Soliton-like Solutions Research Articles

Large-space and large-time asymptotic properties of vector rogon-soliton and soliton-like solutions for n-component NLS equations.

In this paper, we analyze the large-space and large-time asymptotic properties of the vector rogon-soliton and soliton-like solutions of the n-component nonlinear Schrödinger equation with mixed nonzero and zero boundary conditions. In particular, we find that these solutions have different decay velocities along different directions of the x axis, that is, the solutions exponentially and algebraically decay along the positive and negative directions of the x axis, respectively. Moreover, we study the change of the acceleration of soliton moving with the increase in time or distance along the characteristic line (i.e., soliton moving trajectory). As a result, we find that the product of the acceleration and distance square tends to some constant value as time increases. These results will be useful to better understand the related multi-wave phenomena and to design physical experiments.

Large-amplitude internal waves and turbulent mixing in three-layer flows under a rigid lid

We consider a nonlinear long-wave Boussinesq-type model describing the propagation of breaking internal solitary waves in a three-layer flow between two rigid boundaries. The Green–Naghdi-type equations govern the fluid flow in the top and bottom homogeneous layers. In the intermediate hydrostatic layer, the fluid is non-homogeneous, and its flow is described by the depth-averaged shallow water equations for shear flows. The velocity shear in the outer layers can lead to the development of the Kelvin–Helmholtz instability and turbulent mixing. To take this into account, we propose a simple law of vertical mixing, which governs the interaction of these layers. Stationary solutions and non-stationary calculations show the effect of mixing (or breaking) for waves of sufficiently large amplitude. We construct steady-state soliton-like solutions of the three-layer model adjacent to a given constant flow. The obtained theoretical profiles of breaking solitary waves are consistent with laboratory experiments.

Investigating (2+1)-dimensional dissipative long wave system in water waves using three innovative integration norms

This research article investigates the dissipative long wave system in (2+1) dimensions. To analyze this system, three separate integrating strategies were used: the single manifold approach, the unified method, and the generalized projective Ricatti equation method. Novel soliton solutions were effectively obtained by applying these strategies to the suggested model. The resulting solutions are various, including mixed soliton-like solutions, dark solitons, rational solutions, triangular periodic solutions, and combined Jacobi elliptic wave function solutions. Using the aforementioned approaches, an extensive variety of solution types has been discovered for the proposed model. The dynamic character of these closed-form solutions has been adequately demonstrated using both three dimensional and two dimensional graphical representations. This visualization helps to better understand the behavior of the acquired solutions.

Soliton-like solutions supported by refined hydrodynamic-type model of an elastic medium with soft inclusions

A nonlinear elastic medium containing sharp inhomogeneities is considered. The properties of a modified model of such a medium are investigated. The modification consists in including in the asymptotic equation of state those terms that were discarded in the previously considered models. The main purpose of the ongoing research is to analyze the existence, stability, and dynamic properties of soliton-like solutions within the modified model, as well as to compare these solutions with analogous solutions obtained in the previously considered models.

The application of the "inverse problem" method for constructing confining potentials that make N-soliton waveforms exact solutions in the Gross-Pitaevskii equation.

In this work, we discuss an application of the "inverse problem" method to find the external trapping potential, which has particular N trapped soliton-like solutions of the Gross-Pitaevskii equation (GPE) also known as the cubic nonlinear Schrödinger equation (NLSE). This inverse method assumes particular forms for the trapped soliton wave function, which then determines the (unique) external (confining) potential. The latter renders these assumed waveforms exact solutions of the GPE (NLSE) for both attractive (g<0) and repulsive (g>0) self-interactions. For both signs of g, we discuss the stability with respect to self-similar deformations and translations. For g<0, a critical mass Mc or equivalently the number of particles for instabilities to arise can often be found analytically. On the other hand, for the case with g>0 corresponding to repulsive self-interactions which is often discussed in the atomic physics realm of Bose-Einstein condensates, the bound solutions are found to be always stable. For g<0, we also determine the critical mass numerically by using linear stability or Bogoliubov-de Gennes analysis, and compare these results with our analytic estimates. Various analytic forms for the trapped N-soliton solutions in one, two, and three spatial dimensions are discussed, including sums of Gaussians or higher-order eigenfunctions of the harmonic oscillator Hamiltonian.

Breather patterns and other soliton dynamics in (2+1)-dimensional conformable Broer-Kaup-Kupershmit system

&lt;abstract&gt;&lt;p&gt;In this work, the Extended Direct Algebraic Method (EDAM) is utilized to analyze and solve the fractional (2+1)-dimensional Conformable Broer-Kaup-Kupershmit System (CBKKS) and investigate different types of traveling wave solutions and study the soliton like-solutions. Using the suggested method, the fractional nonlinear partial differential equation (FNPDE) is primarily reduced to an integer-order nonlinear ordinary differential equation (NODE) under the traveling wave transformation, yielding an algebraic system of nonlinear equations. The ensuing algebraic systems are then solved to construct some families of soliton-like solutions and many other physical solutions. Some derived solutions are numerically analyzed using suitable values for the related parameters. The discovered soliton solutions grasp vital importance in fluid mechanics as they offer significant insight into the nonlinear behavior of the targeted model, opening the way for a deeper comprehension of complex physical phenomena and offering valuable applications in the associated areas.&lt;/p&gt;&lt;/abstract&gt;

Physically Admissible and Inadmissible Exact Localized Solutions in Problems of Nonlinear Wave Dynamics of Cylindrical Shells

It is shown that, when studying nonlinear longitudinal deformation waves in cylindrical shells, it is possible to obtain physically admissible solitary wave solutions using refined shell models. In the article, a physically admissible exact localized solution based on the Flügge – Lurie – Byrne model is constructed. An analysis of the influence of the external nonlinear elastic medium on the exact solutions obtained is carried out. It is established that the use of quadratic and cubic nonlinear deformation laws leads to nonintegrable equations with exact soliton-like solutions. However, the amplitudes of the exact solutions exceed the values of permissible displacements corresponding to the maximum points on the curves of the deformation laws of the external medium, which leads to the physical inadmissibility of these solutions.

Chains of mini-boson stars

In this paper, we re-investigate the static, soliton-like solutions in the model of the Einstein gravity coupled to a free and complex scalar field, which have been known as mini-boson stars. With the numerical methods, we have discovered a new family of solutions in addition to the typical single mini-boson star solution. These solutions can be interpreted as chains of boson stars, consisting of multiple boson stars along the symmetry axis. We demonstrate the configuration of two types of chains, one with an even number of constituents and the other with an odd number of constituents. Furthermore, we also study the effect of the frequency of the complex scalar field on the ADM mass M and the U(1) scalar charge Q. It is noteworthy that the existence of chains of boson stars does not require the introduction of a complex scalar field with self-interacting potential.

Korteweg–De Vries–Burger Equation with Jeffreys’ Wind–Wave Interaction: Blow-Up and Breaking of Soliton-like Solutions in Finite Time

In this study, the evolution of surface water solitary waves under the action of Jeffreys’ wind–wave amplification mechanism in shallow water is analytically investigated. The analytic approach is essential for numerical investigations due to the scale of energy dissipation near coasts. Although many works have been conducted based on the Jeffreys’ approach, only some studies have been carried out on finite depth. We show that nonlinearity, dispersion, and anti-dissipation are the dominating phenomena, obeying an anti-diffusive and fully nonlinear Serre–Green–Naghdi (SGN) equation. Applying an appropriate perturbation method, the current research yields a Korteweg–de Vries–Burger-type equation (KdV-B), combining weak nonlinearity, dispersion, and anti-dissipation. This derivation is novel. We show that the continuous transfer of energy from wind to water results in the growth over time of the KdV-B soliton’s amplitude, velocity, acceleration, and energy, while its effective wavelength decreases. This phenomenon differs from the classical results of Jeffreys’ approach and is due to finite depth. In this study, it is shown that expansion and breaking occur in finite time. These times are calculated and expressed with respect to soliton- and wind-appropriateparameters and values. The obtained values are measurable in experimental facilities. A detailed analysis of the breaking time is conducted with regard to various criteria. By comparing these times to the experimental results, the validity of these criteria are examined.

Construction of abundant solitons in a coupled nonlinear pendulum lattice through two discrete distinct techniques

In this paper, we deal with two distinct discrete techniques named the discrete Tanh method and the differential-difference Jacobi elliptic functions sub-ODE method to extract abundant soliton solutions for a coupled nonlinear pendulum chains. According to the two straightforward schemes’, we construct various solutions likewise, kink, anti-kink, bright, dark, singular-bright solitary wave, traveling compacton, traveling gray soliton-like, periodic soliton-like have not reported on the most recently studied mechanical model. We extract the sought solutions with less approximation (introduction of a discrete wave transformation) and display some graphical representations showing well known, novel shapes discovered in a coupled nonlinear pendulum lattice. More often, the tediousness computations of discrete techniques lead some researchers to changing a nonlinear differential-difference equation into a nonlinear ordinary differential equation throughout the semi-discrete approximation before introducing a wave transformation. Nonetheless, the semi-discrete approximation involve a Taylor bounded expansion’s such as the threshold order depends on the ability and skills of researchers to get solutions through available methods and those that they can perform. Hence, while an algorithm can be lead us to avoiding approximation (eventual source of errors), tediousness analytical computations, we perform by using it, in addition of software availability. And therefore, we establish as is the case in the present work various soliton-like solutions by means of two direct discrete techniques. The effect of some physical parameters are displayed on the shaped through 2D and 3D graphical simulations. The used methods (tanh, sn, cn, dn Jacobi elliptic functions approaches) previously applied on other physical models (discrete electrical transmission lattices) are presented here in manner to be replicable for exploring more soliton solutions in different nonlinear discrete physical models.

Strongly non-linear Boussinesq-type model of the dynamics of internal solitary waves propagating in a multilayer stratified fluid

We propose a system of first-order balance laws that describe the propagation of internal solitary waves in a multilayer stratified shallow water with non-hydrostatic pressure in the upper and lower layers. The construction of this model is based on the use of additional variables, which make it possible to approximate the Green–Naghdi-type dispersive equations by a first-order system. In the Boussinesq approximation, the governing equations allow one to simulate the propagation of non-linear internal waves, taking into account fine density stratification, a weak velocity shear in the layers, and uneven topography. We obtain smooth steady-state soliton-like solutions of the proposed model in the form of symmetric and non-symmetric waves of mode-2 adjoining to a given multilayer constant flow. Numerical calculations of the generation and propagation of large-amplitude internal waves are carried out using both the proposed first-order system and Green–Naghdi-type equations. It is established that the solutions of these models practically coincide. The advantage of the first-order equations is the simplicity of numerical implementation and a significant reduction in the calculation time. We show that the results of numerical simulation are in good agreement with the experimental data on the evolution of mode-2 solitary waves in tanks of constant and variable height.

Symbolic computation and Novel solitons, traveling waves and soliton-like solutions for the highly nonlinear (2+1)-dimensional Schrödinger equation in the anomalous dispersion regime via newly proposed modified approach

Symbolic computation and Novel solitons, traveling waves and soliton-like solutions for the highly nonlinear (2+1)-dimensional Schrödinger equation in the anomalous dispersion regime via newly proposed modified approach

Different solitary wave solutions and bilinear form for modified mixed-KDV equation

This study we handled (1+1)-dimensional modified-mixed KDV (mmKdV) equation for unique soliton solutions. For this purpose Firstly, auto-Bäcklund and Cole-Hopf transformations are created through the homogeneous balance method. Various soliton-like solutions are given by the trigonometric, hyperbolic, and exponential function waves and that leads to the extraction of novel periodic, dark, and bright soliton solutions. Moreover, the bilinear form of the model is found by utilizing the logarithmic transformation. Density, 3-D, and 2-D graphics are displayed to illustrate the dynamics of the acquired solutions. It is determined that the approach employed to obtain analytic solutions to nonlinear partial differential equation is efficient and powerful.

Emergent soliton-like solutions in the parametrically driven 1-D nonlinear Schrödinger equation

We numerically investigate the long time dynamics of spatially periodic breather solutions of the 1-D nonlinear Schrödinger equation under parametric forcing of the form along with dissipation. In the absence of dissipation, robust soliton-like excitations are observed that travel with constant amplitude and velocity. With dissipation, these solitons lose energy (and amplitude) yet gain speed - a characteristic not observed in an ordinary soliton. Moreover, these novel solitons are found to be stable against random perturbations.

Extraction of new optical solitons and complexitons related to the motion of thermophoresis of wrinkles in graphene sheets

In this paper, the thermophoretic motion equation based on Korteweg–de Vries is utilized to analyze new complexiton and soliton-like solutions. The homogenous balance approach is employed to generate auto-Bäcklund transformation of the concerned problem. This transformation is capitalized to extract abundant explicit and analytic solutions. Moreover, Hirota bilinear form of the concerned equation is taken under consideration to discover complexiton solutions via extended transform rational function approach. 3D visualization of the acquired solutions is also included to discuss its physical behavior.

Modified Exp-Function Method to Find Exact Solutions of Microtubules Nonlinear Dynamics Models

In this paper, we use the modified exp−ψθ-function method to observe some of the solitary wave solutions for the microtubules (MTs). By treating the issues as nonlinear model partial differential equations describing microtubules, we were able to solve the problem. We then found specific solutions to the nonlinear evolution equation (NLEE) covering various parameters that are particularly significant in biophysics and nanobiosciences. In addition to the soliton-like pulse solutions, we also find the rational, trigonometric, hyperbolic, and exponential function characteristic solutions for this equation. The validity of the method we developed and the fact that it provides more solutions are demonstrated by comparison to other methods. We next use the software Mathematica 10 to generate 2D, 3D, and contour plots of the precise findings we observed using the suggested technique and the proper parameter values.

IСНУВАННЯ У ПРОСТОРI ШВАРЦА ТА ВЛАСТИВОСТI РОЗВ’ЯЗКIВ РIВНЯНЬ ТИПУ РIВНЯННЯ ХОПФА ЗI ЗМIННИМИ КОЕФIЦIЄНТАМИ

The issue of the existence of solutions of the Cauchy problem for a first-order quasi-linear differential equation with partial derivatives and variable coefficients is considered. The studied equation is a generalization of the classic Hopf equation, which is used in the study of many mathematical models of hydrodynamics. This equation arises when constructing approximate (asymptotic) solutions of the Korteweg–de Vries equation and other equations with variable coefficients and a singular perturbation, in particular, when finding their asymptotic step-type soliton-like solutions. For the mentioned differential equation of the first order, the solution of the Cauchy problem is obtained in analytical form, and the statement about sufficient conditions for the existence of solutions of the initial problem in the space of rapidly decreasing functions is proved. A similar problem for a first-order differential equation with partial derivatives with variable coefficients and quadratic nonlinearity is considered.

Periodic solutions to a nonlinear Dirac-Klein-Gordon system with concave and convex nonlinearities

Using the variational method, we study the existence and multiplicity of periodic solutions to nonlinear Dirac-Klein-Gordon systems with two types of nonlinearities. Both the Dirac field and the Klein–Gordon field considered here are general nonlinear fields, different from previously considered models. Moreover, results for the regularity of the solutions are given. In addition, we obtain the soliton-like solution of Dirac-Klein-Gordon systems by approximation of periodic solutions.

AN INSIGHT ON THE (2 + 1)-DIMENSIONAL FRACTAL NONLINEAR BOITI–LEON–MANNA–PEMPINELLI EQUATIONS

With the aid of a new fractal derivative, the nonlinear Boiti–Leon–Manna–Pempinelli equation (NBLMPE) with nonsmooth boundary is explored. The variational principle of the fractal NBLMPE is successfully established by fractal wave transformation (FWT) and fractal semi-inverse method (SIM) and strong minimum condition of fractal NBLMPE is proven with the fractal Weierstrass theorem. Based on the two-scale transformation method (TSTM) and homogeneous equilibrium method (HBM), soliton-like solutions for the [Formula: see text]-dimensional (SLS [Formula: see text]D) fractal NBLMPE are acquired. A powerful means of coupling HBM and TSTM to solve fractal differential equations is proposed.

Movement of a Hydrogen Atom through Interstices in a Diamond-Like Lattice

The behavior of hydrogen atom isotopes in a diamond-like lattice without defects is studied. The movement of the hydrogen atom along the interstices is considered. The collective nature of the movement of the hydrogen atom (and its isotopes) is taken into account, which consists in the fact that any movement of the hydrogen atom is accompanied by reversible displacements of the nearest lattice atoms. To take into account this nature of the movement, local chains are built. As a result, a new one-dimensional Lagrangian is derived, the equations of motion from which lead to a soliton-like solution in the form of a kink - Frenkel-Kontorova soliton. The study of the structure of such a kink and the distribution of displacement velocities in it reveals that the kink formed by tritium moves faster through the lattice than the kinks formed by deuterium and hydrogen.