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Article
Mathematics, Interdisciplinary Applications
Run-Fa Zhang et al.
Summary: In this work, new test functions are constructed by setting generalized activation functions in different artificial network models. The explicit solution of a generalized breaking soliton equation is solved using the bilinear neural network method. Rogue waves of the generalized breaking soliton equation are obtained using symbolic computing technology and displayed intuitively with the help of Maple software.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Engineering, Multidisciplinary
Wen-Xiu Ma
Summary: This paper analyzes N-soliton solutions and explores the Hirota N-soliton conditions for scalar (1 + 1)-dimensional equations within the Hirota bilinear formulation. An algorithm is proposed to verify the Hirota conditions by factoring out common factors in N wave vectors of the Hirota function and comparing degrees of involved polynomials. Applications to a class of generalized KdV equations and a class of generalized higher-order KdV equations are made, with proofs of the existence of N-soliton solutions provided for all equations in the two classes.
INTERNATIONAL JOURNAL OF NONLINEAR SCIENCES AND NUMERICAL SIMULATION
(2022)
Article
Mathematics, Applied
Wen-Xiu Ma
Summary: This study proposes a nonlocal real reverse-spacetime integrable hierarchies of PT symmetric matrix AKNS equations, achieved through nonlocal symmetry reductions on the potential matrix, to determine generalized Jost solutions. By applying the Sokhotski-Plemelj formula, the associated Riemann-Hilbert problems are transformed into integral equations of Gelfand-Levitan-Marchenko type. The Riemann-Hilbert problems corresponding to the reflectionless case are explicitly solved, presenting soliton solutions for the resulting nonlocal real reverse-spacetime integrable PT-symmetric matrix AKNS equations.
PHYSICA D-NONLINEAR PHENOMENA
(2022)
Article
Mathematics
Wenxiu Ma
Summary: This paper focuses on establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrodinger (NLS) hierarchies associated with higher-order matrix spectral problems. The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations. A new formulation of solutions to special Riemann-Hilbert problems with the identity jump matrix, corresponding to the reflectionless inverse scattering transforms, is proposed and applied to construction of soliton solutions to each system in the considered nonlocal reverse-time NLS hierarchies.
ACTA MATHEMATICA SCIENTIA
(2022)
Article
Mathematics, Applied
Xueru Wang et al.
Summary: The coupled nonlocal nonlinear Schrodinger equation was studied using the 2x2 Dbar problem, introducing two spectral transform matrices to define associated Dbar problems. Relationships between the coupled nonlocal NLS potential and the Dbar problem solution were constructed, and the spatial transform method was extended to derive the coupled nonlocal NLS equation and its conservation laws. The general nonlocal reduction of the coupled nonlocal NLS equation to the nonlocal NLS equation was discussed in detail, with explicit solutions derived.
STUDIES IN APPLIED MATHEMATICS
(2022)
Article
Mathematics
Wen-Xiu Ma
Summary: Reduced nonlocal matrix integrable modified Korteweg-de Vries (mKdV) hierarchies are obtained through two transpose-type group reductions in the matrix Ablowitz-Kaup-Newell-Segur (AKNS) spectral problems. Riemann-Hilbert problems and soliton solutions are formulated based on the reduced matrix spectral problems.
Article
Mathematics, Applied
Wen-Xiu Ma
Summary: Two nonlocal group reductions were used to generate a class of nonlocal reverse-spacetime integrable mKdV equations from the AKNS matrix spectral problems, leading to soliton solutions through solving corresponding generalized Riemann-Hilbert problems with the identity jump matrix.
JOURNAL OF GEOMETRY AND PHYSICS
(2022)
Article
Mathematics, Applied
Wen-Xiu Ma
Summary: A novel reduced nonlocal integrable mKdV equation of odd order is presented by taking two group reductions of the AKNS matrix spectral problems. Soliton solutions are generated from the corresponding reflectionless Riemann-Hilbert problems based on the distribution of eigenvalues.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Physics, Multidisciplinary
Abdul-Majid Wazwaz
Summary: This study investigates the integrable nonlinear (4+1)-dimensional Fokas equation using the simplified Hirota's method, revealing multiple soliton and complex soliton solutions, and confirming integrability through the Painlev'e integrability in the sense of WTC method. The results show that each set of multiple soliton solutions has a distinct physical structure.
WAVES IN RANDOM AND COMPLEX MEDIA
(2021)
Article
Mathematics, Applied
Jiguang Rao et al.
Summary: This study investigates the doubly localized rogue waves in the Fokas system, which are described by semi-rational solutions. There are two types of doubly localized rogue waves: line segment rogue waves and lump-type ones, both of which are truly localized in two-dimensional space and time. These waves serve as an extension of reported rogue waves in two-dimensional systems and hold great importance in nonlinear science and physical applications.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Lijuan Guo et al.
Summary: In this study, a family of eigenfunction solutions of the Lax pair of the KP1 equation is constructed, and higher-order rogue wave solutions are obtained using the binary Darboux transformation. It is conjectured that these solutions evolve in a triangular extreme wave pattern and provide insights into the fundamental understanding of extreme wave events in various physical systems.
PHYSICA D-NONLINEAR PHENOMENA
(2021)
Article
Mathematics, Applied
Jiguang Rao et al.
Summary: This study investigates doubly-localized two-dimensional rogue waves in the background of dark solitons or a constant, using the Kadomtsev-Petviashvili hierarchy reduction method and Hirota's bilinear technique. These waves are described by semi-rational solutions and illustrate resonant collisions between lumps or line rogue waves and dark solitons, resulting in them being doubly localized in two-dimensional space and time.
JOURNAL OF NONLINEAR SCIENCE
(2021)
Article
Mathematics, Applied
Charles Lester et al.
Summary: In this study, a class of solutions for the Kadomtsev-Petviashvili (KP)-I equation is constructed using a reduced version of the Grammian form of the tau-function. The solutions consist of linear periodic lump chains with distinct group and wave velocities, evolving into linear arrangements of lump chains. These solutions can be seen as the KP-I analogues of line-soliton solutions in KP-II, but with more general linear arrangements that allow for degenerate configurations such as parallel or superimposed lump chains. The interactions between lump chains and individual lumps are also discussed, along with the relationship between solutions obtained using reduced and regular Grammian forms.
STUDIES IN APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Jiguang Rao et al.
Summary: This study examines resonant collisions between localized lumps and line solitons of the KP-I equation, revealing that phase shifts can become indefinitely large during resonant collisions, leading to the emergence of rogue lumps. As the number of lumps increases, collisions become increasingly complex, with multiple lumps detaching from one or different line solitons.
STUDIES IN APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Yulei Cao et al.
Summary: In this paper, the PT-symmetric version of the Maccari system is introduced, along with various exact solutions obtained through different methods. Additionally, a new nonlocal Davey-Stewartson-type equation is derived,and solutions for breathers and rogue waves are obtained. The novel semirational solutions identified enrich the understanding of waves in nonlocal nonlinear systems.
STUDIES IN APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Ioannis Ioannou-Sougleridis et al.
STUDIES IN APPLIED MATHEMATICS
(2020)
Article
Engineering, Mechanical
Yulei Cao et al.
NONLINEAR DYNAMICS
(2020)
Article
Mathematics, Applied
Yulei Cao et al.
APPLIED MATHEMATICS LETTERS
(2018)
Article
Mathematics, Interdisciplinary Applications
Yulei Cao et al.
CHAOS SOLITONS & FRACTALS
(2018)
Article
Mathematics, Applied
Zhang YingNan et al.
SCIENCE CHINA-MATHEMATICS
(2015)
Article
Physics, Multidisciplinary
A. S. Fokas et al.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2012)
Article
Physics, Multidisciplinary
Engui Fan
Article
Physics, Mathematical
A. S. Fokas
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2009)
Article
Physics, Multidisciplinary
Yang Zheng-Zheng et al.
COMMUNICATIONS IN THEORETICAL PHYSICS
(2009)
Article
Mathematics, Applied
Wen-Xiu Ma et al.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2009)
Article
Mathematics, Applied
Xing-Biao Hu et al.
Article
Physics, Multidisciplinary
A. S. Fokas
PHYSICAL REVIEW LETTERS
(2006)
Article
Mathematics, Interdisciplinary Applications
F Lambert et al.
CHAOS SOLITONS & FRACTALS
(2001)
Article
Mathematics, Applied
AS Fokas et al.
PHYSICA D-NONLINEAR PHENOMENA
(2001)
