Exact solutions for the higher-order nonlinear Schördinger equation in nonlinear optical fibres

First, by using the generally projective Riccati equation method, many kinds of exact solutions for the higher-order nonlinear Schördinger equation in nonlinear optical fibres are obtained in a unified way. Then, some relations among these solutions are revealed.     

Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres

A generalized method, which is called the generally projective Riccati equation method, is presented to find more exact solutions of nonlinear differential equations based upon a coupled Riccati equation. As an application of the method, we choose the higher-order nonlinear Schrodinger equation to illustrate the method. As a result more new exact travelling wave solutions are found which include bright soliton solutions, dark soliton solution, new solitary waves, periodic solutions and rational solutions. The new method can be extended to other nonlinear differential equations in mathematical physics.     

Analytical studies of soliton pulses along two-dimensional coupled nonlinear transmission lines

A nonlinear network with many coupled nonlinear LC dispersive transmission lines is considered, each line of the network containing a finite number of cells. In the semi-discrete limit, we apply the reductive perturbation method and show that the wave propagation along the network is governed by a two-dimensional nonlinear partial differential equation (2-D NPDE) of Schrödinger type. Two regimes of wave propagation, the high-frequency and the low-frequency are detected. By the means of exact soliton solution of the 2-D NPDE, we investigate analytically the soliton pulse propagation in the network. Our results show that the network under consideration supports the propagation of kink and dark solitons.     

Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation

In this paper, we consider the nonlinear Kirchhoff-type equation $ u_{tt} + M(\left\| {D^m u(t)} \right\|_2^2 )( - \Delta )^m u + \left| {u_t } \right|^{q - 2} u_t = \left| {u_t } \right|^{p - 2} u $ u_{tt} + M(\left\| {D^m u(t)} \right\|_2^2 )( - \Delta )^m u + \left| {u_t } \right|^{q - 2} u_t = \left| {u_t } \right|^{p - 2} u with initial conditions and homogeneous boundary conditions. Under suitable conditions on the initial datum, we prove that the solution blows up in finite time.     

Traveling-wave solution to a nonlinear equation in semiconductors with strong spatial dispersion

The third-order nonlinear differential equation (u _xx ? u) _t + u _xxx + uu _x = 0 is analyzed and compared with the Korteweg-de Vries equation u _t + u _xxx ? 6uu _x = 0. Some integrals of motion for this equation are presented. The conditions are established under which a traveling wave is a solution to this equation.     

SOLITON SOLUTION OF NONLINEAR SCHRODINGER EQUATION WITH HIGHER ORDER DISPERSION TERMS

It is shown in this paper that if parameters β1,β2 and β3 of a nonlinear Schrodinger equation with higher order dispersion terms (HNLS) satisfy the condition: 6β1-β2-2β3(1- 6β1k)= 0, k a real constant, then the fundamental soliton solutions of the HNLS equation exist. The exact soliton solutions are given and the relation between this condition and the known results inthe literature is also discussed.     

Ground states for the higher-order dispersion managed NLS equation in the absence of average dispersion

The problem of existence of ground states in higher-order dispersion managed NLS equation is considered. The ground states are stationary solutions to dispersive equations with nonlocal nonlinearity which arise as averaging approximations in the context of strong dispersion management in optical communications. The main result of this note states that the averaged equation possesses ground state solutions in the practically and conceptually important case of the vanishing residual dispersions.     

Solitary wave interaction for a higher-order nonlinear Schrodinger equation

Solitary wave interaction for a higher-order version of thenonlinear Schrödinger (NLS) equation is examined. An asymptotictransformation is used to transform a higher-order NLS equationto a higher-order member of the NLS integrable hierarchy, ifan algebraic relationship between the higher-order coefficientsis satisfied. The transformation is used to derive the higher-orderone- and two-soliton solutions; in general, the N-soliton solutioncan be derived. It is shown that the higher-order collisionis asymptotically elastic and analytical expressions are foundfor the higher-order phase and coordinate shifts. Numericalsimulations of the interaction of two higher-order solitarywaves are also performed. Two examples are considered, one satisfiesthe algebraic relationship derived from asymptotic theory, andthe other does not. For the example which satisfies the algebraicrelationship, the numerical results confirm that the collisionis elastic. The numerical and theoretical predictions for thehigher-order phase and coordinate shifts are also in strongagreement. For the example which does not satisfy the algebraicrelationship, the numerical results show that the collisionis inelastic and radiation is shed by the solitary wave collision.As the bed of radiation shed by the waves decays very slowly(like t^–), it is computationally infeasible to calculatethe final phase and coordinate shifts for the inelastic example.An asymptotic conservation law is derived and used to test thefinite-difference scheme for the numerical solutions.     

Analytic approximations for the one-loop soliton solution of the Vakhnenko equation

In this paper, we give an analytic solution for the one-loop soliton solution of the Vakhnenko equation, by the use of the homotopy analysis method and via a fractional basis.     

Global attractivity of a higher-order nonlinear difference equation

The main goal of this paper is to investigate the locally asymptotically stable, period-two solutions, invariant intervals and global attractivity of all negative solutions of the nonlinear difference equation

     

A blow-up result for a higher-order nonlinear Kirchhoff-type hyperbolic equation

In this work we consider a multi-dimensional higher-order Kirchhoff-type wave equation, with Dirichlet boundary conditions. We establish a blow-up result for certain solutions with positive initial energy.     

A generalized nonlinear Schrödinger equation and optical soliton in a gradient index cylindrical media

A generalized form of nonlinear Schrödinger equation is deduced for the propagation of an optical pulse in a fiber with a cylindrical geometry having a gradient in refractive index in the radial direction. The configuration gives a simple model for a fiber with a cladding or multicore fiber. To begin with we have analyzed in detail the modulational instability in terms of Stokes and anti-Stokes side band amplitudes which shows a significant change with respect to the depth parameter L and dispersion constant. Next we have deduced the equations governing the modulation of parameters of a Gaussian pulse as it propagates through it. The moment method is used for the derivation. The gradient of the refractive index leads to the trapping of the pulse, whereas the balance between nonlinearity (Kerr type) and dispersion in the longitudinal direction guides the propagation. Instead of a constant dispersion profile we have considered the standard dispersion map which helps in shaping of the pulse. The numerical simulation of these derived equations shows how the chirp, width, amplitude of the pulse change with type of gradient and the distance travelled.     

Some properties of the Yamabe soliton and the related nonlinear elliptic equation

Firstly we prove the non-existence of positive radially symmetric solution of the nonlinear elliptic equation $\frac{n-1}{m}\Delta v^m+\alpha v+\beta x\cdot \nabla u=0$ in $\mathbb{R }^{n}$ when $n\ge 3$ , $0<m\le \frac{n-2}{n}$ , $\alpha <0$ and $\beta \le 0$ and prove various properties of the solution of the above elliptic equation for other parameter range of $\alpha $ and $\beta $ . Then these results are applied to prove some results on Yamabe solitons including the exact behaviour of the metric of the Yamabe soliton, its scalar curvature and sectional curvature, at infinity. A new proof of a result of Daskalopoulos and Sesum (The classification of locally conformally flat Yamabe solitons, http://arxiv.org/abs/1104.2242) on the positivity of the sectional curvature of Yamabe solitons is also presented.     

Solitary wave solution for the generalized KdV equation with time-dependent damping and dispersion

The solitary wave solution of the generalized KdV equation is obtained in this paper in presence of time-dependent damping and dispersion. The approach is from a solitary wave ansatze that leads to the exact solution. A particular example is also considered to complete the analysis.     

Exact controllability of the nonlinear third-order dispersion equation

Exact controllability of a nonlinear dispersion system has been studied. This work extends the work of Russell and Zhang [D.L. Russell, B.Y. Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain, SIAM J. Control Optim. 31 (1993) 659-676], in which the authors considered a linear dispersion system. We obtain controllability results using two standard types of nonlinearities, namely, Lipschitzian and monotone. We also obtain the exact controllability of the same system through the approach of Integral Contractors which is a weaker condition than Lipschitz condition.     

Regularity of the solution of the Cauchy problem for a higher-order parabolic equation

We prove the unique solvability of the Cauchy problem in a weighted Hölder space for a linear parabolic equation of order 2m under the condition that the lower coefficients and the right-hand side of the equation can have certain growth when approaching the plane that is the support of the initial data, while the higher coefficients do not necessarily satisfy the Dini condition near this plane.We construct a smoothness scale of solutions of the Cauchy problem in the corresponding weighted Hölder spaces.     

Analytical solution for a steady flow of enclosed rotating disks

Low-Reynolds-number flow plays an important role in the centrifugal separation of fluid particles under microgravity conditions and also in micromechanics due to the miniaturization of fluid mechanical parts. In this situation, the governing equations may be simplified. Here an analytical solution is presented for the steady flow of an incompressible viscous fluid between two finite disks enclosed by a cylindrical container for small Reynolds number (Re 10). The general solution is valid for all choices of the aspect ratio () and different cases of disk to cylinder rotation rates (s). An expression for the torque acting on the disk is obtained. The tangential velocity distribution is calculated and presented graphically for different values of ands. Known results in the literature for a single rotating disk and similar problems follow as a particular case of the general solution presented.

---

Zusammenfassung Zahlreiche hydrodynamische Vorgänge unter der Bedingung verminderter Schwerkraft aber auch Vorgänge in der Mikromechanik finden im Bereich kleiner Reynoldszahlen statt. In solchen Situationen können die Bewegungsgleichungen vereinfacht und eventuell analytische Lösungen gefunden werden. In dieser Arbeit wird die stationäre Strömung einer viskosen, inkompressiblen Flüssigkeit für kleine Reynolds- und unterschiedliche Aspektzahlen untersucht. Die Flüssigkeit ist zwischen zwei rotierenden Scheiben und einem zylindrischen Behälter eingeschlossen. Eine analytische Lösung für die Tangentialkomponente des Geschwindigkeitsvektors ist für den allgemeinen Fall, dass die Scheiben und der Behälter unterschiedliche Winkelgeschwindigkeiten besitzen können, dargestellt. Des weiteren wurde eine Beziehung für das Widerstandsmoment der rotierenden Scheibe angegeben. Der Verlauf der Tangentialgeschwindigkeiten für verschiedene Rotations- und Aspektverhältnisse wird graphisch dargestellt und diskutiert. Bereits angegebene Lösungen in der Literatur bezüglich dieser Geometrie können als Sonderfall der hier dargestellten Lösung entwickelt werden.

     

Exact soliton solution for the fourth‐order nonlinear Schrödinger equation with generalized cubic‐quintic nonlinearity

In this paper, we investigate the fourth‐order nonlinear Schrödinger equation with parameterized nonlinearity that is generalized from regular cubic‐quintic formulation in optics and ultracold physics scenario. We find the exact solution of the fourth‐order generalized cubic‐quintic nonlinear Schrödinger equation through modified F‐expansion method, identifying the particular bright soliton behavior under certain external experimental setting, with the system's particular nonlinear features demonstrated. Copyright © 2016 John Wiley & Sons, Ltd.