An Eight Component Integrable Hamiltonian Hierarchy from a Reduced Seventh-Order Matrix Spectral Problem

We present an eight component integrable Hamiltonian hierarchy, based on a reduced seventh order matrix spectral problem, with the aim of aiding the study and classification of multicomponent integrable models and their underlying mathematical structures. The zero-curvature formulation is the tool to construct a recursion operator from the spatial matrix problem. The second and third set of integrable equations present integrable nonlinear Schrödinger and modified Korteweg-de Vries type equations, respectively. The trace identity is used to construct Hamiltonian structures, and the first three Hamiltonian functionals so generated are computed.

An Intelligent MCGDM Model in Green Suppliers Selection Using Interactional Aggregation Operators for Interval-Valued Pythagorean Fuzzy Soft Sets

Green supplier selection is an important debate in green supply chain management(GSCM),attracting global attention from scholars,especially companies and policymakers.Companies frequently search for new ideas and strategies to assist them in realizing sustainable development.Because of the speculative character of human opinions,supplier selection frequently includes unreliable data,and the interval-valued Pythagorean fuzzy soft set(IVPFSS)provides an exceptional capacity to cope with excessive fuzziness,inconsistency,and inexactness through the decision-making procedure.The main goal of this study is to come up with new operational laws for interval-valued Pythagorean fuzzy soft numbers(IVPFSNs)and create two interaction operators-the intervalvalued Pythagorean fuzzy soft interaction weighted average(IVPFSIWA)and the interval-valued Pythagorean fuzzy soft interaction weighted geometric(IVPFSIWG)operators,and analyze their properties.These operators are highly advantageous in addressing uncertain problems by considering membership and non-membership values within intervals,providing a superior solution to other methods.Moreover,specialist judgments were calculated by the MCGDM technique,supporting the use of interaction AOs to regulate the interdependence and fundamental partiality of green supplier assessment aspects.Lastly,a statistical clarification of the planned method for green supplier selection is presented.

RIEMANN-HILBERT PROBLEMS OF A SIX-COMPONENT MKDV SYSTEM AND ITS SOLITON SOLUTIONS

Based on a 4 x 4 matrix spectral problem, an AKNS soliton hierarchy with six potentials is generated. Associated with this spectral problem, a kind of Riemann-Hilbert problems is formulated for a six-component system of mKdV equations in the resulting AKNS hierarchy. Soliton solutions to the considered system of coupled mKdV equations are computed, through a reduced Riemann-Hilbert problem where an identity jump matrix is taken.

Matrix Integrable Fourth-Order Nonlinear Schr?dinger Equations and Their Exact Soliton Solutions

We construct matrix integrable fourth-order nonlinear Schrödinger equations through reducing the Ablowitz-Kaup-Newell-Segur matrix eigenvalue problems.Based on properties of eigenvalue and adjoint eigenvalue problems,we solve the corresponding reflectionless Riemann-Hilbert problems,where eigenvalues could equal adjoint eigenvalues,and formulate their soliton solutions via those reflectionless Riemann-Hilbert problems.Soliton solutions are computed for three illustrative examples of scalar and two-component integrable fourth-order nonlinear Schrödinger equations.

A combined Liouville integrable hierarchy associated with a fourth-order matrix spectral problem

This paper aims to propose a fourth-order matrix spectral problem involving four potentials and generate an associated Liouville integrable hierarchy via the zero curvature formulation.A bi-Hamiltonian formulation is furnished by applying the trace identity and a recursion operator is explicitly worked out,which exhibits the Liouville integrability of each model in the resulting hierarchy.Two specific examples,consisting of novel generalized combined nonlinear Schrodinger equations and modified Korteweg-de Vries equations,are given.

LUMP AND INTERACTION SOLUTIONS TO LINEAR (4+1)-DIMENSIONAL PDES

Taking a class of linear(4+1)-dimensional partial differential equations as examples, we would like to show that there exist lump solutions and interaction solutions in(4+1)-dimensions. We will compute abundant lump solutions and interaction solutions to the considered linear(4+1)-dimensional partial differential equations via symbolic computations,and plot three specific solutions with Maple plot tools, which supplements the existing literature on lump, rogue wave and breather solutions and their interaction solutions in soliton theory.

Lump-type solutions of a generalized Kadomtsev–Petviashvili equation in(3+1)-dimensions

Through the Hirota bilinear formulation and the symbolic computation software Maple, we construct lump-type solutions for a generalized(3+1)-dimensional Kadomtsev-Petviashvili(KP) equation in three cases of the coefficients in the equation. Then the sufficient and necessary conditions to guarantee the analyticity of the resulting lump-type solutions(or the positivity of the corresponding quadratic solutions to the associated bilinear equation) are discussed. To illustrate the generality of the obtained solutions, two concrete lump-type solutions are explicitly presented, and to analyze the dynamic behaviors of the solutions specifically, the three-dimensional plots and contour profiles of these two lump-type solutions with particular choices of the involved free parameters are well displayed.

AN ABLOWITZ-LADIK INTEGRABLE LATTICE HIERARCHY WITH MULTIPLE POTENTIALS

Within the zero curvature formulation,a hierarchy of integrable lattice equations is constructed from an arbitrary-order matrix discrete spectral problem of Ablowitz-Ladik type.The existence of infinitely many symmetries and conserved functionals is a consequence of the Lax operator algebra and the trace identity.When the involved two potential vectors are scalar,all the resulting integrable lattice equations are reduced to the standard Ablowitz-Ladik hierarchy.

Interaction solutions to Hirota-Satsuma-Ito equation in (2+1)-dimensions

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.

N-fold Darboux Transformation for Integrable Couplings of AKNS Equations

For the integrable couplings of Ablowitz-Kaup-Newell-Segur(ICAKNS) equations, N-fold Darboux transformation(DT) TN, which is a 4 × 4 matrix, is constructed in this paper. Each element of this matrix is expressed by a ratio of the(4N + 1)-order determinant and 4N-order determinant of eigenfunctions. By making use of these formulae,the determinant expressions of N-transformed new solutions p^([N ]), q^([N ]), r^([N ])and s^([N ])are generated by this N-fold DT.Furthermore, when the reduced conditions q =-p*and s =-r*are chosen, we obtain determinant representations of N-fold DT and N-transformed solutions for the integrable couplings of nonlinear Schr?dinger(ICNLS) equations.Starting from the zero seed solutions, one-soliton solutions are explicitly given as an example.

Non-Semisimple Lie Algebras of Block Matrices and Applications to Bi-Integrable Couplings

We propose a class of non-semisimple matrix loop algebras consisting of 3×3 block matrices,and form zero curvature equations from the presented loop algebras to generate bi-integrable couplings.Applications are made for the AKNS soliton hierarchy and Hamiltonian structures of the resulting integrable couplings are constructed by using the associated variational identities.

A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential

It is known that the solution to a Cauchy problem of linear differential equations:x'(t)=A(t)x(t),with x(t0)=x0,can be presented by the matrix exponential as exp(∫_(t0)^(t)A(s)ds)x0,if the commutativity condition for the coefficient matrix A(t)holds:[∫_(t0)^(t)A(s)ds,A(t)]=0.A natural question is whether this is true without the commutativity condition.To give a definite answer to this question,we present two classes of illustrative examples of coefficient matrices,which satisfy the chain rule d/dt exp(∫_(t0)^(t)A(s)ds)=A(t)exp(∫_(t0)^(t)A(s)ds),but do not possess the commutativity condition.The presented matrices consist of finite-times continuously differentiable entries or smooth entries.

Integrable nonlocal PT-symmetric generalized so(3,R)-mKdV equations

Based on a soliton hierarchy associated with so(3,R),we construct two integrable nonlocal PT-symmetric generalized mKdV equations.The key step is to formulate two nonlocal reverse-spacetime similarity transformations for the involved spectral matrix,and therefore,integrable nonlocal complex and real reverse-spacetime generalized so(3,R)-mKdV equations of fifth-order are presented.The resulting reduced nonlocal integrable equations inherit infinitely many commuting symmetries and conservation laws.

Dynamical rational solutions and their interaction phenomena for an extended nonlinear equation

In this paper,we analyze the extended Bogoyavlenskii-Kadomtsev-Petviashvili(eBKP)equation utilizing the condensed Hirota's approach.In accordance with a logarithmic derivative transform,we produce solutions for single,double,and triple M-lump waves.Additionally,we investigate the interaction solutions of a single M-lump with a single soliton,a single M-lump with a double soliton,and a double M-lump with a single soliton.Furthermore,we create sophisticated single,double,and triple complex soliton wave solutions.The extended Bogoyavlenskii-Kadomtsev-Petviashvili equation describes nonlinear wave phenomena in fluid mechanics,plasma,and shallow water theory.By selecting appropriate values for the related free parameters we also create three-dimensional surfaces and associated counter plots to simulate the dynamical characteristics of the solutions offered.

Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation

A （2 ＋ 1）-dimensional generalized Bogoyavlensky-Konopelchenko equation that possesses a Hirota bilinear form is considered. Starting with its Hirota bilinear form, a class of explicit lump solutions is computed through conducting symbolic computations with Maple, and a few plots of a specific presented lump solution are made to shed light on the characteristics of lumps. The result provides a new example of （2 ＋ 1）-dimensional nonlinear partial differential equations which possess lump solutions.

Investigation of new solutions for an extended(2+1)-dimensional Calogero-Bogoyavlenskii-Schif equation

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.

Trilinear equations, Bell polynomials, and resonant solutions

A class of trilinear differential operators is introduced through a technique of assigning signs to derivatives and used to create trilinear differential equations. The resulting trilinear differential operators and equations are characterized by the Bell polynomials, and the superposition principle is applied to the construction of resonant solutions of exponential waves. Two illustrative examples are made by an algorithm using weights of dependent variables.

Lump solutions to a generalized Hietarinta-type equation via symbolic computation

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.

Reduced nonlocal integrable mKdV equations of type(-λ, λ) and their exact soliton solutions

We conduct two group reductions of the Ablowitz-Kaup-Newell-Segur matrix spectral problems to present a class of novel reduced nonlocal reverse-spacetime integrable modified Korteweg-de Vries equations. One reduction is local, replacing the spectral parameter with its negative and the other is nonlocal, replacing the spectral parameter with itself. Then by taking advantage of distribution of eigenvalues, we generate soliton solutions from the reflectionless Riemann-Hilbert problems, where eigenvalues could equal adjoint eigenvalues.

Lump Solutions for Two Mixed Calogero-Bogoyavlenskii-Schiff and Bogoyavlensky-Konopelchenko Equations

Based on the Hirota bilinear operators and their generalized bilinear derivatives, we formulate two new(2+1)-dimensional nonlinear partial differential equations, which possess lumps. One of the new nonlinear differential equations includes the generalized Calogero-Bogoyavlenskii-Schiff equation and the generalized BogoyavlenskyKonopelchenko equation as particular examples, and the other has the same bilinear form with different Dp-operators.A class explicit lump solutions of the new nonlinear differential equation is constructed by using the Hirota bilinear approaches. A specific case of the presented lump solution is plotted to shed light on the charateristics of the lump.