Modulation theory for the steady forced KdV–Burgers equation and the construction of periodic solutions

We present a multiple-scale perturbation technique for deriving asymptotic solutions to the steady Korteweg–de Vries (KdV) equation, perturbed by external sinusoidal forcing and Burger’s damping term, which models the near resonant forcing of shallow water in a container. The first order solution in the perturbation hierarchy is the modulated cnoidal wave equation. Using the second order equation in the hierarchy, a system of differential equations is found describing the slowly varying properties of the cnoidal wave. We analyse the fixed point solutions of this system, which correspond to periodic solutions to the perturbed KdV equation. These solutions are then compared to the experimental results of Chester and Bones (1968).     

Several new types of bounded wave solutions for the generalized two-component Camassa–Holm equation

Li and Qiao studied the bifurcations and exact traveling wave solutions for the generalized two-component Camassa–Holm equation $$\begin{aligned} \left\{ \begin{array}{l} m_{t}+\sigma um_{x}-Au_{x}+2m \sigma u_{x}+3(1-\sigma )uu_{x}\\ \quad +\rho \rho _{x}=0, \\ \rho _{t} +(\rho u)_{x}=0, \end{array} \right. \end{aligned}$$ \(m=u-u_{xx}, A>0\) . They showed that there exist solitary wave solutions, cusp wave solutions, and periodic wave solutions for the equation, and their analysis focused on the bifurcations when \(\sigma >0\) . In this paper, we first complement the bifurcations when \(\sigma <0\) by following the same procedure as that of Li, and then show the existence and implicit expressions of several new types of bounded wave solutions, including solitary waves, periodic waves, compacton-like waves, and kink-like waves. In addition, the numerical simulations of the bounded wave solutions are given to show the correctness of our results.     

Stability of the stationary solutions of the Allen–Cahn equation with non-constant stiffness

We study the solutions of a generalized Allen–Cahn equation deduced from a Landau energy functional, endowed with a non-constant higher order stiffness. We assume the stiffness to be a positive function of the field and we discuss the stability of the stationary solutions proving both linear and local non-linear stability.     

Smooth positon solutions of the focusing modified Korteweg–de Vries equation

The n-fold Darboux transformation \(T_{n}\) of the focusing real modified Korteweg–de Vries (mKdV) equation is expressed in terms of the determinant representation. Using this representation, the n-soliton solutions of the mKdV equation are also expressed by determinants whose elements consist of the eigenvalues \(\lambda _{j}\) and the corresponding eigenfunctions of the associated Lax equation. The nonsingular n-positon solutions of the focusing mKdV equation are obtained in the special limit \(\lambda _{j}\rightarrow \lambda _{1}\), from the corresponding n-soliton solutions and by using the associated higher-order Taylor expansion. Furthermore, the decomposition method of the n-positon solution into n single-soliton solutions, the trajectories, and the corresponding “phase shifts” of the multi-positons are also investigated.     

Predicting solutions of the Lotka‐Volterra equation using hybrid deep network

Prediction of Lotka-Volterra equations has always been a complex problem due to their dynamic properties. In this paper, we present an algorithm for predicting the Lotka-Volterra equation and investigate the prediction for both the original system and the system driven by noise. This demonstrates that deep learning can be applied in dynamics of population. This is the first study that uses deep learning algorithms to predict Lotka-Volterra equations. Several numerical examples are presented to illustrate the performances of the proposed algorithm, including Predator nonlinear breeding and prey competition systems, one prey and two predator competition systems, and their respective systems. All the results suggest that the proposed algorithm is feasible and effective for predicting Lotka-Volterra equations. Furthermore, the influence of the optimizer on the algorithm is discussed in detail. These results indicate that the performance of the machine learning technique can be improved by constructing the neural networks appropriately.     

Analytic solutions for the generalized complex Ginzburg–Landau equation in fiber lasers

Nonlinear Dynamics - This paper presents a study on the energy exchange taking place on articulated helicopter main rotor blades. The blades are hinged, and the flap/lag modes are highly coupled....     

Weighted L∞ bounds and uniqueness for the Boltzmann BGK model

We prove rather general L bounds for hydrodynamical fields in terms of weighted L norms of the kinetic density. We use these estimates to prove L bounds and uniqueness for the solution of the BGK Equation, which is a simple relaxation model introduced by Bhatnagar, Gross & Krook to mimic Boltzmann flows.     

New no-traveling wave solutions for the Liouville equation by Bäcklund transformation method

With the aid of the known Bäcklund transformation, starting from some given traveling solutions, we consider new exact no-traveling wave solutions to the Liouville equation, and a series of breather soliton solutions, doubly periodic solutions, two-soliton solutions as well as periodic-soliton solutions are obtained.     

Stability of exact solutions of the cubic-quintic nonlinear Schrödinger equation with periodic potential

The nonlinear Schr?dinger equation with attractive quintic nonlinearity in periodic potential in 1D, modeling a dilute-gas Bose–Einstein condensate in a lattice potential, is considered and one family of exact stationary solutions is discussed. Some of these solutions have an analog neither in the linear Schr?dinger equation nor in the integrable nonlinear Schr?dinger equation. Their stability is examined analytically and numerically.     

A construction of new exact periodic wave and solitary wave solutions for the 2D Ginzburg–Landau equation

Using a uniform algebraic method, new exact solitary wave solutions and periodic wave solutions for 2D Ginzburg–Landau equation are obtained. Moreover, three-dimensional and two-dimensional graphics of some solutions have been plotted.     

N-Fold generalized Darboux transformation and semirational solutions for the Gerdjikov-Ivanov equation for the Alfvén waves in a plasma

Nonlinear Dynamics - Alfvén waves propagating parallel to the ambient magnetic field are modeled via the Gerdjikov-Ivanov equation. With respect to the transverse magnetic field perturbation...     

Nonlinear wave solutions and their relations for the modified Benjamin–Bona–Mahony equation

Nonlinear Dynamics - In this paper, we use the bifurcation method of dynamical systems to investigate the nonlinear wave solutions of the modified Benjamin–Bona–Mahony equation. These...     

Solitary wave solutions to the Kuramoto–Sivashinsky equation by means of the homotopy analysis method

The homotopy analysis method (HAM) is used to find a family of solitary solutions of the Kuramoto–Sivashinsky equation. This approximate solution, which is obtained as a series of exponentials, has a reasonable residual error. The homotopy analysis method contains the auxiliary parameter ℏ, which provides us with a simple way to adjust and control the convergence region of series solution. This method is reliable and manageable.