We discuss the exact number of positive solutions of u+f(u) = 0 with homogeneous Dirichlet boundary condition and on the ball domain. The nonlinearity here includes f(u) = u q + u p for 0 < q < 1 < p n n?2 .

in this paper, exp-function and (g′/g)expansion methods are presented to derive traveling wave solutions for a class of nonlinear space-time fractional differential equations. as a results, some new exact traveling wave solutions are obtained.

The extended homogeneous balance method is used to construct exact traveling wave solutions of the Maccari system, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation. Many exact traveling wave solutions of the Maccari system equation are successfully obtained.

In this paper, a new fractional sub-equation method is proposed for finding exact solutions of fractional partial differential equations (FPDEs) in the sense of modified Riemann-Liouville derivative. With the aid of symbolic computation, we choose the space-time fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation in mathematical physics with a source to illustrate the validity a...

The coherent propagation of four optical pulses through a multilevel resonant medium is investigated theoretically. We present a self-consistent analytic solution without steady-state or adiabatic approximations and use numerical simulations to indicate that the analytic formulas can be used as a guide in an experimental setting.

In this note, we solve the Loewner equation in the upper halfplane with forcing function ξ(t), for the cases in which ξ(t) has a power-law dependence on time with powers 0, 1/2 and 1. In the first case the trace of singularities is a line perpendicular to the real axis. In the second case the trace of singularities can do three things. If ξ(t) = 2 √ κt, the trace is a straight line set at an an...

In this present work, the Kudryashov method and the functional variable method are used to construct exact solutions of the complex KdV equation. The Kudryashov method and the functional variable method are powerful methods for obtaining exact solutions of nonlinear evolution equations.

Let Fn q be a vector space of dimension n over the finite field Fq . A q-analog of a Steiner system (also known as a q-Steiner system), denoted Sq(t,k,n), is a set S of k-dimensional subspaces of Fn q such that each t-dimensional subspace of Fn q is contained in exactly one element of S . Presently, q-Steiner systems are known only for t = 1, and in the trivial cases t = k and k= n. In this pap...

We present a heuristic matching algorithm for the generation of ground states of the short-range * J spin glass in two dimensions. It is much faster than previous heuristic algorithms. I t achieves near optimal solutions in time O( N ) in contrast to the best known exact algorithm which needs a time of O ( N S ” ) . From simulations with lattice sizes of up to 210 x 210 we confirm a phase trans...

 Based on some stationary periodic solutions and stationary soliton solutions, one studies the general solution for the relative lax system, and a number of exact solutions to the Korteweg-de Vries (KdV) equation are first constructed by the known Darboux transformation, these solutions include double and triple singular periodic solutions as well as singular soliton solutions whose amplitude d...