Article
Physics, Multidisciplinary
Asela Abeya, Barbara Prinari, Gino Biondini, Panos G. Kevrekidis
Summary: The article investigates soliton solutions and interactions in a system of coupled nonlinear Schrodinger equations modeling the dynamics in one-dimensional repulsive Bose-Einstein condensates. Multi-soliton solutions are compactly represented using the Inverse Scattering Transform. The study includes a detailed characterization of canonical and non-canonical forms of solutions, as well as the classification of one-soliton solutions and their relationships to spectral data in the Inverse Scattering Transform.
EUROPEAN PHYSICAL JOURNAL PLUS
(2021)
Article
Mathematics, Applied
Nan Liu, Jia-Dong Yu
Summary: The inverse scattering transform approach is applied to investigate the derivative nonlinear Schrodinger equation with nonzero boundary conditions and triple zeros of analytical scattering coefficients. A matrix Riemann-Hilbert problem associated with the equation is constructed based on the analytical, symmetric, and asymptotic properties of eigenfunctions. The reconstruction formula for the potential is obtained by solving the Riemann-Hilbert problem. The explicit expression of the reflectionless potential with triple poles for the nonzero boundary conditions is derived using determinants.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Zong-Bing Yu, Chenghao Zhu, Jian-Shi Zhao, Li Zou
Summary: This work investigates the integrable general three-component nonlinear Schrodinger equation, and formulates a 4 x 4 matrix spectral problem and a Riemann-Hilbert problem to obtain the multi-soliton solutions for this equation.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Junyi Zhu, Xueling Jiang, Xueru Wang
Summary: The Dbar dressing method is expanded to study the focusing/defocusing nonlinear Schrodinger (NLS) equation with nonzero boundary condition. A special type of complex function is considered, which is meromorphic outside an annulus with center 0 and satisfies a local Dbar problem inside the annulus. The theory of such function is extended to construct the Lax pair of the NLS equation with nonzero boundary condition. In this procedure, the relation between the NLS potential and the solution of the Dbar problem is established. A certain distribution for the Dbar problem is introduced to obtain the focusing/defocusing NLS equation and the conservation laws.
ANALYSIS AND MATHEMATICAL PHYSICS
(2023)
Article
Mathematics, Applied
K. T. Gruner
Summary: Soliton solutions for the nonlinear Schrodinger equation on two half-lines connected via integrable defect conditions are reconsidered using the dressing method. N-soliton solutions are explicitly constructed and it is shown that the solitons are transmitted through the defect independently of one another.
Article
Physics, Multidisciplinary
Jin-Jin Mao, Shou-Fu Tian, Tian-Zhou Xu, Lin-Fei Shi
Summary: This paper studies the inverse scattering transforms of the inhomogeneous fifth-order nonlinear Schrodinger equation with different boundary conditions. Bound-state soliton solutions with zero boundary conditions are derived for the first time using the residue theorem and Laurent's series. The Riemann-Hilbert problem for the inhomogeneous fifth-order nonlinear Schrodinger equation with nonzero boundary conditions is revealed by combining with the robust inverse scattering transform. Moreover, new rogue wave solutions are found using the Darboux transformation based on the resulting Riemann-Hilbert problem. The unreported dynamic characteristics of the corresponding solutions are further analyzed and displayed through selected graphs.
COMMUNICATIONS IN THEORETICAL PHYSICS
(2022)
Article
Mathematics, Applied
Nan Liu, Zuxing Xuan, Jinyi Sun
Summary: The triple-pole soliton solutions to the classical DNLS equation with zero boundary conditions at infinity are constructed using the inverse scattering transform method. Detailed analysis of the discrete spectrum in the direct problem is presented under the condition of the scattering coefficient having N-triple zeros. The general solution of the DNLS equation and an explicit N-triple-poles soliton formula for the reflectionless potential are obtained through the matrix Riemann-Hilbert problem with triple poles.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Guixian Wang, Bo Han
Summary: In this work, the discrete modified Korteweg-de Vries equation under nonzero boundary conditions is studied using the robust inverse scattering transform. The Jost solutions, scattering matrix, and their properties are presented based on the Lax pair. Riemann-Hilbert problems and Darboux matrix are constructed using the robust inverse scattering transform, and rational solutions are deduced with detailed graphical characteristics exhibited by choosing suitable parameters. The results are useful for explaining related nonlinear wave phenomena.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Vincent Caudrelier, Nicolas Crampe, Carlos Mbala Dibaya
Summary: This study analyzes the focusing nonlinear Schrodinger equation on the half-line with time-dependent boundary conditions using Backlund transformations and nonlinear method of images. Two possible classes of solutions are identified, one similar to Robin boundary conditions where solitons are reflected at the boundary, and a new regime where one soliton can either be absorbed or emitted by the boundary. The unique feature of time-dependent integrable boundary conditions is demonstrated.
STUDIES IN APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Xinxin Ma, Junyi Zhu
Summary: This article studies the two-component Gerdjikov-Ivanov equation with nonzero boundary conditions using the inverse scattering transform. A set of analytic eigenfunctions is obtained through the associated adjoint problem. Three symmetry conditions are discussed to constrain the scattering data. The behavior of the Jost functions and the scattering matrix at branch points is discussed. The inverse scattering problem is formulated using a matrix Riemann-Hilbert problem. The trace formula and the asymptotic phase difference for the potential are obtained in terms of the scattering data. The solitons classification is described in detail, including the discovery of the dark-dark soliton. The interactions of solitons such as dark-bright, bright-bright, breather-breather, M(-type)-W(-type) solitons are generated when the discrete eigenvalues are off the circle.
STUDIES IN APPLIED MATHEMATICS
(2023)
Article
Engineering, Mechanical
Yu Xiao, Qiaozhen Zhu, Xing Wu
Summary: This paper presents the inverse scattering transform (IST) with Riemann-Hilbert problem (RHP) for a higher-order extended modified Korteweg-de Vries (emKdV) equation with zero/nonzero boundary conditions (Z/NZBC) at infinity. The research results show the formulas for multiple simple poles soliton and higher-order poles solution, as well as the detailed effect of the variable values of the third- and fifth-order dispersion coefficients on the dynamic structures.
NONLINEAR DYNAMICS
(2022)
Article
Physics, Mathematical
Wei-Qi Peng, Yong Chen
Summary: In this work, the double and triple pole soliton solutions for the Gerdjikov-Ivanov type of the derivative nonlinear Schrodinger equation with zero boundary conditions (ZBCs) and nonzero boundary conditions (NZBCs) are studied via the Riemann-Hilbert (RH) method. The RH problem (RHP) is constructed based on the analyticity, symmetry, and asymptotic behavior of the Jost function and scattering matrix under ZBCs and NZBCs. The general precise formulas of N-double and N-triple pole solutions corresponding to ZBCs and NZBCs are derived, and the dynamical behaviors for these solutions are discussed.
JOURNAL OF MATHEMATICAL PHYSICS
(2022)
Article
Mathematics, Applied
Jin-Jie Yang, Shou-Fu Tian, Zhi-Qiang Li
Summary: The Riemann-Hilbert problem is developed to study the focusing behavior of the nonlinear Schrodinger equation with multiple high-order poles. By employing Laurent expansion and solving an algebraic system, soliton solutions corresponding to the transmission coefficient are obtained.
PHYSICA D-NONLINEAR PHENOMENA
(2022)
Article
Mathematics, Applied
Deniz Bilman, Peter D. Miller
Summary: We demonstrate that a universal special solution of the focusing nonlinear Schrödinger equation, which has been shown to emerge in a certain near-field/large-order limit from soliton and Peregrine-like rogue wave solutions, can actually arise from an arbitrary background solution when subjected to a sequence of iterated Backlund transformations.
PHYSICA D-NONLINEAR PHENOMENA
(2022)
Article
Engineering, Mechanical
Yong Chen, Xue-Wei Yan
Summary: This study investigates the Riemann-Hilbert problem and soliton solutions to the high-order nonlinear Schrodinger equation with a matrix version through an equivalent spectral problem. By utilizing inverse scattering, a pair of Jost solutions satisfying the asymptotic conditions and the matrix spectral problem are obtained, leading to the matrix Riemann-Hilbert problem. Different soliton solutions are theoretically and graphically presented based on the two types of zero structures of det(P+) in the case of reflection-less.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics, Applied
Aikaterini Gkogkou, Barbara Prinari, Bao-Feng Feng, A. David Trubatch
Summary: This paper develops the Riemann-Hilbert approach to the inverse scattering transform for the complex coupled short-pulse equation and obtains soliton solutions. The zoology of soliton solutions for the coupled system is richer than in the scalar case, including fundamental solitons, fundamental breathers, and composite breathers.
STUDIES IN APPLIED MATHEMATICS
(2022)
Article
Physics, Multidisciplinary
Asela Abeya, Gino Biondini, Barbara Prinari
Summary: We characterize the initial value problems for the defocusing Manakov system with nonzero background and spatial parity symmetry. The study reveals that the symmetries induced by parity are more complicated than in the scalar case, and the dark solitons and dark-bright solitons have specific symmetric properties.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2022)
Article
Multidisciplinary Sciences
Samuel J. Ryskamp, Mark A. Hoefer, Gino Biondini
Summary: In this study, resonant Y-shaped soliton solutions to the KPII equation are modeled as shock solutions, demonstrating a reduction to a one-dimensional system in the zero dispersion limit. The results have practical applications in analytically describing the dynamics of the Mach reflection problem, showing excellent agreement with direct numerical simulations.
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
(2022)
Article
Mathematics, Applied
Gino Biondini, Dmitri Kireyev
Summary: This paper investigates the periodic, traveling wave solutions of all four versions of the Davey-Stewartson system and describes these solutions in terms of elliptic functions. Special reductions and limiting cases, including harmonic limits, soliton limits, and one-dimensional solutions, are also explicitly discussed.
EAST ASIAN JOURNAL ON APPLIED MATHEMATICS
(2022)
Article
Physics, Multidisciplinary
Gino Biondini, Dmitri Kireyev, Ken-ichi Maruno
Summary: The paper discusses a large class of nonsingular multi-soliton solutions of the defocusing Davey-Stewartson II equation, studying their asymptotics, interaction patterns, and identifying several subclasses of solutions. Many solutions describe phenomena of soliton resonance and web structure, with some exhibiting unique features like V-shape solutions and soliton reconnection.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2022)
Article
Physics, Multidisciplinary
Asela Abeya, Gino Biondini, Mark A. Hoefer
Summary: The Whitham modulation equations for the defocusing nonlinear Schrödinger (NLS) equation are derived using a two-phase ansatz and period-averaging the conservation laws. The resulting equations preserve the symmetries of the NLS equation and can be easily generalized from two to three spatial dimensions. The transformation to Riemann-type variables, harmonic and soliton limits, and reduction to the radial NLS equation are discussed. The extension to higher spatial dimensions is briefly outlined. The obtained NLS-Whitham equations can be applied in studying large amplitude wavetrains in nonlinear photonics and matter waves.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2023)
Article
Mathematics, Applied
Vincent Caudrelier, Aikaterini Gkogkou, Barbara Prinari
Summary: This paper investigates the propagation of ultrashort optical pulses in nonlinear birefringent fibers, using the dressing method and Darboux matrices for various types of solitons. The study reveals the explicit formulas for the polarization shift of fundamental solitons and interactions of other types of solitons.
STUDIES IN APPLIED MATHEMATICS
(2023)
Review
Mathematics, Applied
Barbara Prinari
Summary: This topical review paper provides a survey of classical and more recent results on the inverse scattering transform (IST) for one-dimensional scalar, vector and square matrix NLS systems with physically relevant non-zero boundary conditions at space infinity. It discusses new developments and applications, and offers perspectives on future directions in this field.
JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS
(2023)
Review
Mathematics, Applied
Gino Biondini, Alexander Chernyavsky
Summary: In this paper, we derive the Whitham modulation equations for the Zakharov-Kuznetsov equation and study the stability of its periodic traveling wave solutions. We find that these solutions are linearly unstable with respect to transversal perturbations, and the growth rate of perturbations in the long wave limit can be obtained analytically. The predictions of Whitham modulation theory are validated by numerical calculations and are in excellent agreement with the linearized equation. Furthermore, we extend the stability analysis to periodic waves traveling in arbitrary directions and to perturbations that are not purely transversal, and determine the corresponding domains of stability and instability.
STUDIES IN APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Asela Abeya, Gino Biondini, Barbara Prinari
Summary: This paper develops the inverse scattering transform (IST) for the defocusing Manakov system with non-zero boundary conditions at infinity, involving non-parallel asymptotic polarization vectors. A uniformization variable is used to eliminate the square root branching, and the adjoint Lax pair is applied to overcome the non-analyticity problem of some Jost eigenfunctions. The inverse problem is formulated as a matrix Riemann-Hilbert problem (RHP), and methods for converting the RHP into linear algebraic-integral equations are presented.
EAST ASIAN JOURNAL ON APPLIED MATHEMATICS
(2022)
Article
Optics
A. Romero-Ros, G. C. Katsimiga, S. I. Mistakidis, B. Prinari, G. Biondini, P. Schmelcher, P. G. Kevrekidis
Summary: This study demonstrates the potential for realizing the Peregrine soliton structure through theoretical and numerical simulations based on the experimental realization of the Townes soliton in a two-component Bose-Einstein condensate. By initializing the condensate with a suitable spatial density pattern, the robust emergence of the Peregrine wave is observed in different scenarios. The study also reveals that narrower wave packets can lead to periodic revivals of the Peregrine soliton, while broader ones give rise to a cascade of Peregrine solitons arranged in a specific structure. The persistence of these rogue-wave structures is demonstrated in certain temperature regimes and in the presence of transversal excitations. The findings offer insights into the practical feasibility of generating and observing rogue waves in ultracold atom experimental settings.
Article
Optics
A. Romero-Ros, G. C. Katsimiga, P. G. Kevrekidis, B. Prinari, G. Biondini, P. Schmelcher
Summary: This work extends earlier findings on the creation of dark soliton trains in single-component BECs to two-component BECs, and obtains analytical expressions for the DB soliton solutions produced by a general initial configuration. It is found that the size of the initial box and the amount of filling directly affect the number, size, and velocity of the solitons, while the initial phase determines the parity of the solutions.
Article
Physics, Fluids & Plasmas
Gino Biondini, Antonio Moro, Barbara Prinari, Oleg Senkevich
Summary: In this study, the mean-field analog of the p-star model for homogeneous random networks is compared with the p-star model and its classical mean-field approximation in the thermodynamic regime. It is shown that the partition function of the mean-field model satisfies a sequence of partial differential equations known as the heat hierarchy, and the models connectance is obtained as a solution of a hierarchy of nonlinear viscous PDEs.
