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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]

We study nonlinear Schrödinger equation − Δ u + (λ + V (y)) u = u p ε − 1 in R N , with prescribed mass ∫ R N u 2 = a , where λ is a Lagrange multiplier, V (y) is a real-valued potential, a ∈ R + is a constant, p ε = p ¯ ± ε and p ¯ = 2 + 4 N is the L 2 -critical exponent. Bartsch et al. (2021) [1] proved the existence of a solution with the assumption that V decays at infinity and is non-negative. In this paper, we prove that it is the number of the critical points of the potential V that affects the existence and the number of solutions for this problem. We also prove a local uniqueness for the solutions we construct. [ABSTRACT FROM AUTHOR]

In this paper, we examine generalized nonlinear Schrödinger equation (GNLSE) for lump, breather lump wave, rogue wave, periodic cross kink wave, lump with one kink, lump with two-kink, interaction between lump periodic and kink wave, periodic cross lump wave, periodic wave and multi-wave. Ansatz transformations will be used to study homoclinic breather, kink cross rational, M-shaped rational, periodic cross rational, M-shaped rational with one kink, M-shaped rational with two-kink, M-shaped interaction with rogue and kink and M-shaped interaction with periodic and kink. We will also study the stability of the obtained solutions. Additionally, we present three-dimensional, two-dimensional, contour, and stream plot illustrations for our graphs. [ABSTRACT FROM AUTHOR]

We consider the L 2 -supercritical nonlinear Schrödinger equation with a repulsive Dirac delta potential in one dimensional space. In a previous work, we clarified the global dynamics of even solutions with the same action as the high-frequency ground state standing wave solutions. In that case, there are obvious non-scattering global solutions, i.e., the standing waves. In this paper, we show a scattering and blow-up dichotomy for threshold even solutions in the low-frequency case. We emphasize that this dichotomy still holds at the critical frequency between high and low. [ABSTRACT FROM AUTHOR]

We show new global well-posedness results for mass-critical nonlinear Schrödinger equations on tori in one and two dimensions. For the quintic nonlinear Schrödinger equation on the circle we show global well-posedness for initial data in H s (T) for s > 1 3 and ‖ u 0 ‖ L 2 (T) ≪ 1. In two dimensions we show global well-posedness on possibly irrational tori for s > 3 5. In the focusing case we need to consider small L 2 -norm, whereas in the defocusing case large mass is covered. [ABSTRACT FROM AUTHOR]

We prove that the uniform radius of spatial analyticity σ(t)$$ \sigma (t) $$ of solution at time t$$ t $$ to the one‐dimensional fourth‐order nonlinear Schrödinger equation i∂tu−∂x4u=|u|2u$$ i{\partial}_tu-{\partial}_x^4u={\left|u\right|}^2u $$cannot decay faster than 1/t$$ 1/\sqrt{t} $$ for large t$$ t $$, given that the initial data are analytic with fixed radius σ0$$ {\sigma}_0 $$. The main ingredients in the proof are a modified Gevrey space, a method of approximate conservation law, and a Strichartz estimate for free wave associated with the equation. [ABSTRACT FROM AUTHOR]

This work investigates the non‐global existence of solutions for the inhomogeneous nonlinear Schrödinger problem without gauge invariance. Indeed, initially, we establish the non‐global existence of certain mass‐subcritical solutions. Subsequently, we demonstrate the non‐global existence of energy‐subcritical solutions characterized by small initial data. These results are are acknowledged to be unsuccessful for the classical gauge invariant inhomogeneous nonlinear Schrödinger equation (INLS). [ABSTRACT FROM AUTHOR]

We consider a derivative nonlinear Schrödinger equation with general nonlinearlity: i ∂ t u + ∂ x 2 u + i | u | 2 σ ∂ x u = 0 , In Tang and Xu (J Differ Equ 264(6):4094–4135, 2018), the authors prove the stability of two solitary waves in energy space for σ ∈ (1 , 2) . As a consequence, there exists a solution of the above equation which is close arbitrary to sum of two solitons in energy space when σ ∈ (1 , 2) . Our goal in this paper is proving the existence of multi-solitons in energy space for σ ⩾ 3 2 . Our proofs proceed by fixed point arguments around the desired profile, using Strichartz estimates. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the Cauchy problem for the three dimensional nonlinear Schrödinger equation with a constant magnetic field. We first establish sufficient conditions that ensure the existence of global in time and finite time blow-up solutions. In particular, we derive sharp thresholds for global existence versus blow-up for the equation with mass-critical and mass-supercritical nonlinearities. We next prove the existence and orbital stability of normalized standing waves which extend the previous known results to the mass-critical and mass-supercritical cases. To show the existence of normalized solitary waves, we present a new approach that avoids the celebrated concentration-compactness principle. Finally, we study the existence and strong instability of ground state standing waves which greatly improve the previous literature. [ABSTRACT FROM AUTHOR]

A new generalised non-local nonlinear Schrödinger (NLS) equation is introduced which possesses a Lax pair and is parity–time (PT)-symmetric. Thus, it is confirmed that the generalised non-local NLS equation is integrable. The inverse scattering transform for the generalised non-local NLS equation is developed using a Riemann–Hilbert problem for rapidly decaying initial data and an approach for finding pure soliton solutions is described. The analytical characteristics of the eigenfunctions, scattering data and their symmetries are discussed. Finally, using Mathematica some important two-dimensional plots of the wave solutions are shown to illustrate the dynamics of the model. [ABSTRACT FROM AUTHOR]

In nonlinear Schrödinger equations (NLSEs), the third-order generalized NLSE is a significant sculpture which is utilized for modeling ultrashort pulses in fiber optics. In this study, we obtained wave and soliton solutions by using new extended and modified Rational expansion method to get several types of soliton such as bright solitons, dark solitons, perodic solitons and traveling waves. In three-dimensional and two-dimensional plots, we present graphical representations in dissimilar structures of some solutions to understand the phenomena physically. The development and achievements of computing show the power and effectiveness of current technology. In addition, we are able to resolve various other high-order NLSEs with the assistance of effortless and effectual technique. [ABSTRACT FROM AUTHOR]

In this paper, we examine generalized nonlinear Schrödinger equation (GNLSE) for lump, breather lump wave, rogue wave, periodic cross kink wave, lump with one kink, lump with two kink, interaction between lump periodic and kink wave, periodic cross lump wave, periodic wave, and multiwave. Ansatz transformations will be used to study homoclinic breather, kink cross rational, M-shaped rational, periodic cross rational, M-shaped rational with one kink, M-shaped rational with two kink, M-shaped interaction with rogue and kink and M-shaped interaction with periodic and kink. We will also study the stability of the obtained solutions. Additionally, we present 3D, 2D, contour, and stream plot illustrations for our graphs. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of normalized solutions to the following nonlinear Schrödinger equations with critical growth and potential - Δ u + V (ϵ x) u = λ u + μ | u | q - 2 u + f (u) in R N , ∫ R N | u | 2 d x = a , where a > 0 , μ > 0 , q ∈ (2 , 2 + 4 / N) , ϵ > 0 is a small parameter, λ ∈ R will arise as Lagrange multiplier, the potential V : R N → 0 , + ∞ is a continuous function that satisfies some suitable conditions and f ∈ C (R , R) enjoys critical exponential growth of Trudinger–Moser type when N = 2 , and f (u) = | u | 2 ∗ - 2 u when N ≥ 3 and 2 ∗ = 2 N N - 2 . When ϵ > 0 is sufficiently small, we prove that the existence of normalized solutions by applying truncation techniques, and the relationship between the number of positive solutions and the topology of the set where V attains its minimum is obtained by using Ljusternik–Schnirelmann theory. [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]

The nonlinear Schrödinger equation in the weakly nonlinear regime with random Gaussian fields as initial data is considered. The problem is set on the torus in any dimension greater than two. A conjecture in statistical physics is that there exists a kinetic time scale depending on the frequency localization of the data and on the strength of the nonlinearity, on which the expectation of the squares of moduli of Fourier modes evolve according to an effective equation: the so‐called kinetic wave equation. When the kinetic time for our setup is 1, we prove this conjecture up to an arbitrarily small polynomial loss. When the kinetic time is larger than 1, we obtain its validity on a more restricted time scale. The key idea of the proof is the use of Feynman interaction diagrams both in the construction of an approximate solution and in the study of its nonlinear stability. We perform a truncated series expansion in the initial data, and obtain bounds in average in various function spaces for its elements. The linearized dynamics then involves a linear Schrödinger equation with a corresponding random potential whose operator norm in Bourgain spaces we are able to estimate on average. This gives a new approach for the analysis of nonlinear wave equations out of equilibrium, and gives hope that refinements of the method could help settle the conjecture. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry analysis method is applied to time fractional coupled (2 + 1)-dimensional nonlinear Schrödinger equations. We obtain all the Lie symmetries and reduce the (2 + 1)-dimensional fractional partial differential equations with Riemann-Liouville fractional derivative to (1 + 1)-dimensional counterparts with Erdélyi-Kober fractional derivative. Then we obtain the power series solutions and prove their convergence. In addition, the conservation laws for the governing equations are constructed by using the generalization of the Noether operators and the new conservation theorem. [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]

We study the inhomogeneous nonlinear Schrödinger equation in the L2$$ {L}^2 $$‐critical case p=4−2hN$$ p=\frac{4-2h}{N} $$ (0

Let H = (−Δ)m + V be a higher-order elliptic operator on L2(ℝn), where V is a general bounded decaying potential. This paper focuses on the global Kato smoothing and Strichartz estimates for solutions to Schrödinger-type equation associated with H. In particular, we first establish sharp global Kato smoothing estimates for eitH, based on uniform resolvent estimates of Kato–Yajima type for the absolutely continuous part of H. As a consequence, we also obtain optimal local decay estimates. Using these local decay estimates, we then prove the full set of Strichartz estimates, including the endpoint case. Notably, we derive Strichartz estimates with sharp smoothing effects for higher-order cases with rough potentials, which are applicable to the study of nonlinear higher-order Schrödinger equations. Finally, we introduce new uniform Sobolev estimates of the Kenig–Ruiz–Sogge type, incorporating an additional derivative term, which are crucial for establishing the sharp Kato smoothing estimates. [ABSTRACT FROM AUTHOR]

This paper investigates the nonlinear conformable Schrödinger equation with nonlocal nonlinearities and Kudryashov’s arbitrary refractive index. The study employs the Uniform Method (UM) and the Extended Hyperbolic Function Method (EHFM) to derive various optical soliton solutions for this present conformable equation. To demonstrate the importance of the novel solutions, the paper presents two-dimensional, three-dimensional, and contour plots, illustrating kink-type, wave, bright, bell-shaped, and mixed dark–bright soliton solutions. Further, the impact of the fractional order parameter, temporal parameter, and nonlinearity coefficient on these optical solutions is analyzed, providing valuable insights into the conformable nonlinear Schrödinger model. The behavior of these optical solutions is analyzed through illustrative graphs that account for variations in the conformable order derivative, temporal parameter, and nonlinearity coefficient. The methodologies applied in this research show potential for broader application to various nonlinear Schrödinger equations in fields such as applied mathematics and nonlinear optics. The research suggests that these methods could be applied in the future to investigate other differential equations with fractional and integer orders across different fields of applied sciences. This study expands our understanding of nonlinear optics and has potential practical implications in various fields like optical signal processing, laser technology, and telecommunications. [ABSTRACT FROM AUTHOR]

In this paper, we study the following nonlinear magnetic Schrödinger equation with logarithmic nonlinearity −(∇+iA(x))2u+λV(x)u=|u|q−2u+ulog|u|2,u∈H1(RN,C),$$\begin{equation*} \hspace*{24pt}-(\nabla +\text{i}A(x))^2u+\lambda V(x)u =|u|^{q-2}u+u\log |u|^2,\ u\in H^1(\mathbb {R}^N,\mathbb {C}), \end{equation*}$$where the magnetic potential A∈Lloc2RN,RN$A \in L_{l o c}^2\left(\mathbb {R}^N, \mathbb {R}^N\right)$, 20$20$ is a parameter and the nonnegative continuous function V:RN→R$V: \mathbb {R}^N \rightarrow \mathbb {R}$ has the deepening potential well. Using the variational methods, we obtain that the equation has at least 2k−1$2^k-1$ multi‐bump solutions when λ>0$\lambda >0$ is large enough. [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]

Introduction: The moment quantities associated with the nonlinear Schrödinger equation offer important insights into the evolution dynamics of such dispersive wave partial differential equation (PDE) models. The effective dynamics of the moment quantities are amenable to both analytical and numerical treatments. Methods: In this paper, we present a data-driven approach associated with the "Sparse Identification of Nonlinear Dynamics" (SINDy) to capture the evolution behaviors of such moment quantities numerically. Results and Discussion: Our method is applied first to some well-known closed systems of ordinary differential equations (ODEs) which describe the evolution dynamics of relevant moment quantities. Our examples are, progressively, of increasing complexity and our findings explore different choices within the SINDy library. We also consider the potential discovery of coordinate transformations that lead to moment system closure. Finally, we extend considerations to settings where a closed analytical form of the moment dynamics is not available. [ABSTRACT FROM AUTHOR]

This paper aims to introduce a Kaup–Newell type matrix eigenvalue problem with four potentials, based on a specific matrix Lie algebra, and construct its associated Liouville integrable Hamiltonian hierarchy, through the zero curvature formulation. The Liouville integrability of the resulting hierarchy is shown by determining its recursion operator and bi-Hamiltonian formulation. An illustrative example of combined derivative nonlinear Schrödinger equations with two arbitrary constants is explicitly presented. [ABSTRACT FROM AUTHOR]

In this paper, we study the highly dispersive nonlinear perturbation Schrödinger equation, which has arbitrary form of Kudryashov's with sextic‐power law refractive index and generalized nonlocal laws. For the equation has highly dispersive nonlinear terms and higher order derivatives, it cannot be integrated directly, so we build an integrable factor equation for the approximated equation and apply the trial equation method and the complete discrimination system for polynomial method to create new soliton solutions. On the other hand, we use the bifurcation theory to qualitatively analyze the equation and find the model has periodic solutions, bell‐shaped soliton solutions, and solitary wave solutions via phase diagrams. The topological stability of the solutions with respect to the parameters is explored in order to better understand the effect of parameters perturbations on the stability of the model's solutions. Furthermore, we analyze the modulation instability and give the corresponding linear criterion. After accounting for external perturbation terms, we analyze the chaotic behaviors of the equation through the largest Lyapunov exponents and phase diagrams. [ABSTRACT FROM AUTHOR]

The collisionless unmagnetized five components dusty plasmas have been considered to study the significance of the face‐to‐face collision of single‐ and multi‐solitons, phase shifts due to face‐to‐face collision, and instability of the dust acoustic waves in Jupiter's environment. To study the consequence of face‐to‐face collision extended Poincaré–Lighthill–Kuo method is used to derive the two‐sided nonlinear Korteweg–De Vries (KdV) equation. Besides, to investigate the dust acoustic wave's instability, the nonlinear Schrödinger equation has been derived following the procedure of Slathia et al. The multi‐soliton solutions are determined by employing the Hirota bilinear method. The concerned parameters in the issue play a crucial role in the phase shift (generated by the face‐to‐face collision); the collisional process of single‐, double‐, triple‐, and quadruple‐soliton; and the generation of dust acoustic solitons. Furthermore, the plasma parameters are crucial in the formation of dark solitons and envelop shocks. [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]

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]

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]

In this paper, exact wave propagation patterns and chaotic dynamical behaviors of the extended (3+1)-dimensional nonlinear Schrödinger equation are studied. The topological structure of the dynamic system of the equation is studied by the complete discrimination system for the cubic polynomial method, in which the existence conditions of the soliton solutions and periodic solutions are obtained. Then, by the trial equation method, thirteen exact solutions are obtained, including solitary wave solutions, triangular function solutions, rational solutions and the elliptic function double periodic solutions, especially the elliptic function double periodic solutions. Finally, it is found that the system has chaotic behaviors when given the appropriate perturbations. [ABSTRACT FROM AUTHOR]

A method is proposed for transferring the spectrum of an optical signal from the vicinity of one carrier frequency to the vicinity of another using an off-center chirp. This technique makes it possible to divide one channel into several using a time chirp, the duration of which is a multiple of the bit interval. It is shown that from the point of view of minimizing distortions, it is preferable to split the signal into channels at the beginning of the communication line. [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]

The perturbed nonlinear Schrödinger equation plays a crucial role in various scientific and technological fields. This equation, an extension of the classical nonlinear Schrödinger equation, incorporates perturbative effects that are essential for modeling real-world phenomena more accurately. In this paper, we investigate the traveling wave solutions of the perturbed nonlinear Schrödinger equation using the bifurcation theory of dynamical systems. Graphical presentations of the phase portrait are provided, revealing the traveling wave solutions under various conditions. By employing the auxiliary equation method, we derive a variety of solutions including periodic, dark, singular and bright optical solitons. To provide comprehensive and clearer depiction of the model’s behavior 2D, contour and 3D graphical representations are offered. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. This study expands the scope of known exact solutions and their stability qualities which is offering an extensive analytical technique which enhances previous research. The novelty of our research lies in its examination of bifurcation analysis and the auxiliary equation method within the context of a perturbed nonlinear Schrödinger wave equation for the first time. By integrating these two perspectives, this paper contributes to establishing the complex dynamics and stability characteristics of soliton solutions under perturbations. [ABSTRACT FROM AUTHOR]

In this work, we analyze the asymptotic behaviors of high‐order rogue wave solutions with multiple large parameters and discover novel rogue wave patterns, including modified claw‐like, one triple root (OTR)‐type, modified OTR‐type, two triple roots (TTR)‐type, semimodified TTR‐type, and modified TTR‐type patterns. A correlation is established between these rogue wave patterns and the root structures of the Adler–Moser polynomials with multiple roots. At the positions in the (x,t)$(x,t)$‐plane corresponding to simple roots of the Adler–Moser polynomials, these high‐order rogue wave patterns asymptotically approach first‐order rogue waves. At the positions in the (x,t)$(x,t)$‐plane corresponding to multiple roots of the Adler–Moser polynomials, these rogue wave patterns asymptotically tend toward lower‐order fundamental rogue waves, dispersed first‐order rogue waves, or mixed structures of these rogue waves. These structures are related to the root structures of special Adler–Moser polynomials with new free parameters, such as the Yablonskii–Vorob'ev polynomial hierarchy, among others. Notably, the positions of the fundamental lower‐order rogue waves or mixed structures in these rogue wave patterns can be controlled freely under specific conditions. [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]