Solitons in a generalized space- and time-variable coefficients nonlinear Schrödinger equation with higher-order terms

We introduce an approach that combines a similarity method with several transformations to find analytical solitary wave solutions for a generalized space- and time-variable coefficients of nonlinear Schrödinger equation with higher-order terms with consideration of varying dispersion, higher nonlinearities, gain/loss and external potential. One of these transformations is constructed in such a way that allows study of the width of localized solutions. Solitary-like wave solutions for front, bright and dark are given. The precise expressions of the soliton?s width, peak, and the trajectory of its mass center and the external potential which are symbol of dynamic behavior of these solutions, are investigated analytically. In addition, the dynamical behavior of moving, periodic, quasi-periodic of breathing, and resonant are discussed. Stability of the obtained solutions is analyzed both analytically and numerically.     

The Bäcklund and the Galilei Invariant Transformations Constructed by Similarity Variables for Soliton Equations

Abstract

The Painlevé-test has been applied to checking the integrability of nonlinear PDEs, since similarity solutions of many soliton equations satisfy the Painlevé equation. As is well known, such similarity solutions can be obtained by the infinitesimal transformation, that is, the classical similarity analysis, and also the dimension of the PDEs can be reduced.

In this paper, the KdV, the mKdV, and the nonlinear Schrödinger equations are considered and are transformed into equations with loss and/or nonuniformity by transformations constructed on a basis of the local similarity variables. The transformations include the Bäcklund and the Galilei invariant ones. It should be noticed that the approach is applicable to other PDEs and for nonlocal similarity variables.     

Exact Solutions of Atmospheric(2+1)-Dimensional Nonlinear Incompressible Non-hydrostatic Boussinesq Equations

Exact solutions of the atmospheric(2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq(INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space.     

Exact Solutions of Atmospheric (2+1)-Dimensional Nonlinear Incompressible Non-hydrostatic Boussinesq Equations

Exact solutions of the atmospheric (2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space.     

Localized nonlinear waves in systems with time- and space-modulated nonlinearities

Using similarity transformations we construct explicit nontrivial solutions of nonlinear Schr?dinger equations with potentials and nonlinearities depending both on time and on the spatial coordinates. We present the general theory and use it to calculate explicitly nontrivial solutions such as periodic (breathers), resonant, or quasiperiodically oscillating solitons. Some implications to the field of matter waves are also discussed.     

Vibrations of initially stressed thick,rectangular, orthotropic plates

Equations of motion for antisymmetric cross-ply laminates in a general state of non-uniform initial stress, where the effects of transverse shear and rotary inertia are included, are derived by the virtual work theorem. The equations are adjusted to a generic expression by using proper transformations. Then, the vibrational behaviors of the laminates subjected to a state of uniform tensile or compressive stress plus a uniform bening stress are examined. By considering the transformations, introducing the generalized parameters, and employing a similarity parameter, comprehensive solutions are found to this problem; hence, the curves presented in the text are generic rather than specific. The frequency behavior prior to buckling, undergoes a transition for various mode shape numbers is also investigated.     

Self-phase modulation via similariton solutions of the perturbed NLSE Modulation instability and induced self-steepening

The perturbed nonlinear Schrodinger equation (PNLSE) describes the pulse propagation in optical fibers, which results from the interaction of the higher-order dispersion effect, self-steepening (SS) and self-phase modulation (SPM). The challenge between these aforementioned phenomena may lead to a dominant one among them. It is worth noticing that the study of modulation instability (MI) leads to the inspection of dominant phenomena (DPh). Indeed, the MI triggers when the coefficient of DPh exceeds a critical value and it may occur that the interaction leads to wave compression. The PNLSE is currently studied in the literature, mainly on finding traveling wave solutions. Here, we are concerned with analyzing the similarity solutions of the PNLSE. The exact solutions are obtained via introducing similarity transformations and by using the extended unified method. The solutions are evaluated numerically and they are shown graphically. It is observed that the intensity of the pulses exhibits self steepening which progresses to shock soliton in ultra-short time (or near t = 0). Also, it is found that the real part of the solution exhibits self-phase modulation in time. The study of (MI) determines the critical value for the coefficients of SS, SPM, or high dispersivity to occur.     

Non-autonomous bright matter wave solitons in spinor Bose–Einstein condensates

We investigate the dynamics of bright matter wave solitons in spin-1 Bose–Einstein condensates with time modulated nonlinearities. We obtain soliton solutions of an integrable autonomous three-coupled Gross–Pitaevskii (3-GP) equations using Hirota?s method involving a non-standard bilinearization. The similarity transformations are developed to construct the soliton solutions of non-autonomous 3-GP system. The non-autonomous solitons admit different density profiles. An interesting phenomenon of soliton compression is identified for kink-like nonlinearity coefficient with Hermite–Gaussian-like potential strength. Our study shows that these non-autonomous solitons undergo non-trivial collisions involving condensate switching.     

Heat transfer for boundary layers with cross flow

An analysis is presented to study the dual nature of solutions for the forced convective boundary layer flow and heat transfer in a cross flow with viscous dissipation terms in the energy equation. The governing equations are transformed into a set of three self-similar ordinary differential equations by similarity transformations. These equations are solved numerically using the very efficient shooting method. This study reveals that the dual solutions of the transformed similarity equations for velocity and temperature distributions exist for certain values of the moving parameter, Prandtl number, and Eckert numbers. The reverse heat flux is observed for larger Eckert numbers; that is, heat absorption at the wall occurs.     

Exact solutions for the quintic nonlinear Schrödinger equation with time and space modulated nonlinearities and potentials

In this Letter, by means of similarity transformations, we construct explicit solutions to the quintic nonlinear Schrödinger equation with potentials and nonlinearities depending both on time and on the spatial coordinates. We present the general approach and use it to study some examples and find nontrivial explicit solutions such as periodic (breathers), quasiperiodic and bright and dark soliton solutions.     

Bosonized Supersymmetric Sawada–Kotera Equations: Symmetries and Exact Solutions

The Bosonized Supersymmetric Sawada–Kotera(BSSK) system is constructed by applying bosonization method to a Supersymmetric Sawada–Kotera system in this paper. The symmetries on the BSSK equations are researched and the calculation shows that the BSSK equations are invariant under the scaling transformations, the space-time translations and Galilean boosts. The one-parameter invariant subgroups and the corresponding invariant solutions are researched for the BSSK equations. Four types of reduction equations and similarity solutions are proposed. Period Cnoidal wave solutions, dark solitary wave solutions and bright solitary wave solutions of the BSSK equations are demonstrated and some evolution curves of the exact solutions are figured out.     

Self-similar collapse and blow up of laser beams in nonlinear Kerr media—Revisited

Some features of the blow up and collapse phenomena characteristic of the focusing nonlinear Schrödinger equation are discussed. Particular attention is given to recently found solutions describing self-similar collapse in the form of ring-shaped structures. The characteristic parameters of such collapsing rings are found and particular single-humped initial conditions giving rise to ring formation are also discussed. The possibility of self-similarly collapsing beams with parabolic intensity profiles is investigated and used to infer the blow up distance of initially Gaussian beams. The result is shown to be in good agreement with previous empirical results for the blow up distance.     

New applications of the homogeneous balance principle

The homogeneous balance principle has been widely applied to the exploration of nonlinear transformation, exact solutions (especially solitary wave solution), dromion and similarity reduction to the nonlinear partial differential equations in mathematical physics. In this paper, we use the homogeneous balance principle to derive B?cklund transformations for nonlinear partial differential equations that have more nonlinear terms and more highest-order partial derivative terms. With the aid of the B?cklund transformations derived here, we could obtain exact solutions to the nonlinear partial differential equations. The Davey－Stewartson equation and the Nizhnik－Novikov－Veselov equation are considered as the examples.     

Exact solutions of the Fokker-Planck equations with moving boundaries

By means of time-dependent similarity transformations, we derive exact solutions of the Fokker-Planck equations with moving boundaries in the presence of: (1) a time-dependent linear force and (2) a time-dependent nonlinear force. The method of similarity transformation is simple and can be easily applied to more general Fokker-Planck equations. Furthermore, the knowledge of the exact solutions in closed form can be useful as a benchmark to test approximate numerical or analytical procedures.     

Stability analysis of unsteady stagnation-point gyrotactic bioconvection flow and heat transfer towards the moving sheet in a nanofluid

The present work investigated the unsteady stagnation-point flow and heat transfer of a nanofluid containing gyrotactic microorganisms past a permeable moving surface. The similarity transformations produced the mathematical model in the simpler form, which is in the form of ordinary differential equations, and the collocation method solved it numerically. The dual solutions are observable when the governing parameters vary. The decelerating flow and weak effect of suction at the shrinking sheet delays the boundary layer separation. Stability analysis showed that the upper branch solution is a solution with the stabilizing feature while the lower branch solution is an unstable solution which implies the flow with separation. This theoretical study is significantly relevant to microscopic biological propulsion integrated with the nano-based system.     

Reciprocal transformations of the space-time shifted nonlocal short pulse equations

Reciprocal transformations of the space-time shifted nonlocal short pulse equations are elaborated. Covariance of dependent and independent variables involved in the reciprocal transformations is investigated. Exact solutions of the space-time shifted nonlocal short pulse equations are given in terms of double Wronskians. Realness of independent variables involved in the reciprocal transformations is verified. Dynamics of some obtained solutions are illustrated.     

Three-dimensional mixed convection flow with variable thermal conductivity and frictional heating

In this article,three-dimensional mixed convection flow over an exponentially stretching sheet is investigated.Energy equation is modelled in the presence of viscous dissipation and variable thermal conductivity.Temperature of the sheet is varying exponentially and is chosen in a form that facilitates the similarity transformations to obtain self-similar equations.Resulting nonlinear ordinary differential equations are solved numerically employing the Runge-Kutta shooting method.In order to check the accuracy of the method,these equations are also solved using bvp4c built-in routine in Matlab.Both solutions are in excellent agreement.The effects of physical parameters on the dimensionless velocity field and temperature are demonstrated through various graphs.The novelty of this analysis is the self-similar solution of the threedimensional boundary layer flow in the presence of mixed convection,viscous dissipation and variable thermal conductivity.     

Classification of Equilibrium Solutions of the Cometary Flow Equation and Explicit Solutions of the Euler Equations for Monatomic Ideal Gases

The set of smooth equilibrium solutions of a kinetic model for cometary flows is split into equivalence classes according to similarity transformations. For each equivalence class in the two- and three-dimensional cases a normal form is computed. Each such equilibrium solution gives rise to an explicit solution of the compressible Euler equations for monatomic gases. The set of these solutions is discussed with special emphasis on solutions containing vacuum regions.     

An exact solution of the Einstein-Maxwell equations referring to a magnetic dipole

There is a formal similarity between stationary exterior solutions of the Einstein equations and static magnetic solutions of the Einstein-Maxwell theory. This is particularly evident for axially symmetric fields, and one finds that the sets of equations governing the two cases can be transformed one into the other by simple transformations of the dependent variables.     

Nonequivalent Similarity Reductions and Exact Solutions for Coupled Burgers-Type Equations

Using the machinery of Lie group analysis,the nonlinear system of coupled Burgers-type equations is studied.Using the infinitesimal generators in the optimal system of subalgebra of the said Lie algebras,it leads to two nonequivalent similarity transformations by using it we obtain two reductions in the form of system of nonlinear ordinary differential equations.The search for solutions of these systems by using the G'/G-method has yielded certain exact solutions expressed by rational functions,hyperbolic functions,and trigonometric functions.Some figures are given to show the properties of the solutions.