Combined Soliton Research Articles

Combined periodic wave and solitary wave solutions in two-component Bose-Einstein condensates

In this paper, the Jacobi elliptic function expansion method provides an effective approach to obtain the exact periodic wave solutions of two-component Bose-Einstein condensates. Exact combined bright-bright and dark-dark soliton wave solutions can be achieved in their limit conditions. We also obtain the different formation regions of combined solitons. Our results show that the intraspecies (interspecies) interaction strengths clearly affect the formation of dark-dark, bright-bright and dark-bright soliton solutions in different regions.

Formation of combined solitons in two-component Bose–Einstein condensates

We investigate the combined soliton solutions of two-component Bose–Einstein condensates with external potential. The “phase diagram" is obtained for the formation regions of different combined solitons. Our results show that the intraspecies (interspecies) interaction strengths and the external trapped potential clearly affect the formation of dark-dark, bright-bright, and dark–bright soliton solutions in different regions. Especially, we find that the bright–bright (dark–dark) soliton can exist in the case of both repulsive (attractive) intraspecies interaction strengths in the presence of external potential. This novel phenomenon is completely different from the formation of soliton solution of one-component Bose–Einstein condensates without external potential, and it will be useful for the study of two-component Bose–Einstein condensates.

New soliton and periodic solutions of (1 + 2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity

With the aid of symbolic computation, the new generalized algebraic method is extended to the (1+2)-dimensional nonlinear Schrödinger equation (NLSE) with dual-power law nonlinearity for constructing a series of new exact solutions. Because of the dual-power law nonlinearity, the equation cannot be directly dealt with by the method and require some kinds of techniques. By means of two proper transformations, we reduce the NLSE to an ordinary differential equation that is easy to solve and find a rich variety of new exact solutions for the equation, which include soliton solutions, combined soliton solutions, triangular periodic solutions and rational function solutions. Numerical simulations are given for a solitary wave solution to illustrate the time evolution of the solitary creation. Finally, conditional stability of the solution in Lyapunov’s sense is discussed.

Travelling wave solutions for the generalized Zakharov–Kuznetsov equation with higher-order nonlinear terms

With the aid of symbolic computation and two auxiliary ordinary differential equations, the new generalized algebraic method is extended to the generalized Zakharov–Kuznetsov (GZK) equation with higher-order nonlinear terms for constructing a series of new and more general travelling wave solutions. Because of the higher-order nonlinear terms, the equation can not be directly dealt with by the method and require some kinds of techniques. By means of two proper transformations, we transform the GZK equation to an ordinary differential equation that is easy to solve and find a rich variety of new exact travelling wave solutions for the GZK equation, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, combined Jacobi elliptic function solutions and rational function solutions. The method used here can be also extended to other nonlinear partial differential equations with higher-order nonlinear terms.

Construction of non-travelling wave solutions for the generalized variable-coefficient Gardner equation

With the aid of two first-order nonlinear ordinary differential equations, the new generalized algebraic method is used to study the generalized variable-coefficient Gardner equation. The main idea of the method is to take full advantage of solutions to the first-order ordinary differential equations. More new exact non-travelling solutions, which include soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions and combined Jacobi elliptic function solutions, for the Gardner equation are obtained. The method used here can be also extended to many other nonlinear partial differential equations.

Combined solitons in generalized coupled mode equations of a nonlinear optical Bragg grating

We discuss the existence of combined dark and antidark soliton forms or combined solitons in the generalized coupled mode equations of a nonlinear optical Bragg grating. These solitons are not allowed in the conventional coupled mode equations with uniform nonlinearity and exist outside the linear grating band gap. Their related Hamiltonian phase portrait was briefly reported by de Sterke et al. [Phys. Rev. E 54, 1969 (1996)]. The explicit expressions for the corresponding solitons are presented, as well as their bifurcation process. We demonstrate the unstable propagation of perturbed combined solitons with zero velocity by means of direct numerical integration.

Soliton excitations in pernigraniline-base polymer: Effects of next-nearest-neighbor hopping

Next-nearest-neighbor hopping interaction is considered to investigate the soliton excitations in a pernigraniline-base polymer on the basis of the Baranowski–Büttner–Voit model. Numerical calculations show that the valence band (VB) expands to a larger range and the conduction band (CB) acts to the contrary. For a positive combined soliton, some new shallow levels emerge at the top edge of both the VB and CB with the increase in the next-nearest-neighbor coupling strength. For a negative combined soliton, only one bound state (the soliton state ) appears in the Peierls gap and no other shallow levels exist. We find that, while the coupling strength β takes the value 0.04, the calculated charged soliton excitations (1.15 eV for S + and 1.34 eV for S − ) are in good agreement with the low-energy photo-induced absorption peak (1.0 and 1.3 eV).

Construction of new solutions to the fully nonlinear generalized Camassa-Holm equations by an indirect F function method

An indirect F function method is introduced to solve the generalized Camassa-Holm equation with fully nonlinear dispersion and fully nonlinear convection C(l,n,p). Taking advantage of elliptic equation, this F function is used to map the solutions of the generalized Camassa-Holm equation to those of the elliptic equation. As a result, we can successfully obtain in a unified way and for special values of the parameters of this equation, many exact solutions expressed by various single and combined nondegenerative Jacobi elliptic function solutions and their degenerative solutions (soliton, combined soliton solutions, and triangular solutions) as the modulus m is driven to 1 and 0.

Gap states of charged combined solitons in a pernigraniline-base polymer

Considering the electron–electron interactions, we have investigated the electronic structures of charged combined solitons in a pernigraniline-base polymer on the basis of Baranowski–Büttner–Voit model. The numerical results show that there exist two bound states in the Peierls gap for both positive and negative combined solitons, one is deep level and the other is shallow level. The energy of the deep level (for S + is about 1.03 eV and S − 0.98 eV) agree with the photoinduced absorption data (∼1.0 eV) very well. The shallow one is newly discovered and it is about 0.22 eV under the bottom edge of the conduction band (positive soliton) or above the top edge of the valence band (negative soliton). Some other shallow states outside the energy bands are also predicted.

Exact solutions for the coupled Klein–Gordon–Schrödinger equations using the extended F-expansion method

The extended F-expansion method or mapping method is used to construct exact solutions for the coupled Klein-Gordon Schrodinger equations (K-G-S equations) by the aid of the symbolic computation system Mathematica. More solutions in the Jacobi elliptic function form are obtained, including the single Jacobi elliptic function solutions, combined Jacobi elliptic function solutions, rational solutions, triangular solutions, soliton solutions and combined soliton solutions.

On exact solutions of the coupled Klein–Gordon–Schrödinger and the complex coupled KdV equations using mapping method

The mapping method is used by the aid of symbolic computation system Mathematica for constructing exact solutions to the coupled Klein–Gordon–Schrödinger equations and to the complex coupled KdV equations. The solutions obtained in this paper include Jacobi elliptic solutions, combined Jacobi elliptic solutions, rational solutions, triangular solutions, soliton solutions and combined soliton solutions.