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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]

In this paper, we address various optical soliton solutions and demonstrate the different dynamics of solitary waves to a (3+1)-dimensional nonlinear Schrödinger equation (NLSE) with parabolic law (NLSE) using a newly created powerful and effective method named as the extended generalized Riccati equation mapping method. This technique presents an organized manner to reveal the essential dynamics. There is great significance of the nonlinear Schrödinger equations and their several formulations in numerous fields of science, particularly in nonlinear optics, optical fibers, quantum electronics, and plasma physics. Through the use of numerical simulations and mathematical analysis, we explore the characteristics and behavior of these solitary wave solutions in a variety of scientific contexts. These results demonstrate the essential complexity of the governing equation and yield novel derived solutions. These solutions contribute to a better understanding of nonlinear wave phenomena by highlighting the fundamental dynamics establishing solitary waves in the NLSE. To enhance our wider knowledge, we provide effective graphic representations of the nonlinear wave structures in the derived solutions utilizing a variety of graphs, including 3D, 2D, and density plots. Moreover, a specific transformation has been applied to transform the system into a planar dynamical system, and several phase portraits have been presented to examine its behavior. Furthermore, upon introducing a perturbed term, chaotic behavior has been observed across different parameter values through both two-dimensional and three-dimensional graphics. [ABSTRACT FROM AUTHOR]

By using a Birkhoff normal form, we show that the transfer of mass through a finite barrier is impeded. As a result, we prove localization of arbitrary ℓ 2 data for polynomially long time for the nonlinear quasi-periodic Schrödinger equation on Z d with large disorder and weak nonlinearity. [ABSTRACT FROM AUTHOR]

The Hirota equation takes the form i q t + q x x + 2 | q | 2 q + i α (q x x x + 6 | q | 2 q x) = 0 , which can be regarded as a modified nonlinear Schrödinger equation with high-order dispersion and time-delay corrections to the cubic nonlinearity. In this paper, based on the global existence and L 2 -conservation law of the Hirota equation, we establish the L 2 -orbital stability for a family of Hasimoto soliton solutions of the Hirota equation with a Darboux transformation technique. [ABSTRACT FROM AUTHOR]

This research explores the dual-mode manifestation within the nonlinear Schrödinger equation, elucidating the amplification or absorption of paired waves. This study delves into the simultaneous generation of two distinct waves associated with the dual-mode phenomenon with three crucial parameters: phase velocity, nonlinearity and dispersive factor. The resulting wave phenomena from these solutions have implications across various fields, including fluid dynamics, water wave mechanics, ocean engineering and scientific inquiry. The study employs the modified Sardar sub-equation method to obtain the optical soliton solutions, encompassing various types such as dark, bright, singular, combo dark–singular, periodic singular and dark–bright solitons. The obtained results highlight the reliability and simplicity of the modified Sardar sub-equation method. Additionally, the paper delves into the parametric conditions crucial for shaping and sustaining these solitons. The research explores the interaction of dual waves and the variation in wave speed. Furthermore, dynamic phenomena are illustrated, and the physical implications of the solutions are interpreted using 3D and 2D plots with different parameter values. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem of the nonlinear Schrödinger equation posed on a (d + 1)-dimensional product manifolds M = Z d × S 1 , where Zd denotes the d-dimensional compact manifold under the eigenvalue assumption. We prove the uniform local well-posedness of Hs solution with the defocusing subcritical nonlinearity and extend the results of [Amer. J. Math., 2004, 126(3): 569–605], [Ann. Sci. École Norm. Sup. (4), 2005, 38(2): 255–301] and [Sci. China Math., 2015, 58(5): 1023–1046] to the more general geometry. The main tools in our argument are multilinear estimates and Bony's linearized technique. [ABSTRACT FROM AUTHOR]

We study low regularity local well-posedness of the nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity u ¯ 2 , posed on one-dimensional and two-dimensional tori. While the relevant bilinear estimate with respect to the X s , b -space is known to fail when the regularity s is below some threshold value, we establish local well-posedness for such low regularity by introducing modifications on the X s , b -space. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior in time of solutions to the one dimensional nonlinear Schrödinger equation with a subcritical dissipative nonlinearity λ | u | α u , where 0 < α < 2 , and λ is a complex constant satisfying Im λ > α | Re λ | 2 α + 1 . For arbitrary large initial data, we present the uniform time decay estimates when 4 / 3 ≤ α < 2 , and the large time asymptotics of the solution when 7 + 145 12 < α < 2 . The proof is based on the vector fields method and a semiclassical analysis method. [ABSTRACT FROM AUTHOR]

We construct some new stochastic solutions for the fractional nonlinear Schrödinger type equation. Specifically, we introduce these stochastic solutions via geometric and beta distributions. Through these two statistical distributions, the expectation corresponding to solutions are illustrated to depict the influence of random parameters. Actually, the presented results can be used to explain many interesting phenomena in superfluid, optical fibers, plasma physics and vibration propagation in the ground and in structures. Some graphs are offered to illustrate the wave modes of the expectation corresponding to stochastic solutions. Finally, our analysis can be implemented for many other models in real life. [ABSTRACT FROM AUTHOR]

Some combined high order compact (CHOC) schemes are proposed for non-self-adjoint and nonlinear Schrödinger equation (NSANLSE). There are first order and second order spatial derivatives u x ‾ , u x x in the NSANLSE. If one uses classical high order compact schemes to approximate u x x and u x ‾ separately, it will widen the bandwidth in practical coding due to matrix multiplication. This will partly counteract the advantages of high order compact. To overcome the deficiency, one solves the spatial derivatives simultaneously by combining them. In other words, it solves u x j n and u x x j n simultaneously in terms of u j. The idea is applied to discretize NSANLSE in space. Two efficient numerical schemes are proposed for NSANLSE. The stability and convergence of the new schemes are analyzed theoretically. Numerical experiments are reported to verify the new schemes. [ABSTRACT FROM AUTHOR]

In this work, we investigated the spatial evolution of optical power in a closed-form optical waveguide configuration consisting of six passive waveguides and each of the waveguides exhibits equal strength of Kerr nonlinearity. We considered only nearest neighbor interaction between the waveguides. We found that in the case of low Kerr nonlinearity, spatial evolution of optical power shows identical synchronization. But when we increased the strength of Kerr nonlinearity, we discovered that spatial evolution of optical power in all waveguides exhibits independent characteristics. On the other hand, we have studied the impact of the coupling constant on the synchronization dynamics of our system. Our findings showed us that strong coupling can strengthen the collective dynamics in the presence of strong Kerr nonlinearity. From our results, we can conclude that Kerr nonlinearity in our system plays the role of disorder parameter that destroys as well as alters the synchronization behavior of evolution of optical power in the waveguides and coupling constant, on the other hand, plays the role of an antagonist and restores synchronization in the model. [ABSTRACT FROM AUTHOR]

This study investigates a cubic-quartic nonlinear Schrödinger model with third- and fourth-order dispersions without the disturbance in parabolic law media and group velocity dispersion (GVD). The analytical method is used to produce traveling wave solutions because the inverse scattering transform is unable to solve the Cauchy problem for this equation. The utilized analytical approach is novel and providing generalized families of solutions as compared to other, thus, prior to this study, there was not existing any work in which suck kinds of solutions were exist. The propagating solitons are driven using the unified auxiliary equation approach. As a result, the aforementioned technique is learned in the following areas: singular soliton, shock solution, mixed-singular, mixed trigonometric, mixed shock single solution, plane solutions, exponential, trigonometry, mixed-hyperbolic solution, periodic and mixed periodic solutions, and trigonometry. The stability of the topic under consideration is shown by the modulational instability (MI) study. The appropriate parametric values are shown alongside the graphical representation. [ABSTRACT FROM AUTHOR]

The balance of nonlinearity and dispersion in optical fiber medium gives rise to a constantly propagating pulse. Such distortion less waves have attracted potential interest. The dynamics of optical solitons are governed by the nonlinear Schrödinger's equation (NLSE). A modified form of NLSE which incorporates group velocity dispersion (GVD) and the Kerr law nonlinearity is recently adopted for the study of such waves. Here, we investigate the nonlinear Schrödinger–Hirota's equation (NLSHE) using the Sardar subequation approach. Some novel solutions to the NLSHE corresponding to the bright, dark, kink, and cusp solitons have been reported. Additionally, the spatial and temporal dynamics of these solitons provide deep insight into the behavior of these solutions. The stability study is carried out via modulation instability (MI) concept. Our work might have benefits in the propagation of these pulses in the optical fiber for communication. [ABSTRACT FROM AUTHOR]

In this paper, we study explicit approximations of stochastic nonlinear Schrödinger equations (SNLSEs). We first prove that the classical explicit numerical approximations are divergent for SNLSEs with polynomial nonlinearities. To enhance the stability, we propose a kind of explicit numerical approximations, and establish the regularity analysis and strong convergence rate of the proposed approximations for SNLSEs. There are two key ingredients in our approach. One ingredient is constructing a logarithmic auxiliary functional and exploiting the Bourgain space to prove new regularity estimates of SNLSEs. Another one is providing a dedicated error decomposition formula and presenting the tail estimates of underlying stochastic processes. In particular, our result answers the strong convergence problem of numerical approximation for 2D SNLSEs. [ABSTRACT FROM AUTHOR]

In this paper, a fractional dual-core optical fiber model is considered, which is described as a coupled fractional nonlinear Schrödinger equation. The fractional (e−ϕ(ξ))-expansion method and fractional rational sinh–cosh function method have been successfully employed to derive various new soliton solutions for the coupled fractional nonlinear Schrödinger equations. These obtained solutions are of great significance for a deeper understanding of the corresponding physical phenomena. Finally, some 3D and 2D graphs are provided to elaborate the physical explanation of the obtained solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of solutions for the following nonlinear Schrödinger equation with L 2 -norm constraint { (− Δ) s u = λ u + μ | u | q − 2 u + | u | p − 2 u in Ω , u = 0 on ∂ Ω , ∫ Ω u 2 d x = a 2 , where s ∈ (0 , 1) , μ , a > 0 , N ≥ 3 , 2 < q < p < 2 + 4 s N , (− Δ) s is the fractional Laplacian operator, Ω ⊆ R N is an exterior domain, that is, Ω is an unbounded domain in R N with R N ∖ Ω non-empty and bounded and λ ∈ R is Lagrange multiplier, which appears due to the mass constraint | | u | | L 2 (Ω) = a . In this paper, we use Brouwer degree, barycentric functions and minimax method to prove that for any a > 0 , there exists a positive solution u ∈ H 0 s (Ω) for some λ < 0 if R N ∖ Ω is contained in a small ball. [ABSTRACT FROM AUTHOR]