry reductions, the eCBS equation is reduced to non-solvable nonlinear ODEs using different combinations of optimal Lie vectors. Using the integration method and the Riccati and Bernoulli equation methods, we investigate new analytical solutions to those ODEs. Back substituting to the original variables generates new solutions to the eCBS equation. These results are simulated through three- and two-dimensional plots. [ABSTRACT FROM AUTHOR]

In this research, we will introduce and study the localized interaction solutions and their dynamics of the extended Hirota–Satsuma–Ito equation (HSIe), which plays a key role in studying certain complex physical phenomena. By using the Hirota bilinear method, the lump-type solutions will be firstly constructed, which are almost rationally localized in all spatial directions. Then, three kinds of localized interaction solutions will be obtained, respectively. In order to study the dynamic behaviors, numerical simulations are performed. Two interesting physical phenomena are found: one is the fission and fusion phenomena happening during the procedure of their collisions; the other is the rogue wave phenomena triggered by the interaction between a lump-type wave and a soliton wave. [ABSTRACT FROM AUTHOR]

Localized interaction solutions of the (2+1)-dimensional Ito equation with free parameters are obtained by using a Hirota bilinear transformation. Various plots with particular choices of the involved parameters are made to show energy distributions and dynamical properties of the special exact solutions. This phenomenon may provide us with interesting information on dynamics in the higher-dimensional nonlinear world. [ABSTRACT FROM AUTHOR]

Through the Hirota bilinear formulation, a (2+1)-dimensional combined fourth-order nonlinear equation is proposed, which possesses lump solutions. Two classes of lump solutions are presented explicitly in terms of the coefficients in the combined nonlinear equation. A set of examples of equations is provided to show the diversity of the considered combined nonlinear equation, together with three-dimensional plots, x -curves and y -curves of two specific lump solutions in two cases of the combined equation. [ABSTRACT FROM AUTHOR]

We compute N-soliton solutions and analyze the Hirota N-soliton conditions, in (2+1)-dimensions, based on the Hirota bilinear formulation. An algorithm to check the Hirota conditions is proposed by comparing degrees of the polynomials generated from the Hirota function in N wave vectors. A weight number is introduced while transforming the Hirota function to achieve homogeneity of the resulting polynomial. Applications to three integrable equations: the (2+1)-dimensional KdV equation, the Kadomtsev–Petviashvili equation, the (2+1)-dimensional Hirota–Satsuma–Ito equation, are made, thereby providing proofs of the existence of N-soliton solutions in the three model equations. [ABSTRACT FROM AUTHOR]

We aim to generalize the (2+1)-dimensional Hirota-Satsuma-Ito (HSI) equation, passing the three-soliton test, to a new one which still has diverse solution structures. We add all second-order derivative terms to the HSI equation but demand the existence of lump solutions. Such lump solutions are formulated in terms of the coefficients, except two, in the resulting generalized HSI equation. As an illustrative example, a special completely generalized HSI equation is given, together with a lump solution, and three 3d-plots and contour plots of the lump solution are made to elucidate the characteristics of the presented lump solutions. [ABSTRACT FROM AUTHOR]

An extension of the KPII equation is studied. Adding a new fourth-order derivative term and some second-order derivative terms, we formulate an extended KPII equation. Different types of solutions of the extended equation are obtained by the Hirota bilinear method, and the presented solutions include soliton solutions, lump solutions and interaction solutions. Their dynamical behaviors are analyzed through plots. [ABSTRACT FROM AUTHOR]

In this research, the lump solution, which is rationally localized and decays along the directions of space variables, of a 2D Toda equation is studied. The effective method of constructing the lump solutions of this 2D Toda equation is derived, and the constraint conditions that make the lump solutions analytical and positive are obtained as well. Finally, three classes of lump solutions are constructed, 3D plots, density plots, and contour plots are given to illustrate this proposed method. [ABSTRACT FROM AUTHOR]

Lump solutions are one of important solutions to partial differential equations, both linear and nonlinear. This paper aims to show that a Hietarinta-type fourth-order nonlinear term can create lump solutions with second-order linear dispersive terms. The key is a Hirota bilinear form. Lump solutions are constructed via symbolic computations with Maple, and specific reductions of the resulting lump solutions are made. Two illustrative examples of the generalized Hietarinta-type nonlinear equations and their lumps are presented, together with three-dimensional plots and density plots of the lump solutions. [ABSTRACT FROM AUTHOR]

We present lump-type solutions and interaction solutions to an extended (3 + 1)-dimensional Jimbo–Miwa-like equation. Three classes of lump-type solutions are obtained by the Hirota bilinear method. Interaction solutions are among lump-type solutions, two kink waves and periodic waves, and between two kink waves and a periodic wave are computed. Dynamical characters of the obtained solutions are graphically exhibited. These wave solutions enrich the dynamical theory of higher-dimensional nonlinear dispersive wave equations. [ABSTRACT FROM AUTHOR]

This paper constructs a new (3+1)-dimensional Hirota–Satsuma–Ito equation and employs the Hirota N-soliton condition to determine the conditions for the existence of N-soliton solutions. Under these conditions, the Hirota bilinear method is utilized to derive N-soliton solutions of the equation. Additionally, breather solutions of the equation are obtained through constraints on complex conjugate parameters, while lump solutions are derived using the long-wave limit method. Building upon these results, interaction solutions are also explored. Furthermore, through the multiple-rogue wave method, multi-rogue wave solutions of the (3+1)-dimensional Hirota–Satsuma–Ito equation are investigated. A comparison between lump solutions obtained via the Hirota bilinear method and the multi-rogue wave solutions from multiple-rogue wave method reveals significant discrepancies. [ABSTRACT FROM AUTHOR]

Explicit rational-exponential solutions for the Kadomtsev-Petviashvili-II equation with a self-consistent source (KPIIESCS) are studied by the Hirota bilinear method. One typical feature for this hybrid type of solutions is that they contain two arbitrary functions of time variable t which affect the amplitudes and propagation trajectories. The dynamics of solutions are demonstrated by the three-dimensional figures. The method used here is quite general and can be applied to other equations with self-content sources. [ABSTRACT FROM AUTHOR]

In this paper, explicit representation of general rational solutions for the (3 + 1)-dimensional Mel'nikov equation is derived by employing the Hirota bilinear method. It is obtained in terms of determinants whose matrix elements satisfy some differential and difference relations. By selecting special value of the parameters involved, the first-order and second-order lump solutions are given and their dynamic characteristics are illustrated by two- and three-dimensional figures. [ABSTRACT FROM AUTHOR]

Two hierarchies of isospectral and nonisospectral Gerdjikov–Ivanov equations are constructed. Conservation laws of the isospectral soliton hierarchy are explicitly derived from a specific Riccati equation that a ratio of two eigenfunctions satisfies. The corresponding K-symmetries and τ -symmetries formulated from the isospectral and nonisospectral hierarchies constitute an infinite-dimensional τ -symmetry algebra for the isospectral hierarchy. [ABSTRACT FROM AUTHOR]

A generating scheme for conservation laws of discrete integrable equations, arising as discrete zero curvature equations, is presented from pairs of associated matrix spectral problems. An illustrative application to the Volterra lattice equation is made through a pair of 2 × 2 matrix spectral problems. The general theory is not limited to 2 × 2 matrix spectral problems but for arbitrary-order matrix spectral problems. [ABSTRACT FROM AUTHOR]

In this paper, we study the interaction solutions of the first BKP equation by using the Hirota direct method and taking long-wave limit. In order to obtain the interaction solutions, the multi-soliton solutions are firstly derived using the Hirota direct method, then the interaction solutions are successfully constructed by properly choosing appropriate parameters and taking long-wave limit on the soliton solutions. These parameters have great influences on the propagation directions, shapes as well as energy. Moreover, the dynamic properties of these obtained solutions are illustrated vividly by some graphs. The results in this work could be used to solve nonlinear problems in nonlinear optics and engineering field. [ABSTRACT FROM AUTHOR]

In this paper, a multi-component combined short pulse-mKdV equation is constructed as a nonstreching invariant curve flow in the n -dimensional unit sphere S n (1) = SO (n + 1) / SO(n). The invariant subspace method is used to solve the resulting combined short pulse-mKdV equation with n = 1. Through symbolic computation with the aid of mathematical software Maple, new exact solutions are obtained from its two-dimensional invariant subspaces formed in forms of the exponential functions and trigonometric functions. [ABSTRACT FROM AUTHOR]

Abundant exact interaction solutions, including lump-soliton, lump-kink, and lump-periodic solutions, are computed for the Hirota-Satsuma-Ito equation in (2 + 1)-dimensions, through conducting symbolic computations with Maple. The basic starting point is a Hirota bilinear form of the Hirota-Satsuma-Ito equation. A few three-dimensional plots and contour plots of three special presented solutions are made to shed light on the characteristic of interaction solutions. [ABSTRACT FROM AUTHOR]

Based on Hirota's bilinear structure, we evolute a new protuberance type arrangement of the (3+1)-dimensional Boiti-Boiti-Leon-Manna-Pempinelli equation, which depicts nonlinear wave spreads in incompressible fluid. New lump arrangement is built by applying the bilinear strategy and picking appropriate polynomial. Under various parameter settings, this lump arrangement has three sorts of numerous irregularity waves, blended arrangements including lump waves and solitons are additionally developed. Association practices are seen between lump soliton and soliton. Research demonstrates that soliton can somewhat swallow or release lump waves. The shape and highlights for these subsequent arrangements are portrayed by exploiting the three-dimensional plots and comparing shape plots by picking suitable parameters. The physical significance of these charts is given. [ABSTRACT FROM AUTHOR]

A new generalized Kadomtsev–Petviashvili (GKP) equation is derived from a bilinear differential equation by taking the transformation u = 2 (ln f) x . By symbolic computation with Maple, lump solutions, rationally localized in all directions in the space, to the GKP equation are presented. The obtained lump solutions contain a set of six free parameters, four of which should satisfy a nonzero determinant condition. As special examples, six particular lump solutions are constructed and depicted with t = 1. [ABSTRACT FROM AUTHOR]

Two Sawada–Kotera-like equations are introduced by the generalized bilinear operators D p associated with two prime numbers p = 3 and p = 5 , respectively. Rational solutions of the two presented Sawada–Kotera-like equations are generated by searching polynomial solutions of the corresponding two generalized bilinear equations. [ABSTRACT FROM AUTHOR]

We aim to construct exact and explicit solutions to a generalized Bogoyavlensky-Konopelchenko equation through the Maple computer algebra system. The considered nonlinear equation is transformed into a Hirota bilinear form, and symbolic computations are made for solving both the nonlinear equation and the corresponding bilinear equation. A few classes of exact and explicit solutions are generated from different ansätze on solution forms, including traveling wave solutions, two-wave solutions, and polynomial solutions. [ABSTRACT FROM AUTHOR]

Integrable couplings are associated with non-semisimple Lie algebras. In this paper, we propose a new method to generate new integrable systems through making perturbation in matrix spectral problems for integrable couplings, which is called the "completion process of integrable couplings." As an example, the idea of construction is applied to the Ablowitz-Kaup-Newell-Segur integrable coupling. Each equation in the resulting hierarchy has a bi-Hamiltonian structure furnished by the component-trace identity. [ABSTRACT FROM AUTHOR]

A kind of novel reduced nonlocal integrable mKdV equations of odd order is presented by taking two group reductions of the AKNS matrix spectral problems. One reduction is local, replacing the spectral parameter with its negative and the other is nonlocal, replacing the spectral parameter with its negative complex conjugate. Based on distribution of eigenvalues, soliton solutions are generated from the corresponding reflectionless Riemann–Hilbert problems. [ABSTRACT FROM AUTHOR]

We conduct two nonlocal group reductions of the AKNS matrix spectral problems to generate a class of nonlocal reverse-spacetime integrable mKdV equations. One reduction replaces the spectral parameter with its negative complex conjugate while the other does not change the spectral parameter. Beginning with the specific distribution of eigenvalues, we construct soliton solutions by solving the corresponding generalized Riemann-Hilbert problems with the identity jump matrix, where eigenvalues could equal adjoint eigenvalues. [ABSTRACT FROM AUTHOR]

Based on the Orlov and Shulman’s M operator, the additional symmetries and the string equation of the CKP hierarchy are established, and then the higher order constraints on L l are obtained. In addition, the generating function and some properties are also given. In particular, the additional symmetry flows form a new infinite dimensional algebra $$W^C_{1+\infty}$$ , which is a subalgebra of W 1+∞. [ABSTRACT FROM AUTHOR]