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We study the semilinear equation (Formula presented.) on a Cartan-Hadamard manifold (Formula presented.) of dimension (Formula presented.), and we prove the existence of a nontrivial solution under suitable assumptions on the potential function (Formula presented.). In particular, the decay of (Formula presented.) at infinity is allowed, with some restrictions related to the geometry of (Formula presented.). We generalize some results proved in (Formula presented.) by Alves et al.

Nonlinear waveguides with two distinct domains of anomalous dispersion can support the formation of molecule-like two-color pulse compounds. They consist of two tightly bound subpulses with frequency loci separated by a vast frequency gap. Perturbing such a two-color pulse compound triggers periodic amplitude and width variations, reminiscent of molecular vibrations. With increasing strength of perturbation, the dynamics of the pulse compound changes from harmonic to nonlinear oscillations. The periodic amplitude variations enable coupling of the pulse compound to dispersive waves, resulting in the resonant emission of multi-frequency radiation. We demonstrate that the location of the resonances can be precisely predicted by phase-matching conditions. If the pulse compound consists of a pair of identical subpulses, inherent symmetries lead to degeneracies in the resonance spectrum. Weak perturbations lift existing degeneracies and cause a splitting of the resonance lines into multiple lines. Strong perturbations result in more complex emission spectra, characterized by well separated spectral bands caused by resonant Cherenkov radiation and additional four-wave mixing processes.

This doctoral thesis provides a comprehensive numerical analysis and exploration of several stochastic partial differential equations (SPDEs). More specifically, this thesis investigates time integrators for SPDEs with white noise dispersion. The thesis begins by examining the stochastic nonlinear Schrödinger equation with white noise dispersion (SNLSE), see Paper 1. The investigation probes the performance of different numerical integrators for this equation, focusing on their convergences, L2-norm preservation, and computational efficiency. Further, this thesis thoroughly investigates a conjecture on the critical exponent of the SNLSE, related to a phenomenon known as blowup, through numerical means. The thesis then introduces and studies exponential integrators for the stochastic Manakov equation (SME) by presenting two new time integrators - the explicit and symmetric exponential integrators - and analyzing their convergence properties, see Paper 2. Notably, this study highlights the flexibility and efficiency of these integrators compared to traditional schemes. The narrative then turns to the Lie-Trotter splitting integrator for the SME, see Paper 3, comparing its performance to existing time integrators. Theoretical proofs for convergence in various senses, alongside extensive numerical experiments, shed light on the efficacy of the proposed numerical scheme. The thesis also deep dives into the critical exponents of the SME, proposing a conjecture regarding blowup conditions for this SPDE. Lastly, the focus shifts to the stochastic generalized Benjamin-Bona-Mahony equation, see Paper 4. The study introduces and numerically assesses four novel exponential integrators for this equation. A primary finding here is the superior performance of the symmetric exponential integrator. This thesis also offers a succinct and novel method to depict the order of convergence in probability.

We prove some existence (and sometimes also uniqueness) of weak solutions to some stationary equations associated to the complex Schrödinger operator under the presence of a singular nonlinear term. Among other new facts, with respect some previous results in the literature for such type of nonlinear potential terms, we include the case in which the spatial domain is possibly unbounded (something which is connected with some previous localization results by the authors),the presence of possible non-local terms at the equation, the case of boundary conditions different to the Dirichlet ones and, finally, the proof of the existence of solutions when the right-hand side term of the equation is beyond the usual L2-space., Unión Europea. FP7, DGISPI, Spain, UCM, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

The main goal of this paper is to study the nature of the support of the solution of suitable nonlinear Schrodinger equations, mainly the compactness of the support and its spatial localization. This question touches the very foundations underlying the derivation of the Schrodinger equation, since it is well-known a solution of a linear Schrodinger equation perturbed by a regular potential never vanishes on a set of positive measure. A fact, which reflects the impossibility of locating the particle. Here we shall prove that if the perturbation involves suitable singular nonlinear terms then the support of the solution is a compact set, and so any estimate on its spatial localization implies very rich information on places not accessible by the particle. Our results are obtained by the application of certain energy methods which connect the compactness of the support with the local vanishing of a suitable "energy function" which satisfies a nonlinear differential inequality with an exponent less than one. The results improve and extend a previous short presentation by the authors published in 2006., Unión Europea. FP7, DGISPI (Spain), European program Nonlinear partial differential equations describing front propagation and other singular phenomena., UCM, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

We present a fast method to calculate the significantly large solitonic components of signals with large time-bandwidth products governed by the nonlinear Schrödinger equation, for which the computation typically becomes prohibitively expensive and/or numerically unstable. We partition the full signal in both frequency and time to obtain short signals with a constant number of samples, independent of the size of the full signal. The solitons within each short signal are computed using a conventional nonlinear Fourier transform (NFT) algorithm. The partitioning in general leads to spurious solitons not present in the full signal. We therefore design an acceptance scheme that removes spurious solitons. The remaining solitons are attributed to the full signal. Solitons that are too wide to fit into the short signals cannot be detected by this approach, but since wide solitons must be of low amplitude, the significant solitons will be found. This approach only requires O(N) floating point operations, with N the number of signal samples. It can furthermore be applied to signals with large time-bandwidth products for which conventional NFT algorithms become unreliable or even fail. When applying our proposed method to a signal of 15,000 samples, the significant solitonic components were computed 14 times faster than when considering the whole signal, for which the conventional algorithm furthermore provided wrong results. We found that time-partitioning yields accurate results, while frequency-partitioning causes a small loss in accuracy. Combined frequency-time partitioning leads to the fastest computation, but also suffers from the same loss in accuracy as with frequency-partitioning. As time-partitioning yields a significant speed-up at nearly no loss in accuracy, we regard this as the method of choice in most practical scenarios., Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public., Team Sander Wahls, Team Michel Verhaegen

This doctoral thesis provides a comprehensive numerical analysis and exploration of several stochastic partial differential equations (SPDEs). More specifically, this thesis investigates time integrators for SPDEs with white noise dispersion. The thesis begins by examining the stochastic nonlinear Schrödinger equation with white noise dispersion (SNLSE), see Paper 1. The investigation probes the performance of different numerical integrators for this equation, focusing on their convergences, L2-norm preservation, and computational efficiency. Further, this thesis thoroughly investigates a conjecture on the critical exponent of the SNLSE, related to a phenomenon known as blowup, through numerical means. The thesis then introduces and studies exponential integrators for the stochastic Manakov equation (SME) by presenting two new time integrators - the explicit and symmetric exponential integrators - and analyzing their convergence properties, see Paper 2. Notably, this study highlights the flexibility and efficiency of these integrators compared to traditional schemes. The narrative then turns to the Lie-Trotter splitting integrator for the SME, see Paper 3, comparing its performance to existing time integrators. Theoretical proofs for convergence in various senses, alongside extensive numerical experiments, shed light on the efficacy of the proposed numerical scheme. The thesis also deep dives into the critical exponents of the SME, proposing a conjecture regarding blowup conditions for this SPDE. Lastly, the focus shifts to the stochastic generalized Benjamin-Bona-Mahony equation, see Paper 4. The study introduces and numerically assesses four novel exponential integrators for this equation. A primary finding here is the superior performance of the symmetric exponential integrator. This thesis also offers a succinct and novel method to depict the order of convergence in probability.

Riemann theta functions play a crucial role in the field of nonlinear Fourier analysis, where they are used to realize inverse nonlinear Fourier transforms for periodic signals. The practical applicability of this approach has however been limited since Riemann theta functions are multi-dimensional Fourier series whose computation suffers from the curse of dimensionality. In this paper, we investigate several new approaches to compute Riemann theta functions with the goal of unlocking their practical potential. Our first contributions are novel theoretical lower and upper bounds on the series truncation error. These bounds allow us to rule out several of the existing approaches for the high-dimension regime. We then propose to consider low-rank tensor and hyperbolic cross based techniques. We first examine a tensor-train based algorithm which utilizes the popular scaling and squaring approach. We show theoretically that this approach cannot break the curse of dimensionality. Finally, we investigate two other tensor-train based methods numerically and compare them to hyperbolic cross based methods. Using finite-genus solutions of the Korteweg–de Vries (KdV) and nonlinear Schrödinger equation (NLS) equations, we demonstrate the accuracy of the proposed algorithms. The tensor-train based algorithms are shown to work well for low genus solutions with real arguments but are limited by memory for higher genera. The hyperbolic cross based algorithm also achieves high accuracy for low genus solutions. Its novelty is the ability to feasibly compute moderately accurate solutions (a relative error of magnitude 0.01) for high dimensions (up to 60). It therefore enables the computation of complex inverse nonlinear Fourier transforms that were so far out of reach., Team Sander Wahls

This doctoral thesis provides a comprehensive numerical analysis and exploration of several stochastic partial differential equations (SPDEs). More specifically, this thesis investigates time integrators for SPDEs with white noise dispersion. The thesis begins by examining the stochastic nonlinear Schrödinger equation with white noise dispersion (SNLSE), see Paper 1. The investigation probes the performance of different numerical integrators for this equation, focusing on their convergences, L2-norm preservation, and computational efficiency. Further, this thesis thoroughly investigates a conjecture on the critical exponent of the SNLSE, related to a phenomenon known as blowup, through numerical means. The thesis then introduces and studies exponential integrators for the stochastic Manakov equation (SME) by presenting two new time integrators - the explicit and symmetric exponential integrators - and analyzing their convergence properties, see Paper 2. Notably, this study highlights the flexibility and efficiency of these integrators compared to traditional schemes. The narrative then turns to the Lie-Trotter splitting integrator for the SME, see Paper 3, comparing its performance to existing time integrators. Theoretical proofs for convergence in various senses, alongside extensive numerical experiments, shed light on the efficacy of the proposed numerical scheme. The thesis also deep dives into the critical exponents of the SME, proposing a conjecture regarding blowup conditions for this SPDE. Lastly, the focus shifts to the stochastic generalized Benjamin-Bona-Mahony equation, see Paper 4. The study introduces and numerically assesses four novel exponential integrators for this equation. A primary finding here is the superior performance of the symmetric exponential integrator. This thesis also offers a succinct and novel method to depict the order of convergence in probability.

This doctoral thesis provides a comprehensive numerical analysis and exploration of several stochastic partial differential equations (SPDEs). More specifically, this thesis investigates time integrators for SPDEs with white noise dispersion. The thesis begins by examining the stochastic nonlinear Schrödinger equation with white noise dispersion (SNLSE), see Paper 1. The investigation probes the performance of different numerical integrators for this equation, focusing on their convergences, L2-norm preservation, and computational efficiency. Further, this thesis thoroughly investigates a conjecture on the critical exponent of the SNLSE, related to a phenomenon known as blowup, through numerical means. The thesis then introduces and studies exponential integrators for the stochastic Manakov equation (SME) by presenting two new time integrators - the explicit and symmetric exponential integrators - and analyzing their convergence properties, see Paper 2. Notably, this study highlights the flexibility and efficiency of these integrators compared to traditional schemes. The narrative then turns to the Lie-Trotter splitting integrator for the SME, see Paper 3, comparing its performance to existing time integrators. Theoretical proofs for convergence in various senses, alongside extensive numerical experiments, shed light on the efficacy of the proposed numerical scheme. The thesis also deep dives into the critical exponents of the SME, proposing a conjecture regarding blowup conditions for this SPDE. Lastly, the focus shifts to the stochastic generalized Benjamin-Bona-Mahony equation, see Paper 4. The study introduces and numerically assesses four novel exponential integrators for this equation. A primary finding here is the superior performance of the symmetric exponential integrator. This thesis also offers a succinct and novel method to depict the order of convergence in probability.

The investigation of a nonlinear differential equation (NLDE) integrability is the foundation for understanding and predicting the behavior of systems that are governed by that NLDE. The existence of a Lax pair (LP) representation is a reliable indicator of the indicator of the integrability of an NLDE. If an NLDE admits an LP, it is considered integrable in the LP sense, and this has important implications for the behavior and solutions of the system. The LP plays a crucial role in many powerful analytical solution methods for solving NLDEs. One such method is the Darboux transformation (DT), which allows for the derivation of a series of exact solutions. By using the LP representation, the DT can construct a new solution from a known solution, allowing for the systematic generation of families of solutions to the NLDE. The nonlinear Schrödinger equation (NLSE) with external potentials describes many important physical systems. In the absence of external potentials, the NLSE is completely integrable, and a large number of its exact solutions are known. The presence of the potential renders the NLSE generally to a nonintegrable differential equation. Given the importance of the equation at hand, in this dissertation we consider three main streams. The first stream focuses on the analytical investigation of the NLSE’s integrability in the sense of the existence of LP and the search for new exact solutions, using existing, modified, and new solution methods. The second stream focuses on the development of a high accuracy numerical method based on iterative power series approach, capable of providing the long-time evolution of solutions. The third stream involves numerical simulations to investigate the scattering of different solitons by reflectionless potentials such as the Pöschl-Teller potential, with emphasis on finding the spectrum of bound states. The results of this dissertation are significant contributions to the efforts made towards

The (nonlinear) behaviour of laser beams can be described with Nonlinear Schrödinger equations (NLS). The purpose of this thesis is to shed light on two mathematical papers that give existence and uniqueness results for an NLS equation called the soliton equation. The contribution is to expand the level of detail with which the analysis and some proofs are presented, and to unify the notation where possible., Applied Sciences

Nonlinear Schrödinger equations (NLS) are crucial in quantum physics. Over the past century, they have been widely applied in diverse fields, such as the propagation of laser beams, water waves, and the studies of Bose--Einstein condensates. As the equations can't be solved explicitly, numerical solutions become important as well. Therefore, over the past decades, numerical methods for NLS have been extensively studied in the literature. In this paper, we consider the cubic NLS on a $d$-dimensional torus $\mathbb{T}^d$, i.e. with periodic boundary conditions. As one of the most popular cases of NLS, multitudes of traditional methods are developed. However, these traditional methods require a lot of regularity for the stability argument. Such restriction is not necessary for the local or global well-posedness of the solution. To be specific, $s>\frac d2+2$ is required in traditional methods, where $s$ denotes the regularity of the initial data $u_0$, i.e. $u_0\in H^s(\mathbb{T}^d)$. However, the local well-posedness of the solution only needs $s>\frac d2-1$. Moreover, traditional methods will fail when this restriction is not satisfied, e.g., simulating the Talbot effect with jump discontinuities. The numerical analysis of this low regularity case is still an open question. To overcome such restrictions, we developed a filtered Lie--Trotter splitting method for the cubic NLS on torus. This method shows very good properties in both convergence and long-time behaviours. Moreover, Bourgain developed Bourgain techniques for the analysis of the method, and we adapted these techniques in numerical analysis by constructing discrete Bourgain spaces. These techniques allow us to handle the error analysis for the case $s>\max(\frac d2-1,0)$, which coincides with the regularity restrictions of local well-posedness of the exact solution for $d\geq2$. We also proved that the filtered Lie splitting is convergent in $L^2$, with a convergence order of $\frac s2$ in time and $s$ in s, Dissertation Universität Innsbruck 2023

Due to the ever increasing global connectivity, the demand on the fiber-optic communication infrastructure is projected to keep increasing rapidly. A major factor currently limiting transmission capacity is the fiber nonlinearity. Some researchers have suggested the application of nonlinear Fourier transforms to exploit the fiber nonlinearity rather than ignoring or mitigating it. Nonlinear Fourier transforms allow us to solve certain nonlinear partial differential equations by transforming the complex evolution of the solution in the time-domain to a simple multiplication with a nonlinear frequency response in the nonlinear Fourier domain. This method is analogous to solving linear partial differential equations using the Fourier transform. The nonlinear Schrödinger equation is a suitable model for the propagation of light through a single-mode optical fiber. Its lossless version is solvable through a nonlinear Fourier transform. In recent years, several nonlinear Fourier transform based communication systems have been proposed. Such systems require numerical algorithms to compute the nonlinear Fourier transforms as nonlinear Fourier spectra are known analytically for only a handful of signals, and linear superposition cannot be used to compute the spectrum of a more complex signal. Computationally efficient algorithms are therefore not only essential for the real-time operation of nonlinear Fourier transform based communication systems, but are also important for their simulation. One common way to improve the spectral efficiency of a communication system is to increase the signal power in order to reduce the impact of noise. Another is to increase the signal duration in order to reduce the impact of information-free guard intervals that are inserted between transmissions to deal with the channel memory. Longer signals however require more resources to process them. The numerical problem of computing nonlinear Fourier transforms furthermore gets harder for both hi, Team Sander Wahls

We consider the defocusing cubic nonlinear Schrödinger equation (NLS) on the two-dimensional torus. The equation admits a special family of elliptic invariant quasiperiodic tori called finite-gap solutions. These are inherited from the integrable 1D model (cubic NLS on the circle) by considering solutions that depend only on one variable. We study the long-time stability of such invariant tori for the 2D NLS model and show that, under certain assumptions and over sufficiently long timescales, they exhibit a strong form of transverse instability in Sobolev spaces Hs(T2) (0

In this paper we investigate some qualitative properties of the solutions to the following doubly nonlocal equation \begin{equation}\label{eq_abstract} (- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad \hbox{in $\mathbb{R}^N$} \tag{P} \end{equation} where $N \geq 2$, $s\in (0,1)$, $\alpha \in (0,N)$, $\mu>0$ is fixed, $(-\Delta)^s$ denotes the fractional Laplacian and $I_{\alpha}$ is the Riesz potential. Here $F \in C^1(\mathbb{R})$ stands for a general nonlinearity of Berestycki-Lions type. We obtain first some regularity result for the solutions of \eqref{eq_abstract}. Then, by assuming $F$ odd or even and positive on the half-line, we get constant sign and radial symmetry of the Pohozaev ground state solutions related to equation \eqref{eq_abstract}. In particular, we extend some results contained in \cite{DSS1}. Similar qualitative properties of the ground states are obtained in the limiting case $s=1$, generalizing some results by Moroz and Van Schaftingen in \cite{MS2} when $F$ is odd.

Goal of this paper is to study the following doubly nonlocal equation \begin{equation}\label{eq_abstract} (- \Delta)^s u + \mu u = (I_\alpha*F(u))F'(u) \quad \hbox{in $\mathbb{R}^N$} \tag{P} \end{equation} in the case of general nonlinearities $F \in C^1(\R)$ of Berestycki-Lions type, when $N \geq 2$ and $\mu>0$ is fixed. Here $(-\Delta)^s$, $s \in (0,1)$, denotes the fractional Laplacian, while the Hartree-type term is given by convolution with the Riesz potential $I_{\alpha}$, $\alpha \in (0,N)$. We prove existence of ground states of \eqref{eq_abstract}. Furthermore we obtain regularity and asymptotic decay of general solutions, extending some results contained in \cite{DSS1, MS2}.

We review evolutionary models on quantum graphs expressed by linear and nonlinear partial differential equations. Existence and stability of the standing waves trapped on quantum graphs are studied by using methods of the variational theory, dynamical systems on a phase plane, and the Dirichlet-to-Neumann mappings.

Recently, a nonlinear Fourier transform-based Kerr-nonlinearity identification algorithm was demonstrated for a 1000 km NZDSF link with accuracy of 75%. Here, we demonstrate an accuracy of 99% over 1600 km SSMF. Reasons for improved accuracy are discussed., Accepted Author Manuscript, Team Sander Wahls

Due to the ever increasing global connectivity, the demand on the fiber-optic communication infrastructure is projected to keep increasing rapidly. A major factor currently limiting transmission capacity is the fiber nonlinearity. Some researchers have suggested the application of nonlinear Fourier transforms to exploit the fiber nonlinearity rather than ignoring or mitigating it. Nonlinear Fourier transforms allow us to solve certain nonlinear partial differential equations by transforming the complex evolution of the solution in the time-domain to a simple multiplication with a nonlinear frequency response in the nonlinear Fourier domain. This method is analogous to solving linear partial differential equations using the Fourier transform. The nonlinear Schrödinger equation is a suitable model for the propagation of light through a single-mode optical fiber. Its lossless version is solvable through a nonlinear Fourier transform. In recent years, several nonlinear Fourier transform based communication systems have been proposed. Such systems require numerical algorithms to compute the nonlinear Fourier transforms as nonlinear Fourier spectra are known analytically for only a handful of signals, and linear superposition cannot be used to compute the spectrum of a more complex signal. Computationally efficient algorithms are therefore not only essential for the real-time operation of nonlinear Fourier transform based communication systems, but are also important for their simulation. One common way to improve the spectral efficiency of a communication system is to increase the signal power in order to reduce the impact of noise. Another is to increase the signal duration in order to reduce the impact of information-free guard intervals that are inserted between transmissions to deal with the channel memory. Longer signals however require more resources to process them. The numerical problem of computing nonlinear Fourier transforms furthermore gets harder for both hi, Team Sander Wahls

Nonlinear Schrödinger equations with nonlinearities |u|²ᴷu on the d-dimensional torus are considered for arbitrary positive integers k and d. The solution of the Cauchy problem is shown to be unique in the class CₜH ˢₓ for a certain range of scale-subcritical regularities s, which is almost optimal in the case d ≥ 4 or k ≥ 2. The proof is based on various multilinear estimates and the infinite normal form reduction argument.

In this thesis, we study the behavior of solutions to some semilinear Schr\"odinger equations at short and long time scales. We first consider the nonlinear Schr\"odinger equations with power-type nonlinearity in three dimensions with periodic boundary conditions. We show that this equation is locally well-posed in critically scaling Sobolev spaces $H^s(\bb{T}^3)$. We then investigate the long-time asymptotic behavior of solutions to NLS in Euclidean space with defocusing, mass-subcritical power-type and Hartree nonlinearities. We discuss the divide between the wealth of results on the scattering theory for these equations in weighted $L^2$ spaces and the paucity of analogous results in $L^2(\bb{R}^d)$. To explain this, we show that the scattering problems for these equations are well-posed in weighted $L^2$ spaces in the sense that the scattering operators attain their natural and maximal regularity. Furthermore, we show that these scattering problems are ill-posed in $L^2$ in the sense that the scattering operators cannot be extended to all of $L^2$ without losing a positive (and, in the case of Hartree, infinite) amount of regularity.

Signal propagation in an optical fiber can be described by the nonlinear Schrödinger equation (NLSE). The NLSE has no known closed-form solution when both dispersion and nonlinearities are considered simultaneously. In this paper, we present a novel integral-form approximate model for the nonlinear optical channel, with applications to passive optical networks. The proposed model is derived using logarithmic perturbation in the frequency domain on the group-velocity dispersion (GVD) parameter of the NLSE. The model can be seen as an improvement of the recently proposed regular perturbation (RP) on the GVD parameter. RP and logarithmic perturbation (LP) on the nonlinear coefficient have already been studied in the literature, and are hereby compared with RP on the GVD parameter and the proposed LP model. As an application of the model, we focus on passive optical networks. For a 20 km PON at 10 Gbaud, the proposed model improves the normalized square deviation by 1.5 dB with respect to LP on the nonlinear coefficient. For the same system, histogram-based detectors are developed using the received symbols from the models. The detector obtained from the proposed LP model reduces the uncoded bit-error-rate by up to 5.4 times at the same input power or reduces the input power by 0.4 dB at the same information rate compared to the detector obtained from LP on the nonlinear coefficient.

In this thesis, a variation on the nonlinear Schrödinger (NLS) equation with multiplicative noise is studied. In particular, we consider a stochastic version of the parametrically-forced nonlinear Schrödinger equation (PFNLS), which models the effect of linear loss and the compensation thereof by phase-sensitive amplification in pulse propagation through optical fibers. We establish global existence and uniqueness of mild solutions for initial data in L2(R) and H1(R). The proof is an adaptation of a fixed-point argument employed by de Bouard and Debussche [Comm. Math. Phys., 205:161-181, 1999] for the nonlinear Schrödinger equation with multiplicative noise. The fixed-point argument relies on space-time estimates on the semigroup generated by the linear parametrically-forced Schrödinger operator. We prove these so-called Strichartz estimates, originally proven for the Schrödinger operator, using Fourier methods. A key difference between the Schrödinger operator and its parametrically-forced version is that the latter is not self-adjoint. We overcome this complication by establishing fixed-time estimates on the semigroup and its adjoint, based on their Fourier representations. We also briefly discuss possible future research in the direction of stability of solitary standing wave solutions of the PFNLS equation under the influence of multiplicative noise. Using informal calculations, we demonstrate an approach to track the displacement of a soliton due to small stochastic forcing., Applied Mathematics

We study the defocusing nonlinear Schrodinger equation in the quarter-plane with decaying initial datum and Dirichlet and Neumann boundary values approaching periodic single exponentials at large times. By applying Deift-Zhou steepest descent arguments to an associated Riemann-Hilbert problem, we construct solutions and obtain detailed formulas for their long-time asymptotics., QC 20211112

In this thesis, we study the behavior of solutions to some semilinear Schr\"odinger equations at short and long time scales. We first consider the nonlinear Schr\"odinger equations with power-type nonlinearity in three dimensions with periodic boundary conditions. We show that this equation is locally well-posed in critically scaling Sobolev spaces $H^s(\bb{T}^3)$. We then investigate the long-time asymptotic behavior of solutions to NLS in Euclidean space with defocusing, mass-subcritical power-type and Hartree nonlinearities. We discuss the divide between the wealth of results on the scattering theory for these equations in weighted $L^2$ spaces and the paucity of analogous results in $L^2(\bb{R}^d)$. To explain this, we show that the scattering problems for these equations are well-posed in weighted $L^2$ spaces in the sense that the scattering operators attain their natural and maximal regularity. Furthermore, we show that these scattering problems are ill-posed in $L^2$ in the sense that the scattering operators cannot be extended to all of $L^2$ without losing a positive (and, in the case of Hartree, infinite) amount of regularity.

Motivated by a similar approach for Born-Oppenheimer molecular dynamics, this paper proposes an extended "shadow" Lagrangian density for quantum states of superfluids. The extended Lagrangian contains an additional field variable that is forced to follow the wave function of the quantum state through a rapidly oscillating extended harmonic oscillator. By considering the adiabatic limit for large frequencies of the harmonic oscillator, we can derive the two equations of motions, a Schrodinger-type equation for the quantum state and a wave equation for the extended field variable. The equations are coupled in a nonlinear way, but each equation individually is linear with respect to the variable that it defines. The computational advantage of this new system is that it can be easily discretized using linear time stepping methods, where we propose to use a Crank-Nicolson-type approach for the Schrodinger equation and an extended leapfrog scheme for the wave equation. Furthermore, the difference between the quantum state and the extended field variable defines a consistency error that should go to zero if the frequency tends to infinity. By coupling the time-step size in our discretization to the frequency of the harmonic oscillator we can extract an easily computable consistency error indicator that can be used to estimate the numerical error without additional costs. The findings are illustrated in numerical experiments.

Motivated by a similar approach for Born-Oppenheimer molecular dynamics, this paper proposes an extended "shadow" Lagrangian density for quantum states of superfluids. The extended Lagrangian contains an additional field variable that is forced to follow the wave function of the quantum state through a rapidly oscillating extended harmonic oscillator. By considering the adiabatic limit for large frequencies of the harmonic oscillator, we can derive the two equations of motions, a Schrodinger-type equation for the quantum state and a wave equation for the extended field variable. The equations are coupled in a nonlinear way, but each equation individually is linear with respect to the variable that it defines. The computational advantage of this new system is that it can be easily discretized using linear time stepping methods, where we propose to use a Crank-Nicolson-type approach for the Schrodinger equation and an extended leapfrog scheme for the wave equation. Furthermore, the difference between the quantum state and the extended field variable defines a consistency error that should go to zero if the frequency tends to infinity. By coupling the time-step size in our discretization to the frequency of the harmonic oscillator we can extract an easily computable consistency error indicator that can be used to estimate the numerical error without additional costs. The findings are illustrated in numerical experiments., QC 20210426

We consider the inhomogeneous Dirichlet-boundary value problem for the quadratic nonlinear Schrödinger equations, which is considered as a critical case for the large-time asymptotics of solutions. We present sufficient conditions on the initial and boundary data which ensure asymptotic behavior of small solutions to the equations by using the classical energy method and factorization techniques of the free Schrödinger group.

In this paper we consider the fractional nonlinear Schrödinger equation $$ \eps^{2s}(- \Delta)^s v+ V(x) v= f(v), \quad x \in \R^N$$ where $s \in (0,1)$, $N \geq 2$, $V \in C(\R^N,\R)$ is a positive potential and $f$ is a nonlinearity satisfying Berestycki-Lions type conditions. For $\eps>0$ small, we prove the existence of at least $\cupl(K)+1$ positive solutions, where $K$ is a set of local minima in a bounded potential well and $\cupl(K)$ denotes the cup-length of $K$. By means of a variational approach, we analyze the topological difference between two levels of an indefinite functional in a neighborhood of expected solutions. Since the nonlocality comes in the decomposition of the space directly, we introduce a new fractional center of mass, via a suitable seminorm. Some other delicate aspects arise strictly related to the presence of the nonlocal operator. By using regularity results based on fractional De Giorgi classes, we show that the found solutions decay polynomially and concentrate around some point of $K$ for $\eps$ small.

We study existence of solutions for the fractional problem \begin{equation*} (P_m) \quad \parag{ (-\Delta)^{s} u + \mu u &=g(u) & \; \text{in $\mathbb{R}^N$}, \cr \int_{\mathbb{R}^N} u^2 dx &= m, & \cr u \in H^s_r&(\mathbb{R}^N), & } \end{equation*} where $N\geq 2$, $s\in (0,1)$, $m>0$, $\mu$ is an unknown Lagrange multiplier and $g \in C(\mathbb{R}, \mathbb{R})$ satisfies Berestycki-Lions type conditions. Using a Lagrangian formulation of the problem $(P_m)$, we prove the existence of a weak solution with prescribed mass when $g$ has $L^2$ subcritical growth. The approach relies on the construction of a minimax structure, by means of a \emph{Pohozaev's mountain} in a product space and some deformation arguments under a new version of the Palais-Smale condition introduced in \cite{HT0,IT0}. A multiplicity result of infinitely many normalized solutions is also obtained if $g$ is odd.

We prove the existence of a spherically symmetric solution for a Schr\"odinger equation with a nonlocal nonlinearity of Choquard type, i.e. $$ (- \Delta)^s u + \mu u = (I_\alpha*F(u))f(u) \quad \hbox{in $\mathbb{R}^N$}, $$ where $N\geq 2$, $s \in (0,1)$, $\alpha\in (0,N)$, $I_\alpha(x)=\frac{A_{N,\alpha}}{|x|^{N-\alpha}}$ is the Riesz potential, $\mu>0$ is part of the unknowns, and $F\in C^1(\mathbb{R},\mathbb{R})$, $F' = f$ is assumed to be subcritical and to satisfy almost optimal assumptions. The mass of of the solution, described by its norm in the $L^2$-space, is prescribed in advance by $\int_{\mathbb{R}^N} u^2 \, dx = c$ for some $c>0$. The approach to this constrained problem relies on a Lagrange formulation and new deformation arguments. In addition, we prove that the obtained solution is also a ground state, which means that it realizes minimal energy among all the possible solutions to the problem.

Goal of this paper is to study the following singularly perturbed nonlinear Schr\"odinger equation $$ \varepsilon^{2s}(- \Delta)^s v+ V(x) v= f(v), \quad x \in \mathbb{R}^N,$$ where $s \in (0,1)$, $N \geq 2$, $V \in C(\mathbb{R}^N,\mathbb{R})$ is a positive potential and $f$ is assumed critical and satisfying general Berestycki-Lions type conditions. When $\eps>0$ is small, we obtain existence and multiplicity of semiclassical solutions, relating the number of solutions to the cup-length of a set of local minima of $V$; in particular we improve the result in \cite{HeZo}. Furthermore, these solutions are proved to concentrate in the potential well, exhibiting a polynomial decay. Finally, we prove the previous results also in the limiting local setting $s=1$ and $N\geq 3$, with an exponential decay of the solutions.

Signal propagation in an optical fiber can be described by the nonlinear Schrödinger equation (NLSE). The NLSE has no known closed-form solution when both dispersion and nonlinearities are considered simultaneously. In this paper, we present a novel integral-form approximate model for the nonlinear optical channel, with applications to passive optical networks. The proposed model is derived using logarithmic perturbation in the frequency domain on the group-velocity dispersion (GVD) parameter of the NLSE. The model can be seen as an improvement of the recently proposed regular perturbation (RP) on the GVD parameter. RP and logarithmic perturbation (LP) on the nonlinear coefficient have already been studied in the literature, and are hereby compared with RP on the GVD parameter and the proposed LP model. As an application of the model, we focus on passive optical networks. For a 20 km PON at 10 Gbaud, the proposed model improves the normalized square deviation by 1.5 dB with respect to LP on the nonlinear coefficient. For the same system, histogram-based detectors are developed using the received symbols from the models. The detector obtained from the proposed LP model reduces the uncoded bit-error-rate by up to 5.4 times at the same input power or reduces the input power by 0.4 dB at the same information rate compared to the detector obtained from LP on the nonlinear coefficient.

In this thesis, a variation on the nonlinear Schrödinger (NLS) equation with multiplicative noise is studied. In particular, we consider a stochastic version of the parametrically-forced nonlinear Schrödinger equation (PFNLS), which models the effect of linear loss and the compensation thereof by phase-sensitive amplification in pulse propagation through optical fibers. We establish global existence and uniqueness of mild solutions for initial data in L2(R) and H1(R). The proof is an adaptation of a fixed-point argument employed by de Bouard and Debussche [Comm. Math. Phys., 205:161-181, 1999] for the nonlinear Schrödinger equation with multiplicative noise. The fixed-point argument relies on space-time estimates on the semigroup generated by the linear parametrically-forced Schrödinger operator. We prove these so-called Strichartz estimates, originally proven for the Schrödinger operator, using Fourier methods. A key difference between the Schrödinger operator and its parametrically-forced version is that the latter is not self-adjoint. We overcome this complication by establishing fixed-time estimates on the semigroup and its adjoint, based on their Fourier representations. We also briefly discuss possible future research in the direction of stability of solitary standing wave solutions of the PFNLS equation under the influence of multiplicative noise. Using informal calculations, we demonstrate an approach to track the displacement of a soliton due to small stochastic forcing., Applied Mathematics

We summarize recent progress on long range scattering for nonlinear Schrödinger equation and nonlinear Klein-Gordon equation. We introduce a technique of extracting a resonant part, which has the same oscillation speed as its argument, from non-polynomial nonlinearities, and exhibit its two applications. Firstly, we consider nonlinear Schrödinger equation with a general nonlinearity of the critical order, and investigate the relation between the shape of the nonlinearity and a typical asymptotic behavior of small solutions. Secondly, we consider nonlinear Klein-Gordon equation with a gauge-invariant nonlinearity, and find an asymptotic behavior for both real-valued case and complex-valued case. A slight improvement is seen in the second application.

Communication using the optical fibre channel can be challenging due to nonlinear effects that arise in the optical propagation. These effects represent physical processes that originate from light propagation in optical fibres. To obtain fundamental understandings of these processes, mathematical models are typically used. These models are based on approximations of the nonlinear Schrödinger equation, the differential equation that governs the propagation in an optical fibre. All available models in the literature are restricted to certain regimes of operation. Here, we present an approximate model for the nonlinear optical fibre channel in the weak-dispersion regime, in a noiseless scenario. The approximation is obtained by applying regular perturbation theory on the group-velocity dispersion parameter of the nonlinear Schrödinger equation. The proposed model is compared with three other models using the normalized square deviation metric and shown to be significantly more accurate for links with high nonlinearities and weak dispersion.

We present experimental evidence of formation and persistence of localized waves, breathers, and solitons, occurring in a random sea state and uniformly traveling over non-uniform bathymetry. Recent studies suggest connections between breather dynamics and irregular sea states and between extreme wave formation and breathers, random sea states, or non-uniform bathymetry individually. In this paper, we investigate the joint connection between these phenomena, and we found that breathers and deep-water solitons can persist in more complex environments. Three different sets of significant heights have been generated within a Joint North Sea Wave Observation Project wave spectrum, and the wave heights were recorded with gauges in a wave tank. Statistical analysis was applied to the experimental data, including the space and time distribution of kurtosis, skewness, Benjamin–Feir index, moving Fourier spectra, and probability distribution of wave heights. Stable wave packages formed out of the random wave field and traveling over shoals, valleys, and slopes were compared with exact solutions of the nonlinear Schrödinger equation with a good match, demonstrating that these localized waves have the same structure as deep-water breathers. We identify the formation of rogue waves at moments and over regions where the kurtosis and skewness have local maxima. These results provide insights for understanding of the robustness of Peregrine and higher-order Akhmediev breathers, Kuznetsov–Ma solitons, and rogue waves, and their occurrence in realistic oceanic conditions, and may motivate analogous studies in other fields of physics to identify limitations of exact weakly nonlinear models in non-homogeneous media.

Using a unified framework, the formulation of a super-convergent discontinuous Galerkin (SDG) method and a hybridized discontinuous Galerkin (HDG) version, both applied to a general nonlinear Schrödinger equation is presented. Conservation of the mass and the energy is studied, theoretically for the semi-discrete formulation; and, for the fully discrete method using the Modified Crank–Nicolson time scheme. Conservation of both quantities is numerically validated on two dimensional problems and high order approximations. A numerical study of convergence illustrates the advantages of the new formulations over the traditional Local Discontinuous Galerkin (LDG) method. Numerical experiments show that the approximation of the initial discrete energy converges with order 2k+1, which is better than that obtained by the standard (continuous) finite element, which is only of order 2k when polynomials of degree k are used.

Communication using the optical fibre channel can be challenging due to nonlinear effects that arise in the optical propagation. These effects represent physical processes that originate from light propagation in optical fibres. To obtain fundamental understandings of these processes, mathematical models are typically used. These models are based on approximations of the nonlinear Schrödinger equation, the differential equation that governs the propagation in an optical fibre. All available models in the literature are restricted to certain regimes of operation. Here, we present an approximate model for the nonlinear optical fibre channel in the weak-dispersion regime, in a noiseless scenario. The approximation is obtained by applying regular perturbation theory on the group-velocity dispersion parameter of the nonlinear Schrödinger equation. The proposed model is compared with three other models using the normalized square deviation metric and shown to be significantly more accurate for links with high nonlinearities and weak dispersion.