Theory of passively mode-locked lasers with self-frequency shift and saturable absorber response

A general theoretical model for passively mode-locked lasers is presented, in which both the self-frequency shift and either a fast or a slow saturable absorber response are taken into account. An exact soliton-like solution and condition for its existence are obtained under a definite compatible condition. The stability of the solution is analyzed by using a variational method, and a parameter region, in which the solution is linearly stable, is acquired theoretically. To verify the theoretical predictions, a typical example is given for stable pulse propagation over a long distance. The numerical results show that the soliton-like solution is stable under some perturbations within the linearly stable region and an arbitrary Gaussian pulse converges to the exact soliton-like solution after evolution in a distance.     

A New Solution to Einstein's Field Equations

We construct a new exact solution to the vacuum Einstein field equations. This solution possesses a naked physical singularity. The norm of the Riemann curvature tensor of the solution takes infinity at some points and the solution does not have any event horizon around the singularity. A detailed analysis of this new singularity is also presented.     

Three-dimensional MHD flow over a shrinking sheet: Analytical solution and stability analysis

The magnetohydrodynamic(MHD) steady and unsteady axisymmetric flows of a viscous fluid over a two-dimensional shrinking sheet are addressed. The mathematical analysis is carried out in the presence of a large magnetic field. The steady state problem results in a singular perturbation problem having an infinite domain singularity. The secular term appearing in the solution is removed and a two-term uniformly valid solution is derived using the Lindstedt–Poincaré technique. This asymptotic solution is validated by comparing it with the numerical solution. The solution for the unsteady problem is also presented analytically in the asymptotic limit of large magnetic field. The results of velocity profile and skin friction are shown graphically to explore the physical features of the flow field. The stability analysis of the unsteady flow is made to validate the asymptotic solution.     

Slip Magnetohydrodynamic Viscous Flow over a Permeable Shrinking Sheet

The magnetohydrodynamic （MHD） flow under slip conditions over a shrinking sheet is solved analytically. The solution is given in a closed form equation and is an exact solution of the full governing Navier-Stokes equations. Interesting solution behavior ＆ observed with multiple solution branches for certain parameter domain. The effects of the mass transfer, slip, and magnetic parameters are discussed.     

Nonlinear transport of Bose--Einstein condensates in a double barrier potential

The stable nonlinear transport of the Bose-Einstein condensates through a double barrier potential in a waveguide is studied. By using the direct perturbation method we have obtained a perturbed solution of Cross-Pitaevskii equation. Theoretical analysis reveals that this perturbed solution is a stable periodic solution, which shows that the transport of Bose-Einstein condensed atoms in this system is a stable nonlinear transport. The corresponding numerical results are in good agreement with the theoretical analytical results.     

Nonlinear tunneling through a strong rectangular barrier

Nonlinear tunneling is investigated by analytically solving the one-dimensional Gross–Pitaevskii equation(GPE)with a strong rectangular potential barrier. With the help of analytical solutions of the GPE, which can be reduced to the solution of the linear case, we find that only the supersonic solution in the downstream has a linear counterpart. A critical nonlinearity is explored as an up limit, above which no nonlinear tunneling solution exists. Furthermore, the density solution of the critical nonlinearity as a function of the position has a step-like structure.     

An upper bound for the adiabatic approximation error

In this paper,we derive an upper bound for the adiabatic approximation error,which is the distance between the exact solution to a Schr dinger equation and the adiabatic approximation solution.As an application,we obtain an upper bound for 1 minus the fidelity of the exact solution and the adiabatic approximation solution to a Schrdinger equation.     

Current–phase relations of a ring-trapped Bose–Einstein condensate with a weak link

The current–phase relations of a ring-trapped Bose–Einstein condensate interrupted by a rotating rectangular barrier are extensively investigated with an analytical solution. A current–phase diagram, single and multi-valued relation, is presented with a rescaled barrier height and width. Our results show that the finite size makes the current–phase relation deviate a little bit from the cosine form for the soliton solution in the limit of a vanishing barrier, and the periodic boundary condition selects only the plane wave solution in the case of high barrier. The reason for multi-valued current–phase relation is given by investigating the behavior of soliton solution.     

Topological structure of the solitons solution in SU（3） Dunne-Jackiw-Pi-Trugenberger model

By using C-mapping topological current theory and gauge potential decomposition, we discuss the self-dual equation and its solution in the SU（N） Dunne-Jackiw-Pi-Trugenberger model and obtain a new concrete self-dual equation with a δ function. For the SU（3） case, we obtain a new self-duality solution and find the relationship between the soliton solution and topological number which is determined by the Hopf index and Brouwer degree of C-mapping. In our solution, the flux of this soliton is naturally quantized.     

Noether＇s and Poisson＇s methods for solving differential equation xs^（m） = Fs（t,xk^（m-2） , xk（m-1）

This paper studies integration of a higher-order differential equation which can be reduced to a second-order ordinary differential equation. The solution of the second-order equation can be obtained by the Noether method and the Poisson method. Then the solution of the higher-order equation can be obtained by integrating the solution of the second-order equation.     

Lump,lumpoff and predictable rogue wave solutions to a dimensionally reduced Hirota bilinear equation

We study a simplified(3+1)-dimensional model equation and construct a lump solution for the special case of z=y using the Hirota bilinear method.Then,a more general form of lump solution is constructed,which contains more arbitrary autocephalous parameters.In addition,a lumpoff solution is also derived based on the general lump solutions and a stripe soliton.Furthermore,we figure out instanton/rogue wave solutions via introducing two stripe solitons.Finally,one can better illustrate these propagation phenomena of these solutions by analyzing images.     

Topological structure of the solitons solution in SU(3) Dunne-Jackiw-Pi-Trugenberger model

By using φ-mapping topological current theory and gauge potential decomposition, we discuss the self-dual equation and its solution in the SU(N) Dunne-Jackiw-Pi-Trugenberger model and obtain a new concrete self-dual equation with a δ function. For the SU(3) case, we obtain a new self-duality solution and find the relationship between the soliton solution and topological number which is determined by the Hopf index and Brouwer degree of φ-mapping. In our solution, the flux of this soliton is naturally quantized.     

Grammian Determinant Solution and Pfaffianization for a （3＋1）-Dimensional Soliton Equation

Based on the Pfaffian derivative formulae, a Grammian determinant solution for a （3＋1）-dimensional soliton equation is obtained. Moreover, the Pfaffianization procedure is applied for the equation to generate a new coupled system. At last, a Gram-type Pfaffian solution to the new coupled system is given.     

Novel loop-like solitons for the generalized Vakhnenko equation

A non-traveling wave solution of a generalized Vakhnenko equation arising from the high-frequent wave motion in a relaxing medium is derived via the extended Riccati mapping method.The solution includes an arbitrary function of an independent variable.Based on the solution,two hyperbolic functions are chosen to construct new solitons.Novel single-loop-like and double-loop-like solitons are found for the equation.     

N-Soliton Solution of a Generalized Hirota-Satsuma Coupled KdV Equation and Its Reduction

Based on the Hirota method and the perturbation technique, the N-soliton solution of a generalized Hirota-Satsuma coupled KdV equation is obtained. Further, the N-soliton solution of a complex coupled KdV equation is given by reducing.     

An asymptotic solving method for the periodic solution of a class of disturbed nonlinear evolution equation

<正>A class of disturbed evolution equation is considered using a simple and valid technique.We first introduce the periodic traveling-wave solution of a corresponding typical evolution equation.Then the approximate solution for an original disturbed evolution equation is obtained using the asymptotic method.We point out that the series of approximate solution is convergent and the accuracy of the asymptotic solution is studied using the fixed point theorem for the functional analysis.     

The homotopic mapping solution for the solitary wave for a generalized nonlinear evolution equation

This paper studies a generalized nonlinear evolution equation. Using the homotopic mapping method, it constructs a corresponding homotopic mapping transform. Selecting a suitable initial approximation and using homotopic mapping, it obtains an approximate solution with an arbitrary degree of accuracy for the solitary wave. From the approximate solution obtained by using the homotopic mapping method, it possesses a good accuracy.     

Exact Solution to the One-Dimensional Dirac Equation with Time Varying Mass

We directly use the quantum-invariant operator method to obtain the closed-form solution to the one-dimensional Dirac equation with a time-changing mass with a little manipulation. The solution got is also applicable for the case with time-independence mass.     

Wronskian and Grammian Determinant Solutions for a Variable-Coefficient Kadomtsev-Petviashvili Equation

In this paper, we derive the bilinear form for a variable-coefficient Kadomtsev Petviashvili-typed equation. Based on the bilinear form, we obtain the Wronskian determinant solution, which is proved to be indeed an exact solution of this equation through the Wronskian technique. In addition, we testify that this equation can be reduced to a Jacobi identity by considering its solution as a Grammian determinant by means of Pfaffian derivative formulae.     

Theory of sound field in a room—re-examination

The theory of steady-state sound field in a room is re-examined. It is shown that the normal-mode solution of the wave equation is not the exact solution, and the derivation is incorrect... The exact solution of the wave equation in a reflective room should contain both the free space solution (direct sound field) and the standing wave solution (reverberant sound field), the latter is formed by all the reflected waves to a group of allowed wave types (the normal modes of vibration ).