An investigation of the physical dynamics of a traveling wave solution called a bright soliton

This study examines the 1 + 2 -dimensional Zoomeron equation, which has recently become popular in applied mathematics and physics. Bright soliton (non-topological), kink wave solution and traveling wave solutions are generated with the advantages of the generalized exponential rational function method. With the help of this method, it is aimed to produce different types of solutions for the Zoomeron equation compared to other traditional exponential function methods. The effects of parameters on the amplitude of the wave function are discussed, along with physical explanations backed by simulations. In addition, the advantages and disadvantages of the method for the 1 + 2 -dimensional Zoomeron equation are discussed.

Post-bifurcation behaviour of elasto-capillary necking and bulging in soft tubes

Previous linear bifurcation analyses have evidenced that an axially stretched soft cylindrical tube may develop an infinite-wavelength (localized) instability when one or both of its lateral surfaces are under sufficient surface tension. Phase transition interpretations have also highlighted that the tube admits a final evolved ‘two-phase’ state. How the localized instability initiates and evolves into the final ‘two-phase’ state is still a matter of contention, and this is the focus of the current study. Through a weakly nonlinear analysis conducted for a general material model, the initial sub-critical bifurcation solution is found to be localized bulging or necking depending on whether the axial stretch is greater or less than a certain threshold value. At this threshold value, an exceptionally super-critical kink-wave solution arises in place of localization. A thorough interpretation of the anticipated post-bifurcation behaviour based on our theoretical results is also given, and this is supported by finite-element method simulations.

Lump-Type Wave and Interaction Solutions of the Bogoyavlenskii–Kadomtsev–Petviashvili Equation

Lump-type wave solution of the Bogoyavlenskii–Kadomtsev–Petviashvili equation is constructed by using the bilinear structure and Hermitian quadratic form. The dynamical behaviors of lump-type wave solution are investigated and presented analytically and graphically. Furthermore, we discuss the interaction between a lump-type wave and a kink wave solution. Absorb-emit interaction between two kinds of solitary wave solutions is shown. This kind of interaction solution can be regarded as a lump-type wave which propagates on the kink wave background.

Exact Traveling Wave Solutions and Bifurcations of the Time-Fractional Differential Equations with Applications

This paper presents a method to investigate exact traveling wave solutions and bifurcations of the nonlinear time-fractional partial differential equations with the conformable fractional derivative proposed by [Khalil et al., 2014]. The method is based on employing the bifurcation theory of planar dynamical systems proposed by [Li, 2013]. For the fractional PDEs, up till now, there is no related paper to obtain the exact solutions by applying bifurcation theory. We show how to use this method with applications to two fractional PDEs: the fractional Klein–Gordon equation and the fractional generalized Hirota–Satsuma coupled KdV system, respectively. We find the new exact solutions including periodic wave solution, kink wave solution, anti-kink wave solution and solitary wave solution (bright and dark), which are different from previous works in the literature. This approach can also be extended to other nonlinear time-fractional differential equations with the conformable fractional derivative.

Analytic homoclinic wave and soliton solutions for 2D coupled complex Ginzburg–Landau equations

Analytic wave solutions including homoclinic wave, kink wave and soliton solutions for the 2D coupled complex Ginzburg–Landau equations are obtained using the auxiliary function method, Hirota method and the ansatz function technique under certain constraint conditions of coefficients in equations, respectively. The result shows that there exists a kink-wave solution which tends to one and the same periodic wave solution as time tends to infinite.

A new two-component integrable system with peakon solutions

A new two-component system with cubic nonlinearity and linear dispersion: m t = b u x + 1 2 [ m ( u v − u x v x ) ] x − 1 2 m ( u v x − u x v ) , n t = b v x + 1 2 [ n ( u v − u x v x ) ] x + 1 2 n ( u v x − u x v ) , m = u − u x x , n = v − v x x , where b is an arbitrary real constant, is proposed in this paper. This system is shown integrable with its Lax pair, bi-Hamiltonian structure and infinitely many conservation laws. Geometrically, this system describes a non-trivial one-parameter family of pseudo-spherical surfaces. In the case b =0, the peaked soliton (peakon) and multi-peakon solutions to this two-component system are derived. In particular, the two-peakon dynamical system is explicitly solved and their interactions are investigated in details. Moreover, a new integrable cubic nonlinear equation with linear dispersion m t = b u x + 1 2 [ m ( | u | 2 − | u x | 2 ) ] x − 1 2 m ( u u x ∗ − u x u ∗ ) , m = u − u x x , is obtained by imposing the complex conjugate reduction v = u * to the two-component system. The complex-valued N -peakon solution and kink wave solution to this complex equation are also derived.