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Nonlinear Schrodinger Systems with Non-Zero Background: Inverse Scattering, Solitons and Modulation Theory.

Record Type:	Electronic resources : Monograph/item
Title/Author:	Nonlinear Schrodinger Systems with Non-Zero Background: Inverse Scattering, Solitons and Modulation Theory./
Author:	Abeya Ranasinghe Mudiyanselage, Asela Viraj.
Published:	Ann Arbor : ProQuest Dissertations & Theses, : 2022,
Description:	169 p.
Notes:	Source: Dissertations Abstracts International, Volume: 83-12, Section: B.
Contained By:	Dissertations Abstracts International83-12B.
Subject:	Mathematics. -
ISBN:	9798819393864

Nonlinear Schrodinger Systems with Non-Zero Background: Inverse Scattering, Solitons and Modulation Theory.
Abeya Ranasinghe Mudiyanselage, Asela Viraj.

Nonlinear Schrodinger Systems with Non-Zero Background: Inverse Scattering, Solitons and Modulation Theory. - Ann Arbor : ProQuest Dissertations & Theses, 2022 - 169 p.

Source: Dissertations Abstracts International, Volume: 83-12, Section: B.

Thesis (Ph.D.)--State University of New York at Buffalo, 2022.

This item must not be sold to any third party vendors.

Nonlinear waves are ubiquitous in nature, and are described by nonlinear wave equations, which are nonlinear partial differential equations of evolution type. Scalar, vector and matrix nonlinear Schrodinger (NLS) equations are universal models for the evolution of weakly nonlinear dispersive wave trains. As such, these equations appear in many physical contexts, such as deep water waves, nonlinear optics, acoustics, and Bose-Einstein condensation. These equations are also relevant for describing modulational instability and the formation of rogue waves. It is well known that these equations are completely integrable infinite-dimensional Hamiltonian systems, and possesses an infinite number of conserved quantities. As a result, the initial value problem (IVP) can in principle be solved by the inverse scattering transform (IST), a nonlinear analogue of the Fourier transform. When available, the IST provides an effective tool to study the behavior of solutions. However, the formulation of the IST is still in some cases an open problem.In many physical situations, the relevant models are (1+1)-dimensional equations (i.e., equations in one spatial and one temporal dimension). The (2+1)-dimensional nonlinear Schrodinger (2DNLS) equation (i.e., the NLS equation with two spatial dimensions)is also a universal model for the slowly varying amplitude of a nonlinear wave train in weakly nonlinear dispersive systems, and as such, it arises in many branches of physics.I n contrast to the above mentioned NLS-type equations in one spatial dimension, however, the 2DNLS equation is not integrable, and therefore one cannot make use of the IST to study its solutions. Therefore, to investigate small-dispersion limits and dispersive shock waves of the 2DNLS equation, one need to employ the Whitham modulation theory, which provides an asymptotic method for studying slowly varying periodic waves, and is basically a nonlinear analogue of Wentzel-Kramers-Brillouin (WKB) theory. This dissertation is concerned with the study of NLS-type equations, specifically, the defocusing Manakov system (also known as the two-component vector NLS equation), the matrix NLS equation and the 2DNLS equation. Specifically, this thesis presents the following original contributions:IST for the defocusing Manakov system with non-parallel boundary conditions at infinity.I formulate the IST for the defocusing Manakov system with non-zero boundary conditions at infinity comprising non-parallel states of polarizations. The formalism uses a uniformization variable to map two copies of the spectral plane into a single copy of the complex plane, thereby eliminating square root branching. The "adjoint'' Lax pair is also used to overcome the problem of non-analyticity of some of the Jost eigenfunctions. The inverse problem is formulated in term of a suitable matrix Riemann-Hilbert problem (RHP).The most significant difference in the IST compared to the case of parallel boundary conditions is the asymptotic behavior of the scattering coefficients, which affects the normalization of the eigenfunctions and the sectionally meromorphic matrix in the RHP. When the asymptotic polarization vectors are not orthogonal, two different methods are presented to convert the RHP into a set of linear algebraic-integral equations. When the asymptotic polarization vectors are orthogonal, however, only one of these methods is applicable. Finally, I show that, both in the case of orthogonal and non-orthogonal polarization vectors, no reflectionless potentials can exist, which implies that the problem does not admit pure soliton solutions. This work has been accepted for publication in the East Asian Journal on Applied Mathematics.Solitons and soliton interactions in repulsive spinor Bose-Einstein condensates with non-zero background.I characterize the soliton solutions and their interactions for a system of coupled evolution equations of NLS-type that models the dynamics in one-dimensional repulsive Bose-Einstein condensates with spin one, taking advantage of the representation of such model as a special reduction of a 2X2 matrix NLS system. Specifically, I study in detail the case in which solutions tend to a non-zero background at space infinities. First, I derive a compact representation for the multi-soliton solutions in the system using the IST. I introduce the notion of canonical form of a solution, corresponding to the case when the background as x to infinity is proportional to the identity. I show that solutions for which the asymptotic behavior at infinity is not proportional to the identity, referred to as being in non-canonical form, can be reduced to canonical form by unitary transformations that preserve the symmetric nature of the solution (physically corresponding to complex rotations of the quantization axes). Then I give a complete characterization of the two families of one-soliton solutions arising in this problem, corresponding to ferromagnetic and to polar states of the system, and I discuss how the physical parameters of the solitons for each family are related to the spectral data in the IST. Finally, I discuss two-soliton interactions and presented a complete classification of the possible scenarios that can arise depending on whether either soliton is of ferromagnetic or polar type. This work was published in European Physical Journal Plus.Manakov system with parity symmetry on nonzero background and associated boundary value problems.I characterize the soliton solutions of the Manakov system with parallel non-zero boundary condition when the initial conditions have well-defined spatial parity. In particular, I derive the extra symmetries of the Jost eigenfunctions and the scattering coefficients induced by the parity of the initial conditions. Using the resulting symmetries, I obtain the parity symmetry of the discrete spectrum and the associated norming constants. I show that the discrete eigenvalues come in either quartets on the circle of radius $q_o$ (with $q_o$ being the background amplitude of the solution), or in symmetric octets, and I obtain explicit relations for the norming constants associated with symmetric eigenvalues. Furthermore, I obtain conditions for the existence of self-symmetric eigenvalues, which give arise to an extra symmetry of the corresponding norming constants. I validate the results by comparison with numerical simulations. This work has been accepted for publication in the Journal of Physics A.Whitham modulation theory for the (2+1)-dimensional defocusing NLS equation.I derive the Whitham modulation equations for the defocusing 2DNLS equation using a two-phase ansatz for the periodic solutions and the conservation laws of the 2DNLS equation. The resulting equations are written in vector form, which allows one to immediately verify that the modulation equations preserve the rotational invariance of the 2DNLS equation, as well as the invariance with respect to scaling and Galilean transformations. The transformation to Riemann-type variables is also carried out, and the harmonic and soliton limits are explicitly considered, as is the reduction to the Whitham equations of the radial NLS equation. This work is currently being finalized for submission.

ISBN: 9798819393864Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Integrable systems

Nonlinear Schrodinger Systems with Non-Zero Background: Inverse Scattering, Solitons and Modulation Theory.
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100 1 $a Abeya Ranasinghe Mudiyanselage, Asela Viraj. $3 3689179
245 1 0 $a Nonlinear Schrodinger Systems with Non-Zero Background: Inverse Scattering, Solitons and Modulation Theory.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2022
300 $a 169 p.
500 $a Source: Dissertations Abstracts International, Volume: 83-12, Section: B.
500 $a Advisor: Biondini, Gino;Prinari, Barbara.
502 $a Thesis (Ph.D.)--State University of New York at Buffalo, 2022.
506 $a This item must not be sold to any third party vendors.
520 $a Nonlinear waves are ubiquitous in nature, and are described by nonlinear wave equations, which are nonlinear partial differential equations of evolution type. Scalar, vector and matrix nonlinear Schrodinger (NLS) equations are universal models for the evolution of weakly nonlinear dispersive wave trains. As such, these equations appear in many physical contexts, such as deep water waves, nonlinear optics, acoustics, and Bose-Einstein condensation. These equations are also relevant for describing modulational instability and the formation of rogue waves. It is well known that these equations are completely integrable infinite-dimensional Hamiltonian systems, and possesses an infinite number of conserved quantities. As a result, the initial value problem (IVP) can in principle be solved by the inverse scattering transform (IST), a nonlinear analogue of the Fourier transform. When available, the IST provides an effective tool to study the behavior of solutions. However, the formulation of the IST is still in some cases an open problem.In many physical situations, the relevant models are (1+1)-dimensional equations (i.e., equations in one spatial and one temporal dimension). The (2+1)-dimensional nonlinear Schrodinger (2DNLS) equation (i.e., the NLS equation with two spatial dimensions)is also a universal model for the slowly varying amplitude of a nonlinear wave train in weakly nonlinear dispersive systems, and as such, it arises in many branches of physics.I n contrast to the above mentioned NLS-type equations in one spatial dimension, however, the 2DNLS equation is not integrable, and therefore one cannot make use of the IST to study its solutions. Therefore, to investigate small-dispersion limits and dispersive shock waves of the 2DNLS equation, one need to employ the Whitham modulation theory, which provides an asymptotic method for studying slowly varying periodic waves, and is basically a nonlinear analogue of Wentzel-Kramers-Brillouin (WKB) theory. This dissertation is concerned with the study of NLS-type equations, specifically, the defocusing Manakov system (also known as the two-component vector NLS equation), the matrix NLS equation and the 2DNLS equation. Specifically, this thesis presents the following original contributions:IST for the defocusing Manakov system with non-parallel boundary conditions at infinity.I formulate the IST for the defocusing Manakov system with non-zero boundary conditions at infinity comprising non-parallel states of polarizations. The formalism uses a uniformization variable to map two copies of the spectral plane into a single copy of the complex plane, thereby eliminating square root branching. The "adjoint'' Lax pair is also used to overcome the problem of non-analyticity of some of the Jost eigenfunctions. The inverse problem is formulated in term of a suitable matrix Riemann-Hilbert problem (RHP).The most significant difference in the IST compared to the case of parallel boundary conditions is the asymptotic behavior of the scattering coefficients, which affects the normalization of the eigenfunctions and the sectionally meromorphic matrix in the RHP. When the asymptotic polarization vectors are not orthogonal, two different methods are presented to convert the RHP into a set of linear algebraic-integral equations. When the asymptotic polarization vectors are orthogonal, however, only one of these methods is applicable. Finally, I show that, both in the case of orthogonal and non-orthogonal polarization vectors, no reflectionless potentials can exist, which implies that the problem does not admit pure soliton solutions. This work has been accepted for publication in the East Asian Journal on Applied Mathematics.Solitons and soliton interactions in repulsive spinor Bose-Einstein condensates with non-zero background.I characterize the soliton solutions and their interactions for a system of coupled evolution equations of NLS-type that models the dynamics in one-dimensional repulsive Bose-Einstein condensates with spin one, taking advantage of the representation of such model as a special reduction of a 2X2 matrix NLS system. Specifically, I study in detail the case in which solutions tend to a non-zero background at space infinities. First, I derive a compact representation for the multi-soliton solutions in the system using the IST. I introduce the notion of canonical form of a solution, corresponding to the case when the background as x to infinity is proportional to the identity. I show that solutions for which the asymptotic behavior at infinity is not proportional to the identity, referred to as being in non-canonical form, can be reduced to canonical form by unitary transformations that preserve the symmetric nature of the solution (physically corresponding to complex rotations of the quantization axes). Then I give a complete characterization of the two families of one-soliton solutions arising in this problem, corresponding to ferromagnetic and to polar states of the system, and I discuss how the physical parameters of the solitons for each family are related to the spectral data in the IST. Finally, I discuss two-soliton interactions and presented a complete classification of the possible scenarios that can arise depending on whether either soliton is of ferromagnetic or polar type. This work was published in European Physical Journal Plus.Manakov system with parity symmetry on nonzero background and associated boundary value problems.I characterize the soliton solutions of the Manakov system with parallel non-zero boundary condition when the initial conditions have well-defined spatial parity. In particular, I derive the extra symmetries of the Jost eigenfunctions and the scattering coefficients induced by the parity of the initial conditions. Using the resulting symmetries, I obtain the parity symmetry of the discrete spectrum and the associated norming constants. I show that the discrete eigenvalues come in either quartets on the circle of radius $q_o$ (with $q_o$ being the background amplitude of the solution), or in symmetric octets, and I obtain explicit relations for the norming constants associated with symmetric eigenvalues. Furthermore, I obtain conditions for the existence of self-symmetric eigenvalues, which give arise to an extra symmetry of the corresponding norming constants. I validate the results by comparison with numerical simulations. This work has been accepted for publication in the Journal of Physics A.Whitham modulation theory for the (2+1)-dimensional defocusing NLS equation.I derive the Whitham modulation equations for the defocusing 2DNLS equation using a two-phase ansatz for the periodic solutions and the conservation laws of the 2DNLS equation. The resulting equations are written in vector form, which allows one to immediately verify that the modulation equations preserve the rotational invariance of the 2DNLS equation, as well as the invariance with respect to scaling and Galilean transformations. The transformation to Riemann-type variables is also carried out, and the harmonic and soliton limits are explicitly considered, as is the reduction to the Whitham equations of the radial NLS equation. This work is currently being finalized for submission.
590 $a School code: 0656.
650 4 $a Mathematics. $3 515831
650 4 $a Applied mathematics. $3 2122814
650 4 $a Applied physics. $3 3343996
653 $a Integrable systems
653 $a Inverse scattering transform
653 $a Long-time asymptotics
653 $a Nonlinear waves
653 $a Soliton solutions
653 $a Whitham modulation theory
690 $a 0405
690 $a 0364
690 $a 0215
710 2 0 $a State University of New York at Buffalo. $b Mathematics. $3 1022476
773 0 $t Dissertations Abstracts International $g 83-12B.
790 $a 0656
791 $a Ph.D.
792 $a 2022
793 $a English