Exact Localized Solutions Research Articles

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.

Various types of discrete exact localized wave solutions and dynamical analysis for the discrete complex modified Korteweg-de Vries equation

Under consideration is the second-order integrable discretization of complex modified Korteweg-de Vries (mKdV) equation which is regarded as the discrete counterpart of the mKdV equation having an essential role in describing the propagation behavior of water waves and acoustic waves in nonlinear media. First of all, based on the known linear spectral problem, the discrete generalized (n,N−n)-fold Darboux transformation is constructed to derive various types of discrete exact localized wave solutions, including soliton and semi-rational soliton solutions on vanishing background, breather, rogue wave and hybrid interaction solutions on plane wave background, and rational soliton solution on constant background, and the relevant evolution structures are studied graphically. Secondly, the asymptotic analysis is used to discuss the elastic interaction for two-soliton solutions and limit states for rational soliton solutions. Finally, the numerical simulation is utilized to investigate the dynamical behavior and propagation stability of some exact solutions. The findings presented in this paper may contribute to explaining physical phenomena described by the mKdV equation.

Diverse Forms of Breathers and Rogue Wave Solutions for the Complex Cubic Quintic Ginzburg Landau Equation with Intrapulse Raman Scattering

This manuscript consist of diverse forms of lump: lump one stripe, lump two stripe, generalized breathers, Akhmediev breather, multiwave, M-shaped rational and rogue wave solutions for the complex cubic quintic Ginzburg Landau (CQGL) equation with intrapulse Raman scattering (IRS) via appropriate transformations approach. Furthermore, it includes homoclinic, Ma and Kuznetsov-Ma breather and their relating rogue waves and some interactional solutions, including an interactional approach with the help of the double exponential function. We have elaborated the kink cross-rational (KCR) solutions and periodic cross-rational (KCR) solutions with their graphical slots. We have also constituted some of our solutions in distinct dimensions by means of 3D and contours profiles to anticipate the wave propagation. Parameter domains are delineated in which these exact localized soliton solutions exit in the proposed model.

Space-Time Coupling: Current Concept and Two Examples from Ultrafast Optics Studied Using Exact Solution of EM Equations

Current approach to space-time coupling (STC) phenomena is given together with a complementary version of the STC concept that emphasizes the finiteness of the energy of the considered pulses. Manifestations of STC are discussed in the framework of the simplest exact localized solution of Maxwell’s equations, exhibiting a “collapsing shell”. It falls onto the center, continuously deforming, and then, having reached maximum compression, expands back without losing energy. Analytical solutions describing this process enable to fully characterize the field in space-time. It allowed to express energy density in the center of collapse in the terms of total pulse energy, frequency and spectral width in the far zone. The change of the pulse shape while travelling from one point to another is important for coherent control of quantum systems. We considered the excitation of a two-level system located in the center of the collapsing EM (electromagnetic) pulse. The result is again expressed through the parameters of the incident pulse. This study showed that as it propagates, a unipolar pulse can turn into a bipolar one, and in the case of measuring the excitation efficiency, we can judge which of these two pulses we are dealing with. The obtained results have no limitation on the number of cycles in a pulse. Our work confirms the productivity of using exact solutions of EM wave equations for describing the phenomena associated with STC effects. This is facilitated by rapid progress in the search for new types of such solutions.

Semi-exact local absorbing boundary condition for seismic wave simulation

AbstractAn absorbing boundary condition is necessary in seismic wave simulation for eliminating the unwanted artificial reflections from model boundaries. Existing boundary condition methods often have a trade-off between numerical accuracy and computational efficiency. We proposed a local absorbing boundary condition for frequency-domain finite-difference modelling. The proposed method benefits from exact local plane-wave solution of the acoustic wave equation along predefined directions that effectively reduces the dispersion in other directions. This method has three features: simplicity, accuracy and efficiency. Numerical simulation demonstrated that the proposed method has higher efficiency than the conventional methods such as the second-order absorbing boundary condition and the perfectly matched layer (PML) method. Meanwhile, the proposed method shared the same low-cost feature as the first-order absorbing boundary condition method.

Electromagnetic waves in graded-index planar waveguides

The propagation of guided TE and TM modes through graded-index planar waveguides is reported in this paper. Both a real-valued and complex-valued position-dependent refractive index are supposed for a film layer. It is possible to set various refractive index profiles based on five distribution parameters. For the position-dependent refractive index, the governing equations are transformed to Heun’s differential equation, an exact local solution expressed in terms of local Heun functions. The general nature of these functions is demonstrated based on four degenerate cases of Heun’s equation. The calculation of guided modes requires evaluation of the general solution in the interval containing two regular singular points. For this purpose, the generalized Heun function is introduced and employed in general solutions to the governing equations. The applicability of the generalized solutions is demonstrated by the calculation of guided modes for both the real-valued and complex-valued refractive index.

Collective Neurodynamic Optimization for Image Segmentation by Binary Model with Constraints

Threshold method is an important image segmentation method, which has been widely used in image segmentation. For this method, it is very important to choose a good threshold. The traditional threshold segmentation algorithm is implemented by exhaustive method, which makes the solution efficiency very low. This paper presents a collective neurodynamic optimization algorithm to solve the problem of binary optimization in the image segmentation. The problem of image segmentation based on threshold is transformed into binary optimization with constraints. Then, a collective neurodynamic optimization algorithm is introduced which combined with feedback neural network and particle swarm optimization (PSO) algorithm. And the linear programming relaxation constraint method is used to relax binary constraints. It is proved by numerical simulation that the feedback neural network algorithm can converge to the exact local optimal solution of the model and the PSO algorithm can get a better local optimal solution. Finally, several sets of comparative experiments are presented. The feasibility of our proposed method is verified; the experimental results demonstrate the effectiveness of our approach in image segmentation. In this study, a collective neurodynamic optimization was proposed for the image segmentation problem. In the future, we expect that multiple centralized neurodynamic models and intelligent algorithms can be used to solve the problem and improve the convergence speed of the solved model.

A class of localized soliton and fractal pattern solutions of the (2 + 1)-dimensional modified dispersive long wave model

The variable separation approach is an important tool to obtain exact localized and fractal structure solutions of higher dimensional nonlinear problems. The general separable solutions involving double random functions are obtained by means of Riccati equation and the variable separation hypothesis. By choosing the suitable arbitrary functions, we obtain a class of localized excitations and fractal structures solutions of the (2 + 1)-dimensional modified dispersive long wave (MDLW) model. Such solutions present couple lump waves, breather wave solutions, dromions, oscillating multi-lumps, oscillating multi-dromoins and ring solitons of the model. Moreover, the fractal lump and fractal dromions type excitation-localized structures of the model are presented here. To visualize the dynamics of each excitation structures are demonstrated by the 3D and density plots.

Families of Rational and Semirational Solutions of the Partial Reverse Space-Time Nonlocal Mel′nikov Equation

Exact periodic and localized solutions of a nonlocal Mel′nikov equation are derived by the Hirota bilinear method. Many conventional nonlocal operators involve integration over a spatial or temporal domain. However, the present class of nonlocal equations depends on properties at selected far field points which result in a potential satisfying parity time symmetry. The present system of nonlocal partial differential equations consists of two dependent variables in two spatial dimensions and time, where the dependent variables physically represent a wave packet and an auxiliary scalar field. The periodic solutions may take the forms of breathers (pulsating modes) and line solitons. The localized solutions can include propagating lumps and rogue waves. These nonsingular solutions are obtained by appropriate choice of parameters in the Hirota expansion. Doubly periodic solutions are also computed with elliptic and theta functions. In sharp contrast with the local Mel′nikov equation, the auxiliary scalar field in the present set of solutions can attain complex values. Through a coordinate transformation, the governing equation can reduce to the Schrödinger–Boussinesq system.

Exact Solutions and Numerical Simulation of the Discrete Sawada–Kotera Equation

We investigated an integrable five-point differential-difference equation called the discrete Sawada–Kotera equation. On the basis of the geometric series method, a new exact soliton-like solution of the equation is obtained that propagates with positive or negative phase velocity. In terms of the Jacobi elliptic function, a class of new exact periodic solutions is constructed, in particular stationary ones. Using an exponential generating function for Catalan numbers, Cauchy’s problem with the initial condition in the form of a step is solved. As a result of numerical simulation, the elasticity of the interaction of exact localized solutions is established.

Exact nonlocal solutions of circular nanoplates subjected to uniformly distributed loads and nonlocal concentrated forces

The nonlocal solutions of the bending problem of the circular elastic nanoplates subjected to uniformly distributed loads by using the Eringen’s nonlocal theory are approximate. Seeking for an exact solution of the concerning problem by adding a concentrated force; a specific concentrated force which depends on the nonlocal parameter and, therefore, vanishes in the local case, is obtained. In this context; starting from the local bending solution of the simply supported circular nanoplates under uniformly distributed loads and evolving the solution step by step, exact nonlocal solutions of the bending problem of the simply supported and clamped circular nanoplates subjected to uniformly distributed loads and the concerning nonlocal concentrated forces acting at the centers of the circular nanoplates are obtained with the expectation that the exact nonlocal solution is not very different than the corresponding exact local solution. The closed-form solutions are checked to satisfy exactly the equations of equilibrium, constitutive equations and the boundary conditions of the relevant nonlocal problem. The application of the concerning specific concentrated force, in addition to the uniformly distributed load, is seen to make the radial bending moments for the simply supported circular nanoplate and the deflections for the clamped circular nanoplate free from the effects of the nonlocality.

Localized and complex soliton solutions to the integrable (4+1)-dimensional Fokas equation

We explore a family of exact traveling and localized soliton solutions of the nonlinear integrable (4+1)-dimensional Fokas equation by different approaches, namely, the Jacobian-function method, the Pades type transformation, He’s semi-inverse variational technique and sine–cosine or triangle function approach. Some new exact traveling wave solutions of physical interest involving some constraint conditions are reported. The reported solutions are Jacobi doubly periodic wave solutions, fractional soliton, localized soliton when parameters are taken to be special values of doubly periodic functions and complex Bloch solitons. These obtained solutions may facilitate us in understanding the dynamical behavior of physical phenomenon governed by nonlinear integrable Fokas equation and show the applicability and efficacy of the used approaches that can be applied for higher dimensional nonlinear integrable equations as well as linear ones in mathematical physics. The differences and similarities between the used distinct approaches are discussed. To exhibits, the dynamical behavior of reported solutions, the two and three-dimensional structures are numerically simulated.

Charge and energy transport by Holstein solitons in anharmonic one-dimensional systems

We consider the problem of electron transport and energy transfer in a one-dimensional molecular chain with non-dipole optical phonon mode. We take into account the dispersion of optical phonons, anharmonicity of the lattice on-site potential and electron-lattice interaction. In the lowest order linear approximation such a system admits solutions in the form of the Holstein polaron. Here, within the traveling wave formalism for the corresponding non-linear equations of motion in the long-wave limit, we show the existence of three particular types of exact analytical localized solutions. Two of them, here referred to as Holstein solitons of the first and second kind, respectively, describe a one-hump localized electron wave functions, while the third one displays two humps in the envelope of the wave function. We use the variational approach to reproduce the exact analytical profiles in the three cases of the particular normalized solutions and to variationally predict the existence of branches of normalized solutions for the three types of profiles. We confirm our findings by numerically continuing, in the parameter of velocity of propagation, the analytically exact particular solutions.

Self-similar optical solitons with continuous-wave background in a quadratic–cubic non-centrosymmetric waveguide

We investigate the propagation of self-similar optical solitons on a continuous-wave background through a non-centrosymmetric waveguide with second- and third-order nonlinearities. The generalized inhomogeneous nonlinear Schrödinger equation with quadratic–cubic nonlinearities and gain or loss is used to describe the beam propagation through the waveguide. Three new types of exact self-similar localized solutions propagating on a continuous-wave background are found under certain parametric conditions. The self-similar wave solutions comprise bright solitons, W-shaped, and kink solitons. The dynamic behaviors of these solitons are analyzed for a periodically distributed dispersion system.

Simple Exact Solutions and Asymptotic Localized Solutions of the Two-Dimensional Massless Dirac Equation for Graphene

We construct exact solutions and asymptotic localized solutions of the two-dimensional massless Dirac equation for graphene with a small potential.

Exact localized solutions of the generalized (3+1)-dimensional Landau-Lifshitz equation

Exact localized solutions of the generalized (3+1)-dimensional Landau-Lifshitz equation

Exact Local Solutions for the Formation of Singularities on the Free Surface of an Ideal Fluid

A classical problem of the dynamics of the free surface of an ideal incompressible fluid with infinite depth has been considered. It has been found that the regime of motion of the fluid where the pressure is a quadratic function of the velocity components is possible in the absence of external forces and capillarity. It has been shown that equations of plane potential flow for this situation are linearized in conformal variables and are then easily solved analytically. The found solution includes an arbitrary function specifying the initial shape of the surface of the fluid. The developed approach makes it possible for the first time to locally describe the formation of various singularities on the surface of the fluid—bubbles, drops, and cusps.

Description of waves in inhomogeneous domains using Heun’s equation

There are a number of model equations describing electromagnetic, acoustic or quantum waves in inhomogeneous domains and some of them are of the same type from the mathematical point of view. This isomorphism enables us to use a unified approach to solving the corresponding equations. In this paper, the inhomogeneity is represented by a trigonometric spatial distribution of a parameter determining the properties of an inhomogeneous domain. From the point of view of modeling, this trigonometric parameter function can be smoothly connected to neighboring constant-parameter regions. For this type of distribution, exact local solutions of the model equations are represented by the local Heun functions. As the interval for which the solution is sought includes two regular singular points. For this reason, a method is proposed which resolves this problem only based on the local Heun functions. Further, the transfer matrix for the considered inhomogeneous domain is determined by means of the proposed method. As an example of the applicability of the presented solutions the transmission coefficient is calculated for the locally periodic structure which is given by an array of asymmetric barriers.

Modulation of localized solutions in quadratic-cubic nonlinear Schrödinger equation with inhomogeneous coefficients

We study the presence of exact localized solutions in a quadratic-cubic nonlinear Schrödinger equation with inhomogeneous nonlinearities. Using a specific ansatz, we transform the nonautonomous nonlinear equation into an autonomous one, which engenders composed states corresponding to solutions localized in space, with an oscillating behavior in time. Direct numerical simulations are employed to verify the stability of the modulated solutions against small random perturbations.

Fermi–Pasta–Ulam recurrence and modulation instability

We give a qualitative explanation of the analog of the Fermi-Pasta-Ulam (FPU) recurrence in a one-dimensional focusing nonlinear Schrodinger equation (NLSE). That recurrence can be considered as a result of the nonlinear development of modulation instability. All known exact localized solitons-type solutions describing propagation on the background of the modulationally unstable condensate show the recurrence to the condensate state after its interaction with solitons. The condensate state locally recovers its original form with the same amplitude but a different phase after soliton leave its initial region. This is the analog of the FPU recurrence for the NLSE. Based on the integrability of the NLSE, we demonstrate that the FPU recurrence takes place not only for condensate but for more general solution in the form of the cnoidal wave. This solution is periodic in space and can be represented as a solitonic lattice. That lattice reduces to isolated soliton solution in the limit of large distance between solitons. The lattice transforms into the condensate in the opposite limit of dense soliton packing. The cnoidal wave is also modulationally unstable due to soliton overlapping. This instability at the linear stage does not provide the cnoidal wave recurrence. The recurrence happens at the nonlinear stage of the modulation instability. From the practical point of view the latter property is very important, especially for the fiber communication systems which use soliton as an information carrier.