Nonlinear Schrodinger Equation Research Articles

PINN for solving forward and inverse problems involving the integrable two dimensional nonlocal equations

Abstract In this paper, the physics informed neural network (PINN) deep learning method is applied to solve the two dimensional nonlocal equations including partial reverse space y-nonlocal Mel'nikov equation, partial reverse space-time nonlocal Mel'nikov equation and nonlocal two dimensional NLS equation. By PINN method, we successfully derive data-driven two soliton solution, lump solution and rogue wave solution. Numerical simulation results indicate that the error range between the data-driven solution and the exact solution is relatively small, which verifies the effectiveness of the PINN deep learning method for solving high dimensional nonlocal equation.
Moreover, the parameter discovery of the partial reverse space-time nonlocal Mel'nikov equation is analysed in terms of its soliton solution for the first time.

Nuclear acoustic envelope soliton in a relativistically degenerate magneto-rotating stellar plasma

The study explores a model called Relativistic Degenerate Magneto-Rotating Quantum Plasma (RDMRQP), which comprises a static heavy nucleus, an inertial non-degenerate light nucleus, and warm non-relativistic or ultra-relativistic electrons. The focus is on observing the emergence of Nucleus-Acoustic Envelope Solitons (NAESs). Using the reductive perturbation method, a Nonlinear Schrödinger equation (NLSE) is derived to characterize the properties of NAESs. We have analysed the Peregrine breather soliton solution. The investigation reveals that the temperature of warm degenerate species, plasma system’s rotational speed, and the presence of heavy nucleus species can alter the fundamental features (height and width) of NAESs in the WDMRQP system. The study emphasizes the existence of only positive NA wave potential. Additionally, a phase plane analysis is conducted to gain a deeper understanding of the parametric dependencies. Through detailed mathematical and numerical analysis, the study avoids overloading with complex mathematics while demonstrating parametric dependence via phase portrait analysis. The research augments the envelop soliton model with breather mode solutions and discusses modulation instability using the Benjamin-Feir Index, highlighting the significance of solitons in star formation. Envelop solitons, stable waves within stars and proto-stars, influence energy transport and stellar evolution, playing a crucial role in the accretion process and formation of stable structures. Key findings include the effects of various parameters on NAESs’ generation and propagation in which non-relativistic and ultra-relativistic electrons support NAESs, with amplitude and width influenced by temperature, rotational frequency, inclination angle, and the presence of a static heavy nucleus. The research is applicable to hot white dwarfs and neutron stars, suggesting further exploration of quantum effects and non-planar or arbitrary amplitude NAESs.

Pressure scaling of femtosecond laser filamentation in air: Prospects for long-range atmospheric propagation

The principal trend of the last decades is the use of laser systems generating ultrashort (femtosecond) laser pulses of gigawatt and terawatt power in various spheres of human activity, in particular, in atmospheric researches. One of the important challenges within this problem is the use of the unique nonlinear optical phenomena manifesting in the atmosphere as pulse self-focusing, light-induced ionization of the medium, plasma generation and laser filamentation for remote laser diagnostics of the aerosol-gas constituents and efficient delivery of concentrated optical energy over long distances in the atmosphere. This brings to the forefront the problem of increasing the actual length of the nonlinear interaction of an ultrashort laser pulse with medium to kilometer distances whereas real physical length of most existing optical paths is usually limited by several hundred meters. We propose and theoretically substantiate an effective way to solve the problem of long-range atmospheric filamentation by transforming the propagation medium itself using short laboratory traces and optical cuvettes with gases of increased (over-atmospheric) pressure. Following the scaling laws of the optical characteristics of pressurized medium, the coefficients describing the interaction between optical wave and medium increase along with the increase in gas pressure. By the numerical simulations in the framework of the nonlinear Schrodinger equation, we show that this makes it possible to emulate kilometers-long atmospheric traces for laser beams of centimeter diameter and terawatt pulse power on the laboratory scale (tens of meters).

Elzaki Transform Homotopy Analysis Techniques for Solving Fractional (2+1)-D and (3+1)-D Nonlinear Schrodinger Equations

Elzaki Transform Homotopy Analysis Techniques for Solving Fractional (2+1)-D and (3+1)-D Nonlinear Schrodinger Equations

Numerical simulation of Alfvén dark envelope soliton in Hall-MHD plasmas

This study investigates the propagation of dark Alfvén solitons in low-β magnetized plasma using Hall-magnetohydrodynamic (Hall-MHD) numerical simulations. The rational solution of the nonlinear Schrödinger equation (NLSE) is presented, which is proposed as an effective tool for studying dark envelope solitons in plasma. Our results show a high degree of agreement between numerical simulations and analytical solutions derived from the NLSE via the reductive perturbation method. This agreement validates our modeling and computational approach. In addition, the simulations confirm the existence of dark Alfvén solitons in magnetized plasma. This work demonstrates the effectiveness of Hall-MHD simulations in studying complex plasma phenomena, contributing to the broader understanding of soliton propagation in plasma environments.

Inverse scattering transform for the higher-order integrable discrete nonlinear Schrodinger equation with nonzero boundary conditions

Inverse scattering transform for the higher-order integrable discrete nonlinear Schrodinger equation with nonzero boundary conditions

Normalized solutions for Sobolev critical fractional Schrödinger equation

Abstract In the present study, we investigate the existence of the normalized solutions to Sobolev critical fractional Schrödinger equation: ( − Δ ) s u + λ u = f ( u ) + ∣ u ∣ 2 s * − 2 u , in R N , ( P m ) ∫ R N ∣ u ∣ 2 d x = m 2 , \hspace{14em}\left\{\begin{array}{ll}{\left(-\Delta )}^{s}u+\lambda u=f\left(u)+{| u| }^{{2}_{s}^{* }-2}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{12em}<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" voffset="-2.9ex">\left({P}_{m})</mml:mpadded>\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={m}^{2},\hspace{1.0em}\end{array}\right. where 0 < s < 1 0\lt s\lt 1 , N ≥ 2 N\ge 2 , m > 0 m\gt 0 , 2 s * ≔ 2 N N − 2 s {2}_{s}^{* }:= \frac{2N}{N-2s} , λ \lambda is an unknown parameter that will appear as a Lagrange multiplier, and f f is a mass supercritical and Sobolev subcritical nonlinearity. Under fairly general assumptions about f f , with the aid of the Pohozaev manifold and concentration-compactness principle, we obtain a couple of the normalized solution to ( P m ) \left({P}_{m}) . We mainly extend the results of Appolloni and Secchi (Normalized solutions for the fractional NLS with mass supercritical nonlinearity, J. Differential Equations 286 (2021), 248–283) concerning the above problem from Sobolev subcritical setting to Sobolev critical setting, and also extend the results of Jeanjean and Lu (A mass supercritical problem revisited, Calc. Var. 59 (2020), 174) from classical Schrödinger equation to fractional Schrödinger equation involving Sobolev critical growth. More importantly, our result settles an open problem raised by Soave (Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case, J. Funct. Anal. 279 (2020), 108610), when s = 1 s=1 .

Exploring the dynamic interplay of intermodal and higher order dispersion in nonlinear negative index metamaterials

Our study delves into the intricate interplay of various factors within metamaterials, with a focus on modulation instability. Through our research, we elucidate the intricate dynamics involving intermodal dispersion, self-steepening effect, higher order dispersion, and plane wave amplitude, showcasing their competition and influence on modulation instability phenomena. We aim to explore the impact of intermodal dispersion and higher-order effects by numerically solving the generalized nonlinear Schrödinger equation (NLSE), which models the propagation of a few-cycle pulse in a nonlinear metamaterial. Our modulation instability (MI) analysis captures the complex dynamics these factors introduce. We demonstrate the spatiotemporal evolution of MI under different parameter values, revealing how these variations influence the instability’s development and characteristics. This approach provides a detailed understanding of the system’s behavior across various conditions, highlighting the roles of intermodal dispersion and higher-order effects. We demonstrate that the behavior of modulation instability bands and their reliance on parameters such as self-steepening and wave amplitude is contingent upon the specific characteristics of the optical setup and medium dispersion properties

Dynamic interplay: unveiling inelastic breather collisions and modulation instability enhancement in a periodically gained inhomogeneous fiber optic communication system across temporal frequencies

Examining the impact of inhomogeneity on the propagation of femtosecond ultrafast optical pulses in fiber, we delve into the realm of the modified Hirota nonlinear Schrödinger equation (NLS) with inhomogeneity of variable coefficients (MIH-vc). Employing the Hirota bilinear method, we derive two soliton solutions for the modified Hirota NLS equation and analyze the effect of variable coefficients. The dynamical properties of these soliton solutions come to light as we meticulously analyze the corresponding plots. In our exploration, a noteworthy revelation unfolds as we witness the inelastic collision between two breathers, unleashing profound changes in the trajectory of femtosecond pulses. Furthermore, we showcase a detailed modulation instability analysis, unraveling the gain spectrum for our theoretical model. Through graphical illustrations, we elucidate how inhomogeneous functions intricately shape the modulation instability (MI) gain spectrum. A groundbreaking observation surfaces as, for the first time, we discern the periodic gain enhancement in relation to Group Velocity Dispersion along the fiber and its dynamic interactions.

Bifurcation study, phase portraits and optical solitons of dual-mode resonant nonlinear Schrodinger dynamical equation with Kerr law non-linearity

This study investigates the dynamic characteristics of the dual-mode resonant non-linear Schrodinger equation with a Bhom potential. Hydrodynamics, nonlinear optical fibre communication, elastic media, and plasma physics are just a few of the mathematical physics and engineering applications for this model. The study aims to achieve two main objectives: first, to discuss bifurcation analysis, and second, to extract optical soliton solutions using the extended hyperbolic function method. The study successfully derives various wave solutions, including bright, singular, periodic singular and dark solitons, based on the governing model. The findings conferred in this article show a crucial advancement in understanding the propagation of waves in non-linear media. Additionally, bifurcation of phase portraits of ordinary differential equation consistent with the partial differential equation under consideration is conducted. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. The existing literature shows that these methods are first time applied on this model.

The regularity properties and blow-up of the solutions for nonlocal Schrödinger equations

In this paper, the Cauchy problem for convolution-type nonlocal linear and nonlinear Schrödinger equations (NSEs) is studied. The equations include the general differential operators. The existence, uniqueness, [Formula: see text]-regularity properties of linear problem and the existence, uniqueness, and blow-up at finite time of the nonlinear problem are obtained. By choosing differential operators including in equations, the regularity properties of a different type of nonlocal Schrödinger equation are studied.

From Algebro Geometric Solutions of the Toda Equation to Sato Formulas

We know that the degeneracy of solutions to PDEs, given in terms of theta functions on Riemann surfaces, provides important results about particular solutions, as in the case of the NLS equation. Here, we degenerate the so called finite gap solutions of the Toda lattice equation from the general formulation in terms of abelian functions when the gaps tend to points. This degeneracy allows us to recover the Sato formulas without using inverse scattering theory or geometric or representation theoretic methods.

Analysis of Nonlinear Schrödinger Equations with Time-Dependent Coefficients: Stability and Decay Properties

This study investigates the stability and decay properties of solutions to nonlinear Schrödinger equations (NLSEs) with time-dependent coefficients. Employing a blend of analytical and numerical methods, we delve into how temporal variations in coefficients influence the dynamics of wave functions. Our analysis reveals that time-dependent coefficients significantly affect the stability and decay rates of solutions, uncovering conditions that lead to either enhanced stability or accelerated decay. The findings highlight the critical role of coefficient temporality in dictating the behavior of NLSE solutions. These insights not only advance our theoretical understanding of NLSEs but also bear implications for practical applications in fields modeled by these equations. Our research opens avenues for exploiting time-dependent behaviors in designing systems with desired dynamical properties.

Application of the G’/G Expansion Method for Solving New Form of Nonlinear Schrödinger Equation in Bi-Isotropic Fiber

Bi-isotropic media (chiral and non-reciprocal) present an outstanding challenge for the scientific community. Their characteristics have facilitated the emergence of new and remarkable applications. In this paper, we focus on the novel effect of chirality, characterized through a newly proposed formalism, to highlight the nonlinear effect induced by the magnetization vector under the influence of a strong electric field. This research work is concerned with a new formulation of constitutive relations. We delve into the analysis and discussion of the family of solutions of the nonlinear Schrödinger equation, describing the pulse propagation in nonlinear bi-isotropic media, with a novel approach to constitutive equations. We apply the extended -expansion method with varying dispersion and nonlinearity to define certain families of solutions of the nonlinear Schrödinger equation in bi-isotropic (chiral and non-reciprocal) optical fibers. This clarification aids in understanding the propagation of light with two modes of propagation: a right circular polarized wave (RCP) and a left circular polarized wave (LCP), each having two different wave vectors in nonlinear bi-isotropic media. Various novel exact solutions of bi-isotropic optical solitons are reported in this study. Introduction: The investigation of exact solutions for nonlinear partial differential equations (PDEs) holds significant importance in understanding nonlinear physical phenomena. Nonlinear waves manifest across various scientific domains, notably in optical fibers and solid-state physics. In recent years, several potent methodologies have emerged for identifying solitons and periodic wave solutions of nonlinear PDEs. These include the -expansion method [1-6], the new mapping method [9-10], the method of generalized projective Riccati equations [11-16], and the expansion method [17]. Consequently, an original mathematical approach is proposed to evaluate nonlinear effects in bi-isotropic optical fibers, stemming from magnetization under the influence of a strong electric field [19-20]. The extended -expansion method emerges as a potent technique for deriving solution families of the nonlinear Schrödinger equation in bi-isotropic optical fibers. This method employs a perturbation expansion in powers of the dimensionless parameter and is applicable for both weak and strong nonlinearities. It accommodates varying dispersion and nonlinearity, rendering it suitable for modeling a wide array of optical fibers. Results and Conclusion: This investigate is concerned with a new formulation of constitutive relation linking to the magnetic effect, to understand rigorously the physical nature of biisotropic effects and to generalize the main macroscopic models. We inferred the nonlinear Schrodinger equation for a bi-isotropic medium term with a nonlinear term of magnetizing. In this article, the extended -expansion method is a powerful technique for determining a family of solutions of the nonlinear Schrödinger equation in bi-isotropic optical fibers. This method is based on the use of a perturbation expansion in powers of the dimensionless parameter, and it is valid for both weak and strong nonlinearities. The method allows for the inclusion of varying dispersion and nonlinearity, making it well-suited for modeling a wide range of optical fibres. Overall, the extended -expansion method is a valuable tool for understanding the dynamics of nonlinear optical systems, and it is expected to have a wide range of applications in the field of nonlinear optics.

Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth

Abstract In this article, we are concerned with the nonlinear Schrödinger equation − Δ u + λ u = μ ∣ u ∣ p − 2 u + f ( u ) , in R 2 , -\Delta u+\lambda u=\mu {| u| }^{p-2}u+f\left(u),\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{2}, having prescribed mass ∫ R 2 ∣ u ∣ 2 d x = a 2 > 0 , \mathop{\int }\limits_{{{\mathbb{R}}}^{2}}{| u| }^{2}{\rm{d}}x={a}^{2}\gt 0, where λ \lambda arises as a Lagrange multiplier, μ > 0 \mu \gt 0 , p ∈ ( 2 , 4 ] p\in \left(2,4] , and the nonlinearity f ∈ C 1 ( R , R ) f\in {C}^{1}\left({\mathbb{R}},{\mathbb{R}}) behaves like e 4 π u 2 {e}^{4\pi {u}^{2}} as ∣ u ∣ → + ∞ | u| \to +\infty . For a L 2 {L}^{2} -critical or L 2 {L}^{2} -subcritical perturbation μ ∣ u ∣ p − 2 u \mu {| u| }^{p-2}u , we investigate the existence of normalized solutions to the aforementioned problem. Moreover, the limiting profiles of solutions have been considered as μ → 0 \mu \to 0 or a → 0 a\to 0 . This result can be considered as a supplement to the work of Soave (Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case, J. Funct. Anal. 279 (2020), no. 6, 1–43) and Alves et al. (Normalized solutions for a Schrödinger equation with critical growth in R N {{\mathbb{R}}}^{N} , Calc. Var. Partial Differential Equations 61 (2022), no. 1, 1–24).

Data-driven rogue waves solutions for the focusing and variable coefficient nonlinear Schrödinger equations via deep learning.

In this paper, we investigate the data-driven rogue waves solutions of the focusing and the variable coefficient nonlinear Schrödinger (NLS) equations by the deep learning method from initial and boundary conditions. Specifically, first- and second-order rogue wave solutions for the focusing NLS equation and three deformed rogue wave solutions for the variable coefficient NLS equation are solved using physics-informed memory networks (PIMNs). The effects of optimization algorithm, network structure, and mesh size on the solution accuracy are discussed. Numerical experiments clearly demonstrate that the PIMNs can capture the nonlinear features of rogue waves solutions very well. This is of great significance for revealing the dynamical behavior of the rogue waves solutions and advancing the application of deep learning in the field of solving partial differential equations.

Darboux and generalized Darboux transformations for the fractional integrable derivative nonlinear Schrödinger equation

Analytical methods provide crucial mathematical insights into the stable solitary waves hidden in nonlinear phenomena. The nonlinear Schrödinger (NLS) equation is one of the most important typical integrable soliton models. From a mathematical perspective, the essence of the celebrated derivative NLS (DNLS) equation’s difference from the classical NLS equation lies in its cubic potential being differentiated once by the spatial variable and multiplied by the imaginary unit, which leads to the former having some characteristics that the latter cannot have. This paper extends the DNLS equation to the fractional integrable case with conformable derivative operators, and uses Darboux transformations (DTs) and generalized DT (GDT) to solve it exactly. Specifically, Lax pairs generating the fractional DNLS equation are first given. Based on the given Lax pairs, then the n-fold DTs and GDT for the fractional DNLS equation are derived. Some special exact solutions of the fractional DNLS equation are further obtained by employing the derived n-fold DTs and GDT. Finally, several novel space–time structures and dynamical evolutions of the obtained exact solutions are analyzed. This paper reveals through the DT and GDT methods that the double power-law fractional orders in the exact solutions of the fractional DNLS equation can be used to dominate the variable velocity propagation and anomalous diffusion in fractional dimensional media at different geometric scales.

Soliton robustness, interaction and stability for a variable coefficients Schrödinger(VCNLS) equation with inverse scattering transformation

The inverse scattering transformation(IST) method is an important method for solving soliton equations. In this paper, some novel solutions of a variable coefficients Schrödinger(VCNLS) equation are obtained with IST method. And we find that these waves have not the effected by other external waves, then call them on robustness waves. The VCNLS equation is derived from a non-isospectral AKNS equation hierarchy, and the spectral parameter is determined by an ordinary differential equation with polynomial nonlinearity. The VCNLS equation with the periodic and harmonic external potentials is solved via the inverse scattering transformation(IST) method, which possess the parity-time(PT) symmetric invariance. Unlike the classical NLS equation, the characteristic functions of the CVNLS equation with space–time external potential have two different symmetric reductions and two different backscattering solutions, which extends the application of IST and is helpful for the CVNLS equation. Some novel bright soliton solutions are obtained, which are different to the previous results in the classical case. The stabilities of the bright soliton solutions and the effect of the phase noise on the solutions are analyzed in a wide region through numerically. Especially, the interaction dynamics of between two solitons are investigated through addressing numerically, which have not a effected by other external waves. Thus, the novel features are different from these usual features of solitons, which have not a effected by other external waves.

Plenty of novel soliton molecules and modulation instability in the coherently coupled nonlinear Schrödinger equation

We investigate the coherently coupled nonlinear Schrödinger (NLS) equation, which can describe the propagation of two orthogonally polarized electromagnetic waves in a nonlinear fiber. The multi-soliton solutions are derived by using the Hirota bilinear method. Multi-breather solutions are presented by assuming the relationship between the parameters of the multi-solitons. Based on multi-soliton solutions and multi-breather solutions, some new interaction structures consisting of soliton molecules, breather molecules and soliton-breather molecules are found by applying the velocity resonance condition. The interactions among different types of soliton molecules, which can be observed in marine and oceanic waters, are investigated through numerical simulation. Ultimately, baseband modulation instability of the coherently coupled NLS equation is analyzed. It is of great significance for the study of soliton formation and rogue waves in the coherently coupled NLS equation.

Soliton waves with optical solutions to the three-component coupled nonlinear Schrödinger equation

This study uses the modified Sardar sub-equation method to find novel soliton solutions to the nonlinear three-component coupled nonlinear Schrödinger equation (NLSE), which is used for pulse propagation in nonlinear optical fibers. Multi-component NLSE equations are widely used because they can represent a wide range of complex observable systems and more dynamic patterns of localized wave solutions. The optical solutions proposed in this study are novel and can be described using hyperbolic, trigonometric, and exponential functions. These solutions are categorized as bright, dark, singular, combo bright-singular, and periodic solutions. Some solutions’ dynamic behaviors are demonstrated by selecting appropriate physical parameter values. The results and computational analysis indicate that the techniques provided are simple, effective, and adaptable. They can be applied to a variety of nonlinear evolution equations, whether stable or unstable, and can be used in fields such as mathematics, mathematical physics, and applied sciences.