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Trilinear equations, Bell polynomials, and resonant solutions

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Frontiers of mathematics in China 2013年第5期8卷 1139-1156页

作者： wen-xiu ma Department of mathematics and Statistics University of South FloridaTampa FL 33620-5700 USA

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.

关键词： Trilinear differential equation Bell polynomial superposition principle

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

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Communications in Theoretical Physics 2024年第7期76卷 1-8页

作者： wen-xiu ma Department of mathematics Zhejiang Normal UniversityJinhua 321004China Department of mathematics King Abdulaziz UniversityJeddah 21589Saudi Arabia Department of mathematics and Statistics University of South Florida TampaFL 33620-5700United States of America material Science Innovation and Modelling Department of Mathematical SciencesNorth-West UniversityMafikeng CampusMmabatho 2735South Africa

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 *** specific examples,consisting of novel generalized combined nonlinear Schrodinger equations and modified Korteweg-de Vries equations,are given.

关键词： integrable hierarchy Hamiltonian structure zero curvature equation Lax pair matrix spectral problem

Linear superposition of Wronskian rational solutions to the KdV equation

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Communications in Theoretical Physics 2021年第6期73卷 1-5页

作者： wen-xiu ma Department of mathematics Zhejiang Normal UniversityJinhua 321004ZhejiangChina Department of mathematics King Abdulaziz UniversityJeddah 21589Saudi Arabia Department of mathematics and Statistics University of South FloridaTampaFL 33620-5700United States of America School of mathematical and Statistical Sciences North-West UniversityMafikeng CampusPrivate Bag X2046Mmabatho 2735South Africa

A linear superposition is studied for Wronskian rational solutions to the Kd V equation,which include rogue wave *** is proved that it is equivalent to a polynomial identity that an arbitrary linear combination of two Wronskian polynomial solutions with a difference two between the Wronskian orders is again a solution to the bilinear Kd V *** is also conjectured that there is no other rational solutions among general linear superpositions of Wronskian rational solutions.

关键词： soliton equation Wronskian solution rational solution rogue wave the KdV equation

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

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Chinese Physics Letters 2022年第10期39卷 1-6页

作者： wen-xiu ma Department of mathematics Zhejiang Normal UniversityJinhua 321004China Department of mathematics King Abdulaziz UniversityJeddah 21589Saudi Arabia Department of mathematics and Statistics University of South FloridaTampaFL 33620-5700USA School of mathematical and Statistical Sciences North-West UniversityMafikeng CampusPrivate Bag X2046Mmabatho 2735South Africa

We construct matrix integrable fourth-order nonlinear Schrödinger equations through reducing the Ablowitz-Kaup-Newell-Segur matrix eigenvalue *** 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 *** solutions are computed for three illustrative examples of scalar and two-component integrable fourth-order nonlinear Schrödinger equations.

关键词： eigenvalue integrable adjoint

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

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Frontiers of mathematics in China 2019年第3期14卷 619-629页

作者： wen-xiu ma Department of mathematics Zhejiang Normal University Jinhua 321004 China Department of mathematics King Abdulaziz University Jeddah Saudi Arabia Department of mathematics and Statistics University of South Florida Tampa FL 33620 USA College of mathematics and Physics Shanghai University of Electric Power Shanghai 200090 China College of mathematics and Systems Science Shandong University of Science and Technology Qingdao 266590 China Department of mathematical Sciences North-West University Mafikeng Campus Mmabatho 2735 South Africa

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.

关键词： Symbolic computation lump solution interaction solution

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

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Frontiers of mathematics in China 2020年第3期15卷 435-450页

作者： Sumayah BATWA wen-xiu ma Department of mathematics King Abdulaziz UniversityJeddahSaudi Arabia Department of mathematics Zhejiang Normal UniversityJinhua 321004China Department of mathematics and Statistics University of South FloridaTampaFL 33620USA School of mathematics South China University of TechnologyGuangzhou 510640China

Lump solutions are one of important solutions to partial differential equations,both linear and *** paper aims to show that a Hietarinta-type fourth-order nonlinear term can create lump solutions with second-order linear dispersive *** key is a Hirota bilinear *** solutions are constructed via symbolic computations with maple,and specific reductions of the resulting lump solutions are *** 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.

关键词： Soliton equation lump solution symbolic computation Hirota derivative dispersion relation

AN ABLOWITZ-LADIK INTEGRABLE LATTICE HIERARCHY WITH MULTIPLE POTENTIALS

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Acta mathematica Scientia 2020年第3期40卷 670-678页

作者： wen-xiu ma School of mathematics South China University of TechnologyGuangzhou 510640China Department of mathematics King Abdulaziz UniversityJeddahSaudi Arabia Department of mathematics and Statistics University of South FloridaTampaFL 33620USA Department of mathematics Zhejiang Normal UniversityJinhua 321004China College of mathematics and Systems Science Shandong University of Science and TechnologyQingdao 266590China Department of mathematical Sciences North-West UniversityMafikeng CampusMmabatho 2735South Africa

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

关键词： Integrable lattice discrete spectral problem symmetry and conserved functional

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

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Acta mathematica Scientia 2019年第2期39卷 509-523页

作者： wen-xiu ma Department of mathematics Zhejiang Normal University Department of mathematics King Abdulaziz University Department of mathematics and Statistics University of South Florida College of mathematics and Systems Science Shandong University of Science and Technology College of mathematics and Physics Shanghai University of Electric Power International Institute for Symmetry Analysis and mathematical Modelling Department of Mathematical Sciences North-West University Mafikeng Campus

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.

关键词： integrable hierarchy Riemann-Hilbert problem soliton solution

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

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Acta mathematica Scientia 2019年第2期39卷 498-508页

作者： wen-xiu ma College of mathematics and Physics Shanghai University of Electric Power Department of mathematics King Abdulaziz University Department of mathematics and Statistics University of South Florida Department of mathematics Zhejiang Normal University College of mathematics and Systems Science Shandong University of Science and Technology International Institute for Symmetry Analysis and mathematical Modelling Department of Mathematical Sciences North-West University Mafikeng Campus

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.

关键词： symbolic computation lump solution: interaction solution