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We investigate the propagation characteristics of two temporally separated soliton pulses with the same spectra, under the influence of stimulated Raman scattering, within a single-mode optical fiber. This analysis explores the behavior of the interacting solitons while propagating in different chalcogenide materials, exhibiting new features and promising prospects for soliton transmission in optical communication systems. Our study included all the interaction parameters constituting the nonlinear Schrödinger equation (NLSE). We have examined the relationship between the Kerr nonlinearity, interpulse and intrapulse Raman effects, and material-dependent collision length featuring a key aspect in logic design and phase control in mode-locking systems. We have also systematically shown the manifestation of the Raman response function from the Raman gain curve, which our mathematical model (the Lorentzian model) provides, that exhibits a near agreement with experimental data. Our findings reveal significant differences from the typical behavior of two-soliton interaction only due to Kerr nonlinearity. Furthermore, we have investigated the mechanism of the net energy transfer between the interacting solitons as an integral phenomenon involved in multiple soliton propagation. These results provide an insightful understanding of the associated nonlinear effects in high-power soliton transmission systems and are foreseen to possess the potential for designing advanced optical switches and mode-locked lasers. [ABSTRACT FROM AUTHOR]

A numerical model for a spin wave delay-line active ring resonator is presented. Spin wave dynamics along a one-dimensional strip of magnetic material are modeled using the nonlinear Schrödinger equation. The equation is solved numerically in Fourier space using the fourth-order Runge–Kutta method and yields qualitative agreement with experimental measurements of spin wave dynamics in two different regimes. The model provides a useful tool for performing experiments based on neuromorphic computing and logic gates in traveling spin wave devices. [ABSTRACT FROM AUTHOR]

This paper presents an improved Cham method as an efficient technique for obtaining analytical exact solutions to nonlinear partial differential equations. We apply this method to solve the nonlinear Schrödinger equation in conformable operator, a challenging equation frequently used in various scientific fields. The method enables us to derive different traveling wave solutions, such as kink, coindal waves, breather waves, periodic singular solutions, and periodic multi-wave solitons. We also provide graphical representations of some of the obtained solutions to help understand their dynamic characteristics. These results highlight the effectiveness and adaptability of the method and demonstrate its potential to solve other partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the Fokas–Lenells equation, which is a derivation form of the nonlinear Schrödinger equation and can be used to describe nonlinearity in the propagation of optical pulses. To seek solutions for this nonlinearity, the Logistic method has been proposed in order to drive acceptable and understandable results regarding physical meaning. This method is defined based on the summation of an ordinary differential equation. Moreover, for further study, the Bernoulli (F ′ / F) -expansion method is used by considering hypothetical solutions as a function of the Bernoulli equation. Subsequently, solutions of the Fokas–Lenells equation are successfully acquired, displaying efficient results in hyperbolic, trigonometric, and exponential equations. The importance of these results appears in the definition of the wave function under the consideration of the appropriate coefficients. Hence, computer simulation is used for better understanding of the generated results. The dynamic behaviors of these solutions are demonstrated in 3D graphs, contour maps, and line plots, under applying various parameters, solutions of the waves show different soliton behaviors, including bright-dark, bright, and breather solitons. The results indicate that the exerted methods are novel, reliable, and effective approaches which can be employed in a wide range of nonlinear differential equations. These methods and their begotten results are far from the complexities of mathematical structures and therefore go beyond previous efforts in the literature. [ABSTRACT FROM AUTHOR]

This paper aims to generate a Liouville integrable Hamiltonian hierarchy by introducing a specific matrix eigenvalue problem with four components. The adopted approach is the zero curvature formulation. A bi-Hamiltonian formulation is furnished through applying the trace identity, which shows the Liouville integrability of the resulting hierarchy. Two examples of generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations are presented. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the existence and multiplicity of stationary solutions to the nonlinear Schrödinger equation with concave-convex nonlinearities. The main contribution of this research lies in the fact that zero lies in the boundary of the spectrum of the strongly indefinite Schrödinger operator. Our approach utilizes a novel perturbing method, specifically designed for cases where zero is situated within the spectral gap of the Schrödinger operator. By applying a generalized linking argument concerning the strongly indefinite functional, we establish the existence of two sequences of solutions to the perturbing equation, each corresponding to positive and negative energy states respectively. Finally, we employ the perturbing method to demonstrate the existence of two solutions to our equation. [ABSTRACT FROM AUTHOR]

We consider a Schrödinger equation with a nonlinearity which is a general perturbation of a power nonlinearity. We construct a profile decomposition adapted to this nonlinearity. We also prove global existence and scattering in a general defocusing setting, assuming that the critical Sobolev norm is bounded in the energy-supercritical case. This generalizes several previous works on double-power nonlinearities. [ABSTRACT FROM AUTHOR]

We study a spherical, self‐gravitating fluid model, which finds applications in cosmic structure formation. We argue that since the system features nonlinearity and gravity‐induced dispersion, the emergence of solitons becomes possible. We thus employ a multiscale expansion method to study, in the weakly nonlinear regime, the evolution of small‐amplitude perturbations around the equilibrium state. This way, we derive a spherical nonlinear Schrödinger (NLS) equation that governs the envelope of the perturbations. The effective NLS description allows us to predict a "nonlinear instability" (occurring in the nonlinear regime of the system), namely, the modulational instability, which, in turn, may give rise to spherical soliton states. The latter feature a very slow (polynomial) curvature‐induced decay in time. The soliton profiles may be used to describe the shape of dark matter halos at the rims of the galaxies. [ABSTRACT FROM AUTHOR]

The evolution of a weakly nonlinear, narrow-banded wave packet in an inhomogeneous medium is studied in the context of coastal engineering. Analytically the model is a variable coefficient nonlinear Schrödinger equation (vcNLS) with a gain/loss term, arising from the waves running up a sloping beach. The crucial distinction from previous works is the choice of a 'forcing' boundary condition. Energy then enters the computational domain. This dynamics serves as a crude model for storm surge waves approaching shore. The competition among dispersion, gain and nonlinearity dictates the amplification process. In shallow water (or defocusing regime of vcNLS), waves moving up a sloping beach experience the largest steepening on a convex seafloor, followed by a straight-line seafloor, and then concave seafloor. For a larger spatial domain, a wave packet moving from the open ocean to the shoreline will pass through a critical depth where vcNLS changes from focusing to defocusing. The presence of modulation instability, breather and Fermi–Pasta–Ulam–Tsingou recurrence (FPUT) further complicates the dynamics. Wave amplification in the focusing regime is then most pronounced on a concave seafloor, followed by a straight-line seafloor, and finally a convex seafloor. The growth rate of modulation instability and the magnitude of eigenvalues obtained from Floquet analysis provide quantitative confirmation for these predictions on the dynamics under different seafloor bathymetries. Compared with growth rates of modulation instability, the eigenvalues of the Jacobian matrix in computational analysis provide reasonably accurate predictions of the critical depths. Implications in terms of nonlinear science as well as actual coastal defense mechanisms are discussed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the existence of optimal solutions for control problems involving nonlinear stochastic Schrödinger equations perturbed by a multiplicative fractional Brownian motion. Finite-dimensional approximations and a linearization method are employed and lead to ε-optimal solutions for the considered problems. [ABSTRACT FROM AUTHOR]

The paper aims to study the numerical solution of the Zakharov-Rubenchik equations on unbounded domains, which model the propagation of two kinds of waves in plasma physics, by applying the artificial boundary method. Based on ideas of the operator splitting method and artificial boundary method of hyperbolic system and nonlinear Schrödinger equation, the artificial boundary conditions of the Zakharov-Rubenchik equations are designed on the introduced artificial boundaries to reduce the original problem defined on an unbounded domain into an initial boundary value problem on the bounded computational domain, which can be efficiently solved by finite difference method. A series of auxiliary variables is introduced to rigorously analyze the stability of the reduced initial boundary value problem through energy estimation. Numerical examples are presented to demonstrate the effectiveness of the proposed artificial boundary conditions, validate the theoretical analysis, and simulate the physical phenomena. [ABSTRACT FROM AUTHOR]

In this article, we study a time splitting Fourier pseudo‐spectral (TSFP) method for the nonlinear Schrödinger equation with wave operator (NLSW). The nonlinear strength of the NLSW is characterized by ε∈(0,1]$$ \varepsilon \in \left(0,1\right] $$. Specifically, we propose a coupled system which is equivalent to the NLSW and then apply the TSFP method to this system. As a geometric advantage, the TSFP method has time symmetry and conserves the discrete mass. Rigorous convergence analysis is provided to establish improved error bounds at O(hm+ε2pτ2)$$ O\left({h}^m+{\varepsilon}^{2p}{\tau}^2\right) $$ up to the long‐time at O(1/ε2p)$$ O\left(1/{\varepsilon}^{2p}\right) $$ where m$$ m $$ depends on the smoothness of the solution. Compared with the error bounds O(hm+τ2)$$ O\left({h}^m+{\tau}^2\right) $$ obtained by traditional analysis, our error bounds are greatly improved, especially when the problem presents weak nonlinearity, i.e. 0<ε≪1$$ 0<\varepsilon \ll 1 $$. In error analysis, combining with classical numerical analysis tools, we adopt the regularity compensation oscillation (RCO) technique to study the error accumulation process in detail and then establish the improved error bounds. The numerical experiments support our theoretical analysis. In addition, the numerical results show the long‐term stability of discrete energy. [ABSTRACT FROM AUTHOR]

We propose a second‐order nested Picard iterative integrator sine pseudospectral (NPI‐SP) method for the nonlinear Schrödinger equation with wave operator (NLSW) involving a parameter 0<ε≤1$$ 0<\varepsilon \le 1 $$ and carry out rigorous error estimates. Actually, the equation propagates wave with O(ε2)$$ O\left({\varepsilon}^2\right) $$ wavelength in time, while the amplitude of the leading oscillation is O(ε4)$$ O\left({\varepsilon}^4\right) $$ for well‐prepared initial data, and O(ε2)$$ O\left({\varepsilon}^2\right) $$ for ill‐prepared initial data, respectively. Based on the exponential integrator and nested Picard iteration, the uniformly accurate (w.r.t. ε∈(0,1]$$ \varepsilon \in \left(0,1\right] $$) NPI‐SP scheme is proposed with the optimal uniform error bounds at O(τ2)$$ O\left({\tau}^2\right) $$ in time and spectral accuracy in space for both well‐prepared and ill‐prepared data in H1$$ {H}^1 $$‐norm. This result significantly improves the error bounds of the finite difference methods and exponential wave integrator for the NLSW. Error estimates are rigorously carried out and numerical examples are provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

Solitons of the purely cubic nonlinear Schrödinger equation in a space dimension of n ≥ 2 suffer critical and supercritical collapses. These solitons can be stabilized in a cubic-quintic nonlinear medium. In this paper, we analyze the Crank-Nicolson finite difference scheme for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping. We show that both the discrete solution, in the discrete L 2 -norm, and discrete energy are bounded. By using appropriate settings and estimations, the existence and the uniqueness of the numerical solution are proved. In addition, the error estimations are established in terms of second order for both space and time in discrete L 2 -norm and H 1 -norm. Numerical simulations for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping are conducted to validate the convergence. • The (2+1)D cubic-quintic NLS equation can against 2D-soliton collapses. • A Crank-Nicolson finite difference scheme for the damped (2+1)D cubic-quintic NLS equation is proposed. • The existence and uniqueness of solutions are demonstrated using the boundedness of discrete energy and mass. • The scheme converges second order in time and space. • The numerical experiments are demonstrated to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, by applying the supplementary variable method (SVM), some high-order, conformal structure-preserving, linearized algorithms are developed for the damped nonlinear Schrödinger equation. We derive the well-determined SVM systems with the conformal properties and they are then equivalent to nonlinear equality constrained optimization problems for computation. The deduced optimization models are discretized by using the Gauss type Runge-Kutta method and the prediction-correction technique in time as well as the Fourier pseudo-spectral method in space. Numerical results and some comparisons between this method and other reported methods are given to favor the suggested method in the overall performance. It is worthwhile to emphasize that the numerical strategy in this work could be extended to other conservative or dissipative system for designing high-order structure-preserving algorithms. [ABSTRACT FROM AUTHOR]

The coupled higher-order nonlinear Schrödinger equations (CHNLSEs) are investigated in this paper. This model mimic the actuation of the solitons through multimode fibers. The technique of improved modified extended tanh function is used to conduct the study. Many exact solutions are achieved like bright solitons, dark solitons and singular solitons. Moreover, singular periodic solutions, Weierstrass elliptic function solutions, hyperbolic and Jacobi elliptic solutions are raised. Using 3D and 2D graphs, the physical behavior of the produced solutions is illustrated. [ABSTRACT FROM AUTHOR]

We introduce a simplified model for wave turbulence theory—the Wick nonlinear Schrödinger equation, of which the main feature is the absence of all self‐interactions in the correlation expansions of its solutions. For this model, we derive several wave kinetic equations that govern the effective statistical behavior of its solutions in various regimes. In the homogeneous setting, where the initial correlation is translation invariant, we obtain a wave kinetic equation similar to the one predicted by the formal theory. In the inhomogeneous setting, we obtain a wave kinetic equation that describes the statistical behavior of the wavepackets of the solutions, accounting for both the transport of wavepackets and collisions among them. Another wave kinetic equation, which seems new in the literature, also appears in a certain scaling regime of this setting and provides a more refined collision picture. [ABSTRACT FROM AUTHOR]

ABSTRACT This study focuses on the manipulation of soliton parameters within optical waveguides characterized by a dual‐power law refractive index, drawing similarities with published papers. The precise management of solitons holds significant importance in the field of fiber‐optical communication. While previous research has primarily concentrated on modifying soliton attributes such as amplitude and width, comparatively less attention has been dedicated to controlling the wave speed of solitons. In order to bridge this gap, the researchers adopted the employment of the nonlinear Schrödinger equation with time‐dependent dispersion and two power‐law nonlinearities, mirroring similar approaches found in published literature. By employing this methodology, the researchers successfully achieved active control over the wave speed of solitons in non‐uniform optical waveguides. To effectively handle the inherent complexities of the nonlinear system, two well‐established techniques, namely, the generalized Kudryashov method and the simple equation method, were utilized to derive solutions. These techniques have been previously employed and demonstrated in published papers, further enhancing the validity and reliability of the research findings. We discuss the traveling wave solutions by finding the fixed points of the dynamical planar system. [ABSTRACT FROM AUTHOR]

We consider a nonconservative nonlinear Schrödinger equation (NCNLS) with time‐dependent coefficients, inspired by a water waves problem. This problem does not have mass or energy conservation, but instead mass and energy change in time under explicit balance laws. In this paper, we extend to the particular NCNLS two numerical schemes which are known to conserve energy and mass in the discrete level for the cubic nonlinear Schrödinger equation. Both schemes are second‐order accurate in time, and we prove that their extensions satisfy discrete versions of the mass and energy balance laws for the NCNLS. The first scheme is a relaxation scheme that is linearly implicit. The other scheme is a modified Delfour–Fortin–Payre scheme, and it is fully implicit. Numerical results show that both schemes capture robustly the correct values of mass and energy, even in strongly nonconservative problems. We finally compare the two numerical schemes and discuss their performance. [ABSTRACT FROM AUTHOR]

We consider the stability of bound‐state solutions of a family of regularized nonlinear Schrödinger equations which were introduced by Dumas et al. as models for the propagation of laser beams. Among these bound‐state solutions are ground states, which are defined as solutions of a variational problem. We give a sufficient condition for existence and orbital stability of ground states, and use it to verify that ground states exist and are stable over a wider range of nonlinearities than for the nonregularized nonlinear Schrödinger equation. We also give another sufficient and almost necessary condition for stability of general bound states, and show that some stable bound states exist which are not ground states. [ABSTRACT FROM AUTHOR]

When taking a regular planar polygon of M sides and length 2 π as the initial datum of the vortex filament equation, X t = X s ∧ X s s , the solution becomes polygonal at times of the form t p q = (p / q) (2 π / M 2) , with gcd (p , q) = 1 , and the corresponding polygon has M q sides, if q is odd, and M q / 2 sides, if q is even. Moreover, that polygon is skew (except when q = 1 or q = 2 , where the initial shape is recovered), and the angle ρ between two adjacent sides is a constant. In this paper, we give a rigorous proof of the conjecture that states that, at a time t p q , cos q (ρ / 2) = cos (π / M) , if q is odd, and cos q (ρ / 2) = cos 2 (π / M) , if q is even. Since the transition of one side of the polygon to the next one is given by a rotation in R 3 determined by a generalized Gauss sum, the idea of the proof consists in showing that a certain product of those rotations is a rotation of angle 2 π / M , which is equivalent to proving that some exponential sums with arithmetic content are purely imaginary. [ABSTRACT FROM AUTHOR]

This paper studies the inhomogeneous bi-harmonic nonlinear Schrödinger equation i u ˙ + Δ 2 u ± | x | − τ | u | p − 1 u = 0 , where u : = u (t , x) : R × R N → C , τ > 0 and p > 1 . One proves the existence of global solutions with datum in Lebesgue space L ϱ (R N) , for a certain 1 < ϱ < 2 . This work complements the known global well-posedness result in the mass-sub-critical regime, namely 1 < p < 1 + 8 − 2 τ N and with finite mass. This work aims to develop a local theory in non-Hilbert Lebesgue spaces. Moreover, one complements the local solution to a global one under some suitable assumptions on the parameters. To this end, one uses a Strichartz-type estimate based on L ϱ . Moreover, one breaks down the data into two parts: the first is in L 2 , and the second has a small Strichrtz norm under the free bi-harmonic Schrödinger propagator. Then, one uses a standard fix point argument coupled with Strichartz estimates. First, one resolves the above problem in L 2 and then takes the considered equation with a perturbed source term. The local solution is complemented by a global one with induction. Since the free associated kernel e i ⋅ Δ 2 is unitary in L 2 , the Schrödinger equations seem more adapted to be mathematically studied in functional spaces based on L 2 , such as the Sobolev spaces W s , 2 (R N) . The main novelty here is investigating the above inhomogeneous bi-harmonic nonlinear Schrödinger problem in some Lebesgue spaces different from L 2 . One essential challenge is overcoming the lack of conservation laws, namely the mass and the energy, representing standard tools in the Schrödinger context. The second technical difficulty is the presence of an inhomogeneous source term, namely the singular decaying term | x | − τ , which causes some serious complications. In order to deal with the inhomogeneous term, one uses Lorentz spaces with the property | x | − τ ∈ L N τ , ∞ . [ABSTRACT FROM AUTHOR]

In this article, we propose a split-step finite element method (FEM) for the two-dimensional nonlinear Schrödinger equation (NLS) with Riesz fractional derivatives in space. The space-fractional NLS is first spatially discretized by finite element scheme and the semi-discrete variational scheme is obtained. We prove that it maintains the mass and energy conservation laws. Then, we establish a fully discrete split-step finite element scheme for the considered problem, which avoids the iteration at each time layer, thereby significantly reducing computational cost. The discrete mass conservation property and error estimate of this split-step finite element scheme is derived. Finally, illustrative tests and the numerical simulation of dynamic of wave solutions are included to confirm its effectiveness and capability. [ABSTRACT FROM AUTHOR]

This paper explores the oceanic wave characteristics exhibited by a modified integrable generalized (2+1)-dimensional nonlinear Schrödinger system of equations through variable coefficients. Two newly modified methods, specifically the improved nonlinear Ricatti equation method and the improved sub-equation method, have been proposed to investigate the aforementioned nonlinear system. Through the utilization of these methods, we successfully obtain traveling and solitary waves solutions for this nonlinear system. We emphasize several constraint conditions that serve to guarantee the existence of these solutions. More comprehensive information about the physical dynamical representation of some of the solutions presented is illustrated through graphical depictions. The Mathematica software package is employed to produce both three-dimensional and their corresponding contour plots, thereby improving the visualization and comprehension of the solutions. This paper illustrates that the two proposed approaches provide straightforward and efficient means of acquiring various types of soliton, rational, trigonometric, hyperbolic and exponential solutions. Moreover, they present a more potent mathematical tool for addressing a variety of other nonlinear partial differential equations that hold significance in the field of applied science and engineering. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of soliton interactions in higher-order nonlinear Schrödinger equation, which are commonly used to model multimode wave propagation in various physical scenarios, including nonlinear optics and shallow water. We constructed new exact solitary solutions in generalized forms of generalized higher-order nonlinear Schrödinger equation by using the extended generalized Riccati equation mapping method through symbolic computation. These wave solutions play a crucial role in engineering and various applied sciences. By assigning appropriate values to certain parameters in these solutions, novel graphical structures are generated, enhancing our understanding of the underlying physical phenomena in this model. These solutions shed light on the complex physical phenomena described by this dynamical model, and our computational approach is demonstrated to be simple, versatile, powerful, and effective. Furthermore, this method can also be applied to solve other complex higher-order NLSEs encountered in mathematical physics. [ABSTRACT FROM AUTHOR]

Given the advantages of machine learning method with accuracy and high efficiency, when compared with classical numerical methods, the cell-average based neural network (CANN) method is proposed to opens up a new field for solving and simulating solutions of nonlinear Schrödinger equations numerically. Inspired by the finite volume method for solving fluid flow problems, the CANN method seeks to explore shallow and fast neural network solvers to approximate the solution average difference between two consecutive time levels. The CANN method can be considered as a time discretization scheme, which is an explicit one-step method that evolves the solution forward in time. The CANN method is a network method of the finite volume type, which means that it is mesh dependent and is a local solver. The experimental results demonstrate that the CANN method yields satisfactory outcomes at different time steps within the specified time window. Once the neural network has been effectively trained, it can be applied to solve the NLS equation with different initial conditions. Furthermore, the CANN method demonstrates strong generalization ability in processing low-quality data with noise. To enhance the utilization of the CANN method in partial differential equations, we carry out numerical experiments on the NLS equation, which show the high practicality and accuracy of this method. [ABSTRACT FROM AUTHOR]

This study dynamically investigates the mathematical Ivancevic option pricing governing system in terms of conformable fractional derivative, which illustrates a confined Brownian motion identified with a non-linear Schrödinger type equation. This model describes the controlled Brownian motion that comes with a non-linear Schrödinger type equation. The solution to comprehend the market price fluctuations for the suggested model is developed through the application of a mathematical strategy. The modified Kudryashov analytical method is applied to find the fractional analytical exact soliton solution. The restrictions on the parameters required for these solutions to exist were also the result of this approach. The dynamical insights are examined and significant aspects of the phenomenon under study are discussed through the use of the bifurcation analysis. In the related dynamical system, the phase portraits of market price fluctuations are displayed at equilibrium points and for different parameter values. Additionally, the chaos analysis was carried out to show the quasi-periodic and periodic chaotic patterns. In order to track changes in market price, the sensitivity analysis of the studied model is also looked at and presented at different initial conditions. It was discovered that the model experienced price fluctuations as a result of minute changes in initial conditions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discover some new exact solutions for two kinds of nonlinear Schrödinger equation (NLSE), including the (2+1)-dimensional NLSE and the derivative NLSE, through the use of the variational principle method and amplitude ansatz method. We will find the solutions to these equations by selecting a trial function with a single nontrivial variational parameter, and it is continuous at all intervals in three cases from a region of a rectangular box, then using this trial function to obtain the functional integral and the Lagrangian of the system without any loss. After that, we approximate this trial function by quadratic polynomials with two free parameters rather than a piecewise linear ansatz function. Similarly, this trial function will be approximated by the tanh function. Additionally, we accept several new types of soliton solutions, including bright solitons, dark solitons, bright–dark solitary wave solutions, rational dark–bright solutions, and periodic solitary wave solutions. Also, conditions for the stability of the solutions will be proposed. These explanations are of great importance in the fields of applied science and engineering. The results will be shown through different types of graphs, including 2D, 3D, and contour plots. [ABSTRACT FROM AUTHOR]

We investigate the Cauchy problem for the fractional nonlinear Schrödinger (fNLS) equation. First, we establish the local well-posedness in the fractional Sobolev spaces Hγ with γ> 0 for the fNLS equation under less regularity assumptions for the nonlinear term than previous work. Based on the local well-posedness result, we then prove that the fNLS equation is globally wellposed in Hγ> with γ> > 0 if 4s/d < n > nc(g) and the initial data is sufficiently small. [ABSTRACT FROM AUTHOR]

We will explore, in any space dimension d ≥ 4 , the decay in the energy space for the damped magnetic Schrödinger equation with non-local nonlinearity and radial initial data in H 1 (R d) . We will also display new Morawetz identities and corresponding localized Morawetz estimates. [ABSTRACT FROM AUTHOR]

The Gross–Pitaevskii (GP) equation is a model for the description of the dynamics of Bose–Einstein condensates. Here, we consider the GP equation in a two‐dimensional setting with an external periodic potential in the x$x$‐direction and a harmonic oscillator potential in the y$y$‐direction in the so‐called tight‐binding limit. We prove error estimates which show that in this limit the original system can be approximated by a discrete nonlinear Schrödinger equation. The paper is a first attempt to generalize the results from [19] obtained in the one‐dimensional setting to higher space dimensions and more general interaction potentials. Such a generalization is a non‐trivial task due to the oscillations in the external periodic potential which become singular in the tight‐binding limit and cause some irregularity of the solutions which are harder to handle in higher space dimensions. To overcome these difficulties, we work in anisotropic Sobolev spaces. Moreover, additional non‐resonance conditions have to be satisfied in the two‐dimensional case. [ABSTRACT FROM AUTHOR]

We consider a model initial‐ and Dirichlet boundary–value problem for a nonlinear Schrödinger equation in two and three space dimensions. The solution to the problem is approximated by a conservative numerical method consisting of a standard conforming finite element space discretization and a second‐order, linearly implicit time stepping, yielding approximations at the nodes and at the midpoints of a nonuniform partition of the time interval. We investigate the convergence of the method by deriving optimal‐order error estimates in the L2$L^2$ and the H1$H^1$ norm, under certain assumptions on the partition of the time interval and avoiding the enforcement of a Courant‐Friedrichs‐Lewy (CFL) condition between the space mesh size and the time step sizes. [ABSTRACT FROM AUTHOR]

In the present paper, we study the existence and concentration of multiple normalized solutions to the following nonlinear biharmonic Schrödinger equation: ε 4 Δ 2 u + V ( x ) u = λ u + h ( u ) , x ∈ R N , ∫ R N | u | 2 d x = c 2 ε N , x ∈ R N , where ε > 0 is a positive parameter, λ ∈ R is unknown and appears as a Lagrange multiplier, and V is a positive potential such that inf Λ V < inf ∂ Λ V for some open bounded subset Λ ⊂ R N ( N ⩾ 5 ). Applying the penalization techniques and Ljusternik–Schnirelmann theory, we obtain multiple mormalized solutions u ε . When ε → 0, these solutions concentrates around a local minimum of V.This paper extends the results of Alves and Thin (2023), which considered the nonlinear Schrödinger equations with general nonlinearities, to the biharmonic Schrödinger equations. We develop a truncated skill to obtain the minimum via careful analysis. Moreover, we also obtain orbital stability of the solutions. [ABSTRACT FROM AUTHOR]

The different types of optical soliton solutions are investigated for the 2-dimensional nonlinear Schrödinger equation with an instantaneous Kerr nonlinearity. This equation contains the beam propagation in a multimode optical fiber with a parabolic index profile. Different soliton solutions are examined, such as Breather waves, Lump waves, Mixed types, Periodic cross kink, Multiwave, M-shape, M-shape rational one, two kinks, and rogue waves. To obtain these optical solitons, we apply the Hirota bilinear transformation approach. These results are spectral, temporal, and spatial features of ultrashort light pulses that can be controlled in novel ways by this kind of nonlinear multimode optical fiber, which is gaining popularity. Moreover, the 3-dimensional, 2-dimensional, and contour plots are drawn for some solutions that show their physical interactions by selecting different parameters. These results are very helpful for further study of dynamical systems. [ABSTRACT FROM AUTHOR]

In this paper, we mainly concern with the system of the coupled nonlinear Schrödinger equation with initial data in critical spaces in dimension 3. We first obtain the global well-posedness of the solution to the coupled nonlinear Schrödinger equation with initial data in a critical space $ W^{\frac {11}{7}, \frac 76}({\mathbb {R}}^3) $ W 11 7 , 7 6 ( R 3). The key is to derive a uniform bound of a modified energy $ \mathcal {E}(t) $ E (t) based on a decomposition for the solution as in Dodson [Global well-posedness for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space. Rev Mat Iberoam. 2022;38:1087–1100]. In addition, inspired by Dodson [Scattering for the defocusing, cubic nonlinear Schrödinger equation with initial data in a critical space. International mathematics research notices; 2023], we make a new decomposition for the solution and show the scattering to the coupled nonlinear Schrödinger equation with radially symmetric initial data in a critical Besov space $ B_{1,1}^2({\mathbb {R}}^3) $ B 1 , 1 2 ( R 3). [ABSTRACT FROM AUTHOR]

We consider semiclassically scaled, weakly nonlinear Schrödinger equations with external confining potentials and additional angular-momentum rotation term. This type of model arises in the Gross–Pitaevskii theory of trapped, rotating quantum gases. We construct asymptotic solutions in the form of semiclassical wave packets, which are concentrated in both space and in frequency around an classical Hamiltonian phase-space flow. The rotation term is thereby seen to alter this flow, but not the corresponding classical action. [ABSTRACT FROM AUTHOR]

In this paper, we employ the auxiliary equation method (AEM) to show the leverage of the nearest neighbors hopping and the boson–boson interaction in a zig zag optical lattice on the stressed optical solitons (os) and the Modulation Instability (MI) gain of cold bosonic atoms. The pedestal of the work is set on the continuum approximation of a cold bosonic atoms in a zig zag optical lattice which has demonstrated in detailed in refs. [Tala-Tebue E, Rezazadeh H, Djoufack ZI, et al. Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44; Djoufack ZI, TalaTebue E, FotsaNgafo F, et al. Quantum breathers associated with modulational instability in 1D ultracold boson in optical lattices involving nextnearest neighbor interactions. Optik 164(2018):575–589]. The results collected have enclosed bright, dark optical solitons and the combined bright-dark and bright-bright optical solitons. Compared the acquired results with Tala-Tebue et al. [Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44], further the os including U-shaped and W-shaped profile have been drop in and the powerful of the nearest neighbors hopping and the boson–boson interaction has been exposed across the spatiotemporal and plot evolution of the bright and dark os. Therewith, the MI bands have been stressed and it is observed some new attitude of the MI gain spectra under the induction of the model parameters. The collected results are new in literature in our knowledge and will improve the previous results acquired in refs. [Tala-Tebue E, Rezazadeh H, Djoufack ZI, et al. Optical solutions of cold bosonic atoms in a zig–zag optical lattice. Opt Quant Electron 53(2021):44; Djoufack ZI, TalaTebue E, FotsaNgafo F, et al. Quantum breathers associated with modulational instability in 1D ultracold boson in optical lattices involving nextnearest neighbor interactions. Optik 164(2018):575–589; Zhao XH, Tian B, Liu DY, et al. Dark solitons interaction for a (2+1)-dimensional nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain. Superlattices Microstruct 100(2016):587–595; Uddin MF, Hafez MG, Hammouch Z, et al. Periodic and rogue waves for Heisenberg models of ferromagnetic spin chains with fractional beta derivative evolution and obliqueness. Waves Random Complex Media 2020. https://doi.org/10.1080/17455030.2020.1722331; Inc M, Aliyu AI, Yusuf A. Optical solitons to the nonlinear Shrdingers equation with spatio-temporal dispersion using complex amplitude ansatz. J Mod Opt 2017;64(21):2273–2280]. [ABSTRACT FROM AUTHOR]

We study the nonlinear dynamics of perturbed, spectrally stable T-periodic stationary solutions of the Lugiato–Lefever equation (LLE), a damped nonlinear Schrödinger equation with forcing that arises in nonlinear optics. It is known that for each N ∈ N , such a T-periodic wave train is asymptotically stable against NT-periodic, i.e. subharmonic, perturbations, in the sense that initially nearby data will converge at an exponential rate to a (small) spatial translation of the underlying wave. Unfortunately, in such results both the allowable size of initial perturbations as well as the exponential rates of decay depend on N and, in fact, tend to zero as N → ∞ , leading to a lack of uniformity in the period of the perturbation. In recent work, the authors performed a delicate decomposition of the associated linearized solution operator and obtained linear estimates which are uniform in N. The dynamical description suggested by this uniform linear theory indicates that the corresponding nonlinear iteration can only be closed if one allows for a spatio-temporal phase modulation of the underlying wave. However, such a modulated perturbation is readily seen to satisfy a quasilinear equation, yielding an inherent loss of regularity. We regain regularity by transferring a nonlinear damping estimate, which has recently been obtained for the LLE in the case of localized perturbations to the case of subharmonic perturbations. Thus, we obtain a nonlinear, subharmonic stability result for periodic stationary solutions of the LLE that is uniform in N. This in turn yields an improved nonuniform subharmonic stability result providing an N-independent ball of initial perturbations which eventually exhibit exponential decay at an N-dependent rate. Finally, we argue that our results connect in the limit N → ∞ to previously established stability results against localized perturbations, thereby unifying existing theories. [ABSTRACT FROM AUTHOR]

The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schrödinger equation $$-\Delta u+\lambda u+V(x)u=|u|^{p-2}u,\quad u\in H^1_0(\Omega),\quad\int_\Omega u^2{\rm d}\,x=\rho^2,\quad\lambda\in\mathbb{R},$$ where $\Omega =\mathbb {R}^N$ or $\mathbb {R}^N\setminus \Omega$ is a compact set, $\rho >0$ , $V\ge 0$ (also $V\equiv 0$ is allowed), $p\in (2,2+\frac 4 N)$. The existence of a positive solution $\bar u$ is proved when $V$ verifies a suitable decay assumption (D ρ), or if $\|V\|_{L^q}$ is small, for some $q\ge \frac N2$ ($q>1$ if $N=2$). No smallness assumption on $V$ is required if the decay assumption (D ρ) is fulfilled. There are no assumptions on the size of $\mathbb {R}^N\setminus \Omega$. The solution $\bar u$ is a bound state and no ground state solution exists, up to the autonomous case $V\equiv 0$ and $\Omega =\mathbb {R}^N$. [ABSTRACT FROM AUTHOR]

Resorting to the spectral analysis of the 4 × 4 matrix spectral problem, we construct a 4 × 4 matrix Riemann–Hilbert problem to solve the initial value problem for the Hermitian symmetric space derivative nonlinear Schrödinger equation. The nonlinear steepest decent method is extended to study the 4 × 4 matrix Riemann–Hilbert problem, from which the various Deift–Zhou contour deformations and the motivation behind them are given. Through some proper transformations between the corresponding Riemann–Hilbert problems, the basic Riemann–Hilbert problem is reduced to a model Riemann–Hilbert problem, by which the long-time asymptotic behavior to the solution of the initial value problem for the Hermitian symmetric space derivative nonlinear Schrödinger equation is obtained with the help of the asymptotic expansion of the parabolic cylinder function and strict error estimates. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of multiple normalized solutions to the following class of elliptic problems - ϵ 2 Δ u + V (x) u = λ u + f (u) , in R N , ∫ R N | u | 2 d x = a 2 ϵ N , where a , ϵ > 0 , λ ∈ R is an unknown parameter that appears as a Lagrange multiplier, V : R N → [ 0 , ∞) is a continuous function, and f is a continuous function with L 2 -subcritical growth. It is proved that the number of normalized solutions is related to the topological richness of the set where the potential V attains its minimum value. In the proof of our main result, we apply minimization techniques, Lusternik-Schnirelmann category and the penalization method due to del Pino and Felmer (Calc. Var. Partial Differential Equations 4, 121–137 1996). [ABSTRACT FROM AUTHOR]

We study the Calogero–Moser derivative nonlinear Schrödinger NLS equation i∂tu+∂xxu+(D+|D|)(|u|2)u=0$$\begin{equation*} i\partial _t u +\partial _{xx} u + (D+|D|)(|u|^2) u =0 \end{equation*}$$posed on the Hardy–Sobolev space H+s(R)$H^s_+(\mathbb {R})$ with suitable s>0$s>0$. By using a Lax pair structure for this L2$L^2$‐critical equation, we prove global well‐posedness for s≥1$s \ge 1$ and initial data with sub‐critical or critical L2$L^2$‐mass ∥u0∥L22≤2π$\Vert u_0 \Vert _{L^2}^2 \le 2 \pi$. Moreover, we prove uniqueness of ground states and also classify all traveling solitary waves. Finally, we study in detail the class of multi‐soliton solutions u(t)$u(t)$ and we prove that they exhibit energy cascades in the following strong sense such that ∥u(t)∥Hs∼s|t|2s$\Vert u(t)\Vert _{H^s} \sim _s |t|^{2s}$ as t→±∞$t \rightarrow \pm \infty$ for every s>0$s > 0$. [ABSTRACT FROM AUTHOR]

We consider the nonlinear Schrödinger equation with a repulsive Dirac delta potential in one dimensional Euclidean space. We classify the global dynamics of even solutions with the same action as the high-frequency ground state standing wave solutions. [ABSTRACT FROM AUTHOR]

This paper presents an investigation into solutions of two types of nonlinear Schrödinger equations of logarithmic form. This model is of particular interest to researchers due to its vast application in the field of the implication of nanofibers. Through a comprehensive analysis, it was found that the solutions take the form of Gaussian-type solitary solutions. These solutions possess a variety of new structures due to the presence of many arbitrary constants and time-dependent functions in their general expressions. By fixing these constants and functions appropriately, Gaussian solitary waves for nonlinear perturbed Schrödinger equations can be represented, which could be beneficial in understanding the dynamics of these equations in nanooptical fibers. The results obtained from this study may provide a useful tool for further research into nonlinear Schrödinger equations and can be extended to fit more applications. [ABSTRACT FROM AUTHOR]

We are considered with the following nonlinear Schrödinger equation: −Δu+(λa(x)+1)u=f(u),x∈V,$$ -\Delta u+\left(\lambda a(x)+1\right)u=f(u),x\in V, $$on a locally finite graph G=(V,E)$$ G=\left(V,E\right) $$, where V$$ V $$ denotes the vertex set, E$$ E $$ denotes the edge set, λ>1$$ \lambda >1 $$ is a parameter, f(s)$$ f(s) $$ is asymptotically linear with respect to s$$ s $$ at infinity, and the potential a:V→[0,+∞)$$ a:V\to \left[0,+\infty \right) $$ has a nonempty well Ω$$ \Omega $$. By using variational methods, we prove that the above problem has a ground state solution uλ$$ {u}_{\lambda } $$ for any λ>1$$ \lambda >1 $$. Moreover, we show that as λ→∞$$ \lambda \to \infty $$, the ground state solution uλ$$ {u}_{\lambda } $$ converges to a ground state solution of a Dirichlet problem defined on the potential well Ω$$ \Omega $$. [ABSTRACT FROM AUTHOR]

For vector nonlocal reverse-space nonlinear Schrödinger equations, a binary Darboux transformation is formulated by using two sets of eigenfunctions and adjoint eigenfunctions. The resulting binary Darboux transformation has been decomposed into an N -fold product of single binary Darboux transformations. An application starting from zero seed potentials generates a class of soliton solutions. [ABSTRACT FROM AUTHOR]

This paper investigates a generalized form of the nonlinear Schrödinger equation characterized by a logarithmic nonlinearity. The nonlinear Schrödinger equation, a fundamental equation in nonlinear wave theory, is applied across various physical systems including nonlinear optics, Bose–Einstein condensates, and fluid dynamics. We specifically explore a logarithmic variant of the nonlinear Schrödinger equation to model complex wave phenomena that conventional polynomial nonlinearities fail to capture. We derive four distinct forms of the nonlinear Schrödinger equation with logarithmic nonlinearity and provide exact solutions for each, encompassing bright, dark, and kink-type solitons, as well as a range of periodic solitary waves. Analytical techniques are employed to construct bounded and unbounded traveling wave solutions, and the dynamics of these solutions are analyzed through phase portraits of the associated dynamical systems. These findings extend the scope of the nonlinear Schrödinger equation to more accurately describe wave behaviors in complex media and open avenues for future research into non-standard nonlinear wave equations. [ABSTRACT FROM AUTHOR]

In this work, we provide the analytical solution of several physical problems, such as heat conduction, Schröndinger, and space time vehicular equations, using the local fractional Laplace variational iteration approach. Examples have been provided to show the correctness and competency of the procedure, which have been validated using graphs. [ABSTRACT FROM AUTHOR]

Fractional calculus highlights its importance and has been expanded to many fields. This paper focuses on fractional integrable systems and their novel structural solutions to comply with the forefront development of nonlinear waves. Specifically, a system of fractional integrable coupled nonlinear Schrödinger (CNLS) equations are first derived from the associated linear spectral problem equipped with space-time conformable fractional derivatives of different orders. Then, by extending Darboux transformation (DT) and its generalized version to the derived fractional CNLS equations, we obtained fractional single-soliton solutions and double- and triple-semirational solutions that obey the dual power laws of independent variables. Under the dominant influences of space-time fractional orders, some novel solution structures are finally revealed based on the obtained solutions. In terms of characteristics, these spatial structures exhibit inclination, symmetry and spikes. Observing from the perspective of propagation, some of these obtained solutions have time-varying velocities and widths. From the perspective of spatiotemporal structures, some solutions form collapsed quadrangular pyramids. All of these novel solution structures cannot be supported by the integer-order CNLS equations. [ABSTRACT FROM AUTHOR]

The Schrödinger equation depends on the physical circumstance, which describes the state function of a quantum‐mechanical system and gives a characterization of a system evolving with time. The essential focus of proposed research is to observe the solution for fractional generalized nonlinear Schrödinger (FGNS) equation using q‐homotopy analysis transform method (q‐HATM). The fractional order derivative is taken in the Atangana‐Baleanu (AB) sense. The physical behaviours of achieved solution for FGNS equation are discussed and sketch out graphically. The existence of the solution for the FGNS equation is presented through theorems 4.1 to 4.3. The proposed numerical simulations confirm the advantages of the AB derivative through q‐HATM. Few numerical experiments were carried out to validate the proposed method. Moreover, numerical simulations are carried out to verify efficiency and robustness of the derived results by considering two cases. [ABSTRACT FROM AUTHOR]