Lakshmanan Porsezian Daniel Equation Research Articles

On optical soliton solutions of the higher-order Lakshmanan-Porsezian-Daniel model having the cubic-quintic-septic law in the presence of spatio-temporal and chromatic dispersion

The higher-order Lakshmanan-Porsezian-Daniel equation (LPDE) with the cubic-quintic-septic (CQS) law having spatiotemporal and chromatic dispersion terms (STD-CD) is examined to derive new optical soliton solutions. To accomplish this aim, we operated on a simple version of the new extended auxiliary equation method (SAEM26). The optical soliton solutions of the LPDE with CQS as well as STD-CD are constructed in detail. Moreover, 3D-surface, contour, and 2D plots are presented for the bright and periodic singular soliton solutions. Additionally, the effects of diverse model parameters on the bright soliton structure are surveyed, and these effects are displayed with 2D graphics. The findings established in this work can positively contribute to research in nonlinear optics, while the SAEM26 can be effectively applied to similar nonlinear models.

Bifurcations Analysis and Monotonicity of the Period Function of the Lakshmanan–Porsezian–Daniel Equation with Kerr Law of Nonlinearity

Bifurcations Analysis and Monotonicity of the Period Function of the Lakshmanan–Porsezian–Daniel Equation with Kerr Law of Nonlinearity

Riemann–Hilbert Approach and Multiple Arbitrary-Order Pole Solutions for the Lakshmanan–Porsezian–Daniel Equation with Finite Density Initial Data

Riemann–Hilbert Approach and Multiple Arbitrary-Order Pole Solutions for the Lakshmanan–Porsezian–Daniel Equation with Finite Density Initial Data

Riemann–Hilbert approach, dark solitons and double‐pole solutions for Lakshmanan–Porsezian–Daniel equation in an optical fiber, a ferromagnetic spin or a protein

AbstractInverse scattering transform for the defocusing Lakshmanan–Porsezian–Daniel equation with nonzero boundary condition is constructed via the Riemann–Hilbert approach. Since poles of the associated reflection coefficient are simple, ‐dark soliton solutions corresponding to simple poles are constructed. For ‐dark soliton solutions, results show that the soliton amplitude and width are not affected by the strength of the higher‐order linear and nonlinear effects , but soliton velocity has a linear correlation with ; the interactions between the two‐dark solitons and among the three‐dark solitons are elastic and experience phase and position shifts. Besides, asymptotic analysis of the double‐pole solutions for the focusing Lakshmanan–Porsezian–Daniel equation with nonzero boundary condition is presented. Different from ‐dark soliton solutions which locate in the straight lines and experience position shift after the interaction, the double‐pole solutions diverge from each other logarithmically and experience no position shift after the interaction.

Dispersive shock wave structure analysis for the defocusing Lakshmanan–Porsezian–Daniel equation

Via the finite-gap integration theory, we study the defocusing Lakshmanan–Porsezian–Daniel (LPD) equation. Meanwhile, we derive the degenerate forms for the single-phase periodic solution. In addition, we obtain the basic and combined structures of the dispersive shock wave through the Whitham modulation equation parameterized by the Riemann invariant.

Long-time asymptotics for the integrable nonlocal Lakshmanan–Porsezian–Daniel equation with decaying initial value data

In this work, we study the Cauchy problem of integrable nonlocal Lakshmanan-Porsezian-Daniel equation with rapid attenuation of initial data. The basic Riemann–Hilbert problem of integrable nonlocal Lakshmanan-Porsezian-Daniel equation is constructed from Lax pair. Using Deift-Zhou nonlinear steepest descent method, the explicit long-time asymptotic formula of integrable nonlocal Lakshmanan-Porsezian-Daniel equation is derived, which is different from the local model. Besides, compared to the nonlocal nonlinear Schrödinger equation, since the increase of real stationary phase points, the long-time asymptotic formula for nonlocal Lakshmanan-Porsezian-Daniel equation becomes more complex.

Implicit quiescent optical solitons For Lakshmanan–Porsezian–Daniel model having nonlinear chromatic dispersion and power-law of self-phase modulation by lie symmetry

AbstractThis paper recovers implicit quiescent optical solitons for the Lakshmanan–Porsezian–Daniel equation that is studied with nonlinear chromatic dispersion and power-law of self-phase modulation. The Lie symmetry analysis has made this retrieval possible. An interesting observation has been made with the results that was not recoverable from the prequel paper.

Bifurcation, Phase Portrait and Traveling Wave Solutions of the Coupled Fractional Lakshmanan–Porsezian–Daniel Equation

Bifurcation, Phase Portrait and Traveling Wave Solutions of the Coupled Fractional Lakshmanan–Porsezian–Daniel Equation

Investigation of the optical solitons for the Lakshmanan–Porsezian–Daniel equation having parabolic law

Investigation of the optical solitons for the Lakshmanan–Porsezian–Daniel equation having parabolic law

Soliton molecules, multi-breathers and dynamical behaviors of the Lakshmanan–Porsezian–Daniel equation

Soliton molecules, multi-breathers and dynamical behaviors of the Lakshmanan–Porsezian–Daniel equation

Highly dispersive optical solitons in fiber Bragg gratings for stochastic Lakshmanan–Porsezian–Daniel equation with spatio-temporal dispersion and multiplicative white noise

Nonlinear partial differential equations serve as key components for the mathematical representation of engineering phenomena across several domains within the known universe. In the field of engineering difficulties, the most significant considerations are invariably systems composition and physics of components. In this regard, the impact of noise emerges as a crucial aspect. Building on this foundational understanding, this study aims to present a novel version of the stochastic Lakshmanan–Porsezian–Daniel equation with higher order dispersion in fiber Bragg gratings. The equation incorporates multiplicative white noise in the Itô sense. To investigate this new model, we employ a couple of integration techniques which are the enhanced Kudryashov method and the generalized sine–Gordon equation method. Bright and singular solitons are achieved through the enhanced Kudryashov method, while straddled bright–dark solitons and straddled singular–singular solitons are obtained using the generalized sine–Gordon equation method. Solutions in the context of Jacobi’s elliptic functions are also considered. The emergence of all possible solutions using the proposed techniques has been noticed, each accompanied by specific constraints that guarantee their existence. Moreover, the effect of noise intensity on soliton dynamics is investigated and interpreted by simulating the graphical representations.

Stability analysis and dynamical behavior of solitons in nonlinear optics modelled by Lakshmanan–Porsezian–Daniel equation

Stability analysis and dynamical behavior of solitons in nonlinear optics modelled by Lakshmanan–Porsezian–Daniel equation

Construction of new solitons and other wave solutions for a concatenation model using modified extended tanh-function method

In this work, we consider a concatenation model that is a concatenated version of the familiar nonlinear Schrödinger’s equation, Lakshmanan–Porsezian–Daniel equation and the Sasa–Satsuma equation. With the help of the improved modified extended tanh-function method (IMETFM), new families of solutions are extracted for the proposed model. Bright solitons, dark solitons, combo bright-dark solitons, singular solitons, periodic solutions, exponential solutions, rational solutions and Jacobi elliptic solutions are obtained. The IMETFM enables to recover a wide spectrum of solutions to the governing model. The extracted solutions confirmed the efficacy and strength of the current technique. Moreover, the graphs for some solutions are presented to demonstrate their physical nature.

Exact solutions and Darboux transformation for the reverse space–time nonlocal Lakshmanan–Porsezian–Daniel equation

The integrable reverse space–time nonlocal Lakshmanan–Porsezian–Daniel (LPD) equation is proposed by nonlocal reduction of the AKNS scattering problem. The 1-fold and N-fold Darboux transformations of the nonlocal LPD equation are constructed, from which some interesting nonlinear wave solutions including soliton and rogue wave solutions are obtained, whose dynamical properties are discussed by evolution plots. It is worth mentioning that the real and imaginary parts of the soliton solutions show the characteristics of the breathing solution. These results may be useful in nonlinear optics and related fields.

Quiescent Optical Solitons for the Concatenation Model with Nonlinear Chromatic Dispersion

This paper recovers quiescent optical solitons that are self-sustaining, localized wave packets that maintain their shape and amplitude over long distances due to a balance between nonlinearity and dispersion. When a soliton is in a state of quiescence, it means that it is stationary in both space and time. Quiescent optical solitons are typically observed in optical fibers, where nonlinearity and dispersion can lead to the formation of solitons. The concatenation model is considered to understand the behavior of optical pulses propagating through nonlinear media. Here, we consider the familiar nonlinear Schrödinger equation, the Lakshmanan–Porsezian–Daniel equation, and the Sasa–Satsuma equation. The current paper also addresses the model with nonlinear chromatic dispersion, a phenomenon that occurs in optical fibers and other dispersive media, where the chromatic dispersion of the material is modified by nonlinear effects. In the presence of nonlinearities, such as self-phase modulation and cross-phase modulation, the chromatic dispersion coefficient becomes a function of the optical intensity, resulting in nonlinear chromatic dispersion. A full spectrum of stationary optical solitons, along with straddled stationary solitons, are obtained. There are four integration schemes that made this retrieval possible. The numerical simulations are also included for these solitons. The parameter constraints also indicate the existence criteria for these quiescent solitons.

Multi-vector with nonlocal and non-singular kernel ultrashort optical solitons pulses waves in birefringent fibers

This research study investigates different structures and novel solitary wave solutions of to the nonlinear fractional Lakshmanan–Porsezian–Daniel equation LPDE. The investigated model describes the dynamical and physical properties of the wave pulse in birefringent fibers which incorporates two vector solitons. A new fractional derivative with nonlocal and non-singular kernel is used to convert the model’s fractional form into ordinary differential equation with an integer order. Three recent computational schemes (novel generalized Kudryashov (NKud) method, Khater II (Khat II) method, Sardar Sub-equation (SSE) method) are employed to construct some novel solitary wave solution of the LPDE in various structures. The outcome results are visually demonstrated through some distinct graphs in 2-, 3-dimensional, density, and polar plots to explain some novel properties of the investigated model. Mathematica 13.1 checks all inputs and outputs against the original model for further confidence.

Dark and singular cubic�quartic optical solitons with Lakshmanan�Porsezian�Daniel equation by the improved Adomian decomposition scheme

Dark and singular cubic�quartic optical solitons with Lakshmanan�Porsezian�Daniel equation by the improved Adomian decomposition scheme

NOVEL BRIGHT AND KINK OPTICAL SOLITON SOLUTIONS OF FRACTIONAL LAKSHMANAN–PORSEZIAN–DANIEL EQUATION WITH KERR LAW NONLINEARITY IN CONFORMABLE SENSE

The fractional Lakshmanan–Porsezian–Daniel equation (LPD) is a significant complex model for the fractional Schrödinger family which arises in quantum physics. This paper explores new bright and kink soliton solutions of the space-time fractional LPD equation with the Kerr law of nonlinearity. By considering the conformable derivatives, the governing model is translated into integer-order differential equations with the aid of an appropriate complex traveling wave transformation. Dynamic behavior and phase portrait of traveling wave solutions are investigated. Further, various types of bright and kinked soliton solutions under definite parametric settings are discussed. Moreover, graphical representations of the obtained solution of the diverse fractional order are depicted to naturally illustrate the constructed solution.

Mixed localized waves and their dynamics for a matrix Lakshmanan–Porsezian–Daniel equation

Interactions between different localized waves are of great significance to physical systems. In this paper, we study the mixed localized waves and their dynamics based on the matrix Lakshmanan–Porsezian–Daniel equation. First, we construct the Nth-order mixed localized solutions describing the interactions between the (N−1) th-order rogue waves and breathers. Using these solutions, we discuss the second- and third-order mixed localized waves, as well as their dynamics. Furthermore, we describe five types of interactions between rogue waves and breathers: between the first-order rogue waves and temporal period breathers, the first-order rogue waves and spatial period breathers, the first-order rogue waves and spatial-temporal period breathers, the second-order rogue waves and temporal period breathers, and the second-order rogue waves and spatial period breathers. These results may be useful for the study of nonlinear wave interactions in physical systems.

Bifurcations and dispersive optical solitons for the cubic–quartic nonlinear Lakshmanan–Porsezian–Daniel equation in polarization-preserving fibers

The main attention of this work is to study the bifurcations and optical solitons for the cubic–quartic nonlinear Lakshmanan–Porsezian–Daniel equation in polarization-preserving fibers. Firstly, the NLLPD equation was converted into an equivalent ordinary differential traveling wave system by using the traveling wave transformations. Then, we derive the system’s Hamiltonian and orbits phase portraits. In addition to, we construct dark and singular solitons of this model by using the bifurcation theory of dynamical system method. As you can see, some other dispersive optical solitons have been deduced via the complete discriminant system technique and symbolic computation. It is notable that we give the classification of single traveling wave solutions for NLLPD model at the same time. The obtained conclusions may improve or complement previous conditions. Lastly, in order to further explore the propagation of NLLPD model in nonlinear optics, two-dimensional and three-dimensional graphs are drawn by selecting appropriate parameters of the optical soliton solutions.