Approximate Soliton Research Articles

Canonical Equations of Hamilton with Symmetry and Their Applications

Two systems of mathematical physics are defined by us, which are the first-order differential system (FODS) and the second-order differential system (SODS). Basing on the conventional Legendre transformation, we obtain a new kind of canonical equations of Hamilton (CEH) with some kind of symmetry. We show that the FODS can only be expressed by the new CEH, but do not by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs, on basis of the same conventional Legendre transformation. As an example, we prove that the nonlinear Schrödinger equation can be expressed with the new CEH in a consistent way. Based on the new CEH, the approximate soliton solution of the nonlocal nonlinear Schrödinger equation is obtained, and the soliton stability is analysed analytically as well. Furthermore, because the symmetry of a system is closely connected with certain conservation theorem of the system, the new CEH may be useful in some complicated systems when the symmetry considerations are used.

Soliton approximation in continuum models of leader-follower behavior.

Complex biological processes involve collective behavior of entities (bacteria, cells, animals) over many length and time scales and can be described by discrete models that track individuals or by continuum models involving densities and fields. We consider hybrid stochastic agent-based models of branching morphogenesis and angiogenesis (new blood vessel creation from preexisting vasculature), which treat cells as individuals that are guided by underlying continuous chemical and/or mechanical fields. In these descriptions, leader (tip) cells emerge from existing branches and follower (stalk) cells build the new sprout in their wake. Vessel branching and fusion (anastomosis) occur as a result of tip and stalk cell dynamics. Coarse graining these hybrid models in appropriate limits produces continuum partial differential equations(PDEs) for endothelial cell densities that are more analytically tractable. While these models differ in nonlinearity, they produce similar equationsat leading order when chemotaxis is dominant. We analyze this leading order system in a simple quasi-one-dimensional geometry and show that the numerical solution of the leading order PDE is well described by a soliton wave that evolves from vessel to source. This wave is an attractor for intermediate times until it arrives at the hypoxic region releasing the growth factor. The mathematical techniques used here thus identify common features of discrete and continuum approaches and provide insight into general biological mechanisms governing their collective dynamics.

Dissipative solitons characterization in singly resonant optical parametric oscillators: A variational formalism

In this work, the emergence of single-peak temporal dissipative solitons in singly-resonant degenerate optical parametric oscillators is investigated analytically. Applying the Kantarovitch optimization method, through a Lagrangian variational formalism, an approximate analytical soliton solution is computed using a parameter-dependent ansatz. This permits to obtain analytical estimations for the DS energy, peak power, and boundaries of the DS existence region, which are of great value for experimentalist. To confirm the validity of this procedure, these analytical results are compared with a numerical study performed in the context of pure quadratic systems, showing a good agreement.

Soliton solutions of the equations of hydrodynamics in describing collisions and oscillations of atomic nuclei

The hydrodynamic approach was used to find an analytical solution of the hydrodynamic equations in the soliton approximation for layer collisions in the one- and two-dimensional cases. The compression stage, the decompression stage, and the expansion stage are considered within the framework of a single formula for layers with energies on the order of ten MeV per nucleon. Our generalization to the two-dimensional case leads to the idea of the formation of a rarefied bubble region at the stage of expansion. The approach can be used in other fields for calculations of the nonlinear dynamics of oscillations of complex systems.

Investigating soliton dynamics: Contemporary computational and numerical approaches for analytical and approximate solutions of the CDG model

This investigation employs contemporary computational and numerical techniques to derive analytical and approximate soliton solutions for the Caudrey–Dodd–Gibbon model, which represents a significant variation of the fifth-order Korteweg–de Vries equation. Diverse analytical solutions are constructed, employing distinct formats such as exponential, trigonometric, and hyperbolic functions. Simulations, including two-dimensional, three-dimensional, contour, polar, and discrete plots, are presented to illustrate the real-world behavior of a single soliton. Furthermore, these solutions are utilized to evaluate the essential conditions for implementing the proposed numerical scheme. The agreement between the computed and approximate solutions is demonstrated through various techniques. These results unequivocally establish the superiority of these methods for solving nonlinear mathematical physics problems.

Physics-informed Neural Network method for predicting soliton dynamics supported by complex PT-symmetric potentials

Abstract We examine the deep learning technique referred to as the physics-informed neural network method for approximating non-linear Schrödinger equation under considered parity time symmetric potentials and obtaining multifarious soliton solutions. For the first time, neural networks founded principally physical information are adopted to figure out the solution the examined non-linear partial differential equation and generate six different types of soliton solutions, which are basic, dipole, tripole, quadruple, pentapole and sextupole solitons we consider. We make comparisons between the predicted and actual soliton solutions to see whether deep learning is capable of seeking the solution the partial differential equation described before. We may assess whether physics-informed neural network is capable of effectively providing approximate soliton solutions through the evaluation of squared error between the predicted and numerical results. Besides, we also scrutinize how different activation mechanisms and network architectures impact the capability of selected deep learning technique works. Through the findings we can prove that the neural networks model we established can be utilized to accurately and effectively approximate non-linear Schrödinger equation under consideration and predict the dynamics of soliton solution.

Physics-Informed Neural Network Method for Predicting Soliton Dynamics Supported by Complex Parity-Time Symmetric Potentials

We examine the deep learning technique referred to as the physics-informed neural network method for approximating the nonlinear Schrödinger equation under considered parity-time symmetric potentials and for obtaining multifarious soliton solutions. Neural networks to found principally physical information are adopted to figure out the solution to the examined nonlinear partial differential equation and to generate six different types of soliton solutions, which are basic, dipole, tripole, quadruple, pentapole, and sextupole solitons we consider. We make comparisons between the predicted and actual soliton solutions to see whether deep learning is capable of seeking the solution to the partial differential equation described before. We may assess whether physics-informed neural network is capable of effectively providing approximate soliton solutions through the evaluation of squared error between the predicted and numerical results. Moreover, we scrutinize how different activation mechanisms and network architectures impact the capability of selected deep learning technique works. Through the findings we can prove that the neural networks model we established can be utilized to accurately and effectively approximate the nonlinear Schrödinger equation under consideration and to predict the dynamics of soliton solution.

Unraveling the interplay of gravity and surface tension in driving waves on water’s surface

The aim of this paper is to incorporate modern computational and numerical methods to give analytical and approximate soliton solutions for the generalized fractional Hietarinta–type (GFH) model in (2+1) dimensions. Solitary wave solutions for the suggested model, which mathematically shows how gravity and surface tension make waves move again on the water’s surface, are identified and illustrated. The phase velocity of waves with different wavelengths is described by the difference in frequency in fluid dynamics. A new modern numerical technique is used to show how accurate the computational solution is, along with analytical solutions. Hamiltonian system features are also utilized to regulate the solutions’ stability. All findings are double-checked by reinserting them into Mathematica 13.1’s core model.

A new perspective for analytical and numerical soliton solutions of the Kaup–Kupershmidt and Ito equations

In present study, it is aimed to implement the analytical method and a numerical approach based on finite element method to get approximate soliton solutions of two special forms of fifth-order KdV equation (fKdV) that are of particular importance: Kaup–Kupershmidt (K–K) and Ito equations. To reach these aims, the simplest equation method and septic B-spline collocation method are introduced. L2 and L∞ error norms are computed for single soliton solutions to demonstrate the proficiency and accuracy of the present method. The method is shown to be unconditionally stable by performing the von-Neumann stability analysis. The obtained numerical results have been displayed with tables. Additionally, to show the analytical and numerical behaviors of the single soliton, all figures are drawn in 2D and 3D. Analytical and numerical results ensure that the methods are more suitable and systematically handle the solution procedures of nonlinear equations arising in mathematical physics.

Landau damping of electron-acoustic waves due to multi-plasmon resonances

The linear and nonlinear theories of electron-acoustic waves (EAWs) are studied in a partially degenerate quantum plasma with two-temperature electrons and stationary ions. The initial equilibrium of electrons is assumed to be given by the Fermi–Dirac distribution at finite temperature. By employing the multi-scale asymptotic expansion technique to the one-dimensional Wigner–Moyal and Poisson equations, it is shown that the effects of multi-plasmon resonances lead to a modified complex Korteweg–de Vries (KdV) equation with a new nonlocal nonlinearity. In addition to giving rise to a nonlocal nonlinear term, the wave–particle resonance also modifies the local nonlinear coupling coefficient of the KdV equation. The latter is shown to conserve the number of particles; however, the wave energy decays with time. A careful analysis shows that the two-plasmon resonance is the dominant mechanism for nonlinear Landau damping of EAWs. An approximate soliton solution of the KdV equation is also obtained, and it is shown that the nonlinear Landau damping causes the wave amplitude to decay slowly with time compared to the classical theory.

The phase shift analysis of the colliding dissipative KdV solitons

In this work, the head-on collisions of the non-stationary dissipative soliton in ultracold neutral plasmas (UNPs) are investigated. The extended Poincare-Lighthill-Kuo (PLK) approach is adopted for reducing the fluid equations of the UNPs to two-counterpropagating damped Korteweg-de Vries (dKdV) equations. The dKdV equation is not an integrable Hamiltonian system, i.e., does not have an exact solution. Thus, one of the main goal of this paper is to find a new general approximate analytical solution to the dKdV equation for investigating the mechanism of the propagation and interaction of the non-stationary dissipative solitons. The residual error is estimated for checking the accuracy of the new obtained solution. The approximate analytical soliton solutions are adopted for deriving the temporal phase shifts after the collision. The impact of physical parameters on the nonstationary dissipative soliton profile and the temporal phase shifts is discussed. The obtained results will contribute to understand the mechanism of propagation and interaction of many nonlinear phenomena in different nonlinear mediums such as ocean, sea, optical fiber, plasma physics, etc.

Solitary waves and modulation instability with the influence of fractional derivative order in nonlinear left-handed transmission line

The resolution of the reduced fractional nonlinear Schrödinger equation obtained from the model describing the wave propagation in the left-handed nonlinear transmission line presented by Djidere et al recently, allowed us in this work through the Adomian decomposition method (ADM) to highlight the behavior and to study the propagation process of the dark and bright soliton solutions with the effect of the fractional derivative order as well as the Modulation Instability gain spectrum (MI) in the LHNLTL. By inserting fractional derivatives in the sense of Caputo and in order to structure the approximate soliton solutions of the fractional nonlinear Schrödinger equation reduced, ADM is used. The pipe is obtained from the bright and dark soliton by the fractional derivatives order. By the bias of MI gain spectrum the instability zones occur when the value of the fractional derivative order tends to 1. Furthermore, when the fractional derivative order takes small values, stability zones appear. These results could bring new perspectives in the study of solitary waves in left-handed metamaterials as the memory effect could have a better future for the propagation of modulated waves because in this paper the stabilization of zones of the dark and bright solitons which could be described by a fractional nonlinear Schrödinger equation with small values of fractional derivatives order has been revealed. In addition, the obtained significant results are new and could find applications in many research areas such as in the field of information and communication technologies.

Collapse arrest in the space-fractional Schrödinger equation with an optical lattice* *Project supported by the National Natural Science Foundation of China (Grant No. 11947122), the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2019A1515110935), the Research Start-up Foundation of Dongguan University of Technology, the Guangdong Science and Technology Planning Program (Grant No. 2017A010102019), the Guangdong Province Natural Science Foundation

The soliton solution and collapse arrest are investigated in the one-dimensional space-fractional Schrödinger equation with Kerr nonlinearity and optical lattice. The approximate analytical soliton solutions are obtained based on the variational approach, which provides reasonable accuracy. Linear-stability analysis shows that all the solitons are linearly stable. No collapses are found when the Lévy index 1 < α ≤ 2. For α = 1, the collapse is arrested by the lattice potential when the amplitude of perturbations is small enough. It is numerically proved that the energy criterion of collapse suppression in the two-dimensional traditional Schrödinger equation still holds in the one-dimensional fractional Schrödinger equation. The physical mechanism for collapse prohibition is also given.

Approximate bright-soliton solution of the higher-order nonlinear Schrödinger equation

In this paper, approximate bright-soliton solutions of the higher-order nonlinear Schrödinger equation are constructed by treating the higher-order terms as small perturbations. The first-, second-, and third-order asymptotic solutions are obtained. The errors between the asymptotic solutions and the numerical/analytical solutions are discussed, which gives a high accuracy of the approximate solutions. It is pointed that the asymptotic solutions can be used as the initial value to improve the accuracy of the numerical solutions. This paper may be helpful for undergraduate and graduate students in mathematics and physics to understand the approximate soliton solutions of the higher-order nonlinear Schrödinger equation.

Vortex soliton solutions of a (3 + 1)-dimensional Gross–Pitaevskii equation with partially nonlocal distributed coefficients under a linear potential

A (3 + 1)-dimensional Gross–Pitaevskii equation with partially nonlocal distributed coefficients under a linear potential is followed with interest. The mapping procedure from the distributed-coefficient to the constant-coefficient equations is provided. Via the procedure with solution of constant-coefficient NLSE from the variational principle, an approximate vortex soliton solution is reported. With the increase of value for topological charge m, torus-shaped vortex soliton turns into ring vortex soliton, and central region of ring gradually enlarges. The layer of vortex solitons and their phase along the z-direction is related to the number of $$q+1$$ with the Hermite parameter q. The branch number of spiral phase structures for vortex solitons is related to the value of topological charge m. The stability test from the direct numerical simulation indicates that vortex solitons with $$m=1,2,3,\, q=0$$ are stable, otherwise, vortex solitons with other values of m, q are unstable.

The evolution equation of non-linear waves and its exact solutions by subsidiary ordinary differential equation method

In this work, we investigate the dynamics of the equatorial Rossby waves by including the complete Coriolis force, external source and dissipation. The amplitude evolution of equatorial Rossby waves is described as an extended non-linear mKdV–Burgers equation from a potential vorticity equation and it is unlike the standard mKdV–Burgers equation. Built on the obtained model, the corresponding physical phenomena related to the non-linear Rossby waves are analyzed. Also, the subsidiary ordinary differential equation method is employed to solve the solitary solution of the mKdV equation. By analyzing the solution, we find that the horizontal component of Coriolis parameter works on the amplitude of the Rossby waves. Meanwhile, we use the Adomian decomposition method to obtain the approximate soliton solution of the model.

Longitudinal bulk strain solitons in a hyperelastic rod with quadratic and Cubic nonlinearities

We study long nonlinear longitudinal bulk strain waves in a hyperelastic rod of circular cross section within the scope of the general weakly-nonlinear elasticity leading to a model with quadratic and cubic nonlinearities. We systematically derive the extended Boussinesq and Korteweg - de Vries - type equations and construct a family of approximate weakly-nonlinear soliton solutions with the help of near-identity transformations. These solutions are compared with the results of direct numerical simulations of the original nonlinear problem formulation, showing excellent agreement within the range of their asymptotic validity (waves of small amplitude) and extending their relevance beyond it (to the waves of moderate amplitude) as a very good initial guess. In particular, we were able to observe a stably propagating "table-top" soliton..

Dynamical behaviour of non‐linear quantum ion acoustic wave in weakly magnetized electron–positron–ion plasma

AbstractIn order to investigate the propagation characteristics of linear and non‐linear ion acoustic waves (IAWs) in electron–positron–ion quantum plasma in the presence of external weak magnetic field, we have used a quantum hydrodynamic model, and degenerate statistics for the electrons and positrons are taken into account. It is found that the linear dispersion relation of the IAW was modified by the externally applied magnetic field. By using the reductive perturbation technique, a gyration‐modified Korteweg‐de Vries equation is derived for finite amplitude non‐linear IAWs. Time‐dependent numerical simulation shows the formation of an oscillating tail in front of the ion acoustic solitons in the presence of a weak magnetic field. It is also seen that the amplitude and width of solitons and oscillating tails are affected by the relevant plasma parameters such as quantum diffraction, positron concentration, and magnetic field. We have performed our analysis by extending it to account for approximate soliton solution by asymptotic perturbation technique and non‐linear analysis via a dynamical system approach. The analytical results show the distortion of the shape of the localized soliton with time, and the non‐linear analysis confirms the generation of oscillating tails.

Patterns of water in light

The intricate patterns emerging from the interactions between soliton stripes of a two-dimensional defocusing nonlinear Schrödinger (NLS) model with a non-local nonlinearity are considered. We show that, for sufficiently strong non-locality, the model is asymptotically reduced to a Kadomtsev–Petviashvilli-II (KPII) equation, which is a common model arising in the description of shallow water waves, as such patterns of water may indeed exist in light (this non-local NLS finds applications in nonlinear optics, modelling beam propagation in media featuring thermal nonlinearities, in plasmas, and in nematic liquid crystals). This way, approximate antidark soliton solutions of the NLS model are constructed from the stable KPII line solitons. By means of direct numerical simulations, we demonstrate that non-resonant and resonant two- and three-antidark NLS stripe soliton interactions give rise to wave configurations that are found in the context of the KPII equation. Thus, our study indicates that patterns which are usually observed in water can also be found in optics.

On soliton solutions of time fractional form of Sawada–Kotera equation

Soliton solutions are of utmost importance among traveling wave solutions since they act as a bridge between mathematics and physics. This paper investigates time fractional form of fifth-order Sawada–Kotera equation resulting in different forms of exact and approximate soliton solutions including singular, periodic and dark soliton solutions. The time fractional derivative is used in the Caputo sense throughout the study. The effectiveness of trial equation method is seen in determining exact solutions, whereas residual power series method is employed for its high precision and reliability in calculating approximate solutions of the model. Multiple soliton solutions are also determined by the aid of Hirota’s method. The results are compared with a numerical method, and the graphical representation of all the obtained solutions is shown for different values of the fractional parameter. The time evolution of all solutions is also represented in 2D plots. These findings are highly encouraging to explore other nonlinear fractional models.