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In this paper, the semi-rational solutions that causes vector rogue waves and breathers can be obtained by using the Darboux dressing transformation. We studied vector rogue waves and the interaction between rogue waves and light-dark solitons, and observed that during the interaction, due to the interference between the light-dark components of the solitons, a respiration-like structures appears. Besides, it can be observed that the rogue waves and soliton merge together. Moreover, the main characteristics of the interactions between the breathers and bright-dark solitons are displayed with some graphics.

In this paper, we study the time-fractional nonlinear Korteweg-de Vries (KdV) equation. By using the theory of semigroups, we prove the well-posedness of the time-fractional nonlinear KdV equation. Moreover, we present the boundary controllability result for the problem.

This paper is concerned with a negative order modified Korteweg-de Vries (nmKdV) equation which is in the negative flow of the mKdV hierarchy. We construct the breather solutions by Hirota's bilinear method and derive the infinite conservation laws through the Lax pair of the nmKdV equation. By constructing a new Lyapunov function with the conservation laws, we obtain the nonlinear stability of the breather solutions.

In this paper, the semi-rational solutions that causes vector rogue waves and breathers can be obtained by using the Darboux dressing transformation. We studied vector rogue waves and the interaction between rogue waves and light-dark solitons, and observed that during the interaction, due to the interference between the light-dark components of the solitons, a respiration-like structures appears. Besides, it can be observed that the rogue waves and soliton merge together. Moreover, the main characteristics of the interactions between the breathers and bright-dark solitons are displayed with some graphics.

This article concerns an approximate solution for the Benjamin-Bona-Mahony (BBM) equation with a bilinear control in slowly varying medium. By a sharp estimation of the error term, a suitable approximate solution for this equation is established.

In this paper, we study the stability of modified Korteweg–de Vries equation breather. By using variable separation method, we obtain the exact breather-type soliton solutions. Moreover, this kind of solutions are globally stable in H2 topology, and we describe a simple, mathematical proof of the orbital stability and asymptotic stability of breather-type soliton solutions under a class of small perturbation.

In this paper, we discuss two main problems. In the first section, we establish a new global distributed exact controllability of the periodic two-component $\mu\rho$-Hunter-Saxton system on the circle by means of a distributed control. And in the second section, we present corresponding result of the asymptotic stabilization problem about the periodic two-component $\mu\rho$-Hunter-Saxton system. By presenting concrete form of the feedback law, an equivalent system is got.

In this paper, we present an integrable semi-discretization of the Boussinesq equation. Different from other discrete analogues, we discretize the ‘time’ variable and get an integrable differential-difference system. Under a standard limitation, the differential-difference system converges to the continuous Boussinesq equation such that the discrete system can be used to design numerical algorithms. Using Hirota's bilinear method, we find a Backlund transformation and a Lax pair of the differential-difference system. For the case of ‘good’ Boussinesq equation, we investigate the soliton solutions of its discrete analogue and design numerical algorithms. We find an effective way to reduce the phase shift caused by the discretization. The numerical results coincide with our analysis.

This paper is concerned with the coupled modified Korteweg-de Vries (cmKdV) equations. We derive infinite conservation laws through the Lax pair of the cmKdV equations. Through the analysis of the spectral stability with the conservation laws, we obtain the nonlinear stability of breather solutions to the cmKdV equations.

All possible exotic and smooth solitary wave solutions to the two-component Dullin–Gottwald–Holm equation are investigated. We classify this equation in specified regions of the parametric space. Moreover, we give the limiting relations of all different solitary waves as the parameters trend to some special values. All solitary waves suffer from external perturbations, and these solutions turn to the chaotic state easily. In view of the variation of the control coefficient, the smooth solitary wave is the easiest one to be controlled into a stable state and the cusped solitary wave is the most difficult to be controlled under the same controller condition.

In this paper we prove that the Camassa–Holm equation with quartic nonlinearity is non-integrable via the Painleve method. The orbital stability of solitary waves for this equation is investigated by constructing a functional extremum problem. This result demonstrates that the resulting solitary wave is unstable when its speed lies in the narrow region of the critical value that connects with the bifurcation condition. In contrast when the speed surpasses the narrow region, the solitary wave is stable. In addition, the stable solitary wave turns into a chaotic state when is driven externally. If a damping term controller is added to the perturbed equation, the solitary wave can also propagate stably under a certain condition. Finally our numerical results show that the perturbed equation is not well controlled when a certain resonant-frequency occurs and is well controlled with a smaller wave speed as well as a higher nonlinear convection.

In this paper, we found a new symptom (called energy rogue) of a four-dimensional energy resource demand–supply system. Energy rogue waves denote drastic changes at a certain moment. We found that the energy rogue wave is a symptom before chaos appearing. Furthermore, the energy rogue wave turns to chaos by changing some coefficients. Based on the bifurcation theory and the phase portrait analysis, we explain the occurrence mechanism of energy rogue waves which connects with one fold bifurcation and one Hopf bifurcation. Based on Lyapunov stability theory, an adaptive control law is derived to eliminate energy rogue waves. Numerical simulations are presented to show these results.

In this Letter, we obtain a two-peakon solution to a special Dullin–Gottwald–Holm equation explicitly by direct computation, and then discuss the peakon–antipeakon interaction in this equation. Our results show that, like the Camassa–Holm equation, during the soliton absorption time, the peakon–antipeakon in this equation is essentially a single-peaked, double-cornered wave. However, the two corners may travel in opposite direction for some ω , which is different from that in the Camassa–Holm equation.

In this paper we study a generalized weakly dissipative two-component Camassa–Holm system. We show that this system can exhibit the wave-breaking phenomenon. We also determine the exact blow-up rate of such solutions.

This paper is concerned with the attractor for a viscous two-component generalization of the Camassa–Holm equation subject to an external force, where the viscosity term is given by a second order differential operator. The global existence of solution to the viscous two-component Camassa–Holm equation with the periodic boundary condition is studied. We obtain the compact and bounded absorbing set and the existence of the global attractor in H 2 × H 2 for the viscous two-component Camassa–Holm equation by uniform prior estimate and many inequalities.

For the four generalized Benjamin–Bona–Mahony equations that occurred in [4] , [5] , we apply the theory of Weierstrass elliptic function to obtain all the solutions in [4] , [5] . The traveling wave solutions are expressed by the hyperbolic functions and trigonometric functions. The method is direct and concise, and free of the symbolic calculation. Moreover, it is more effective than the method in [5] and is suitable to other equations with any order.

A two-component Fornberg–Whitham equation is introduced as a model for water waves. The bifurcations of traveling wave solutions are studied. Parametric conditions to smooth soliton solution, kink solution, antikink solution and uncountable infinite many smooth periodic wave solutions are given. Some expressions for those solutions are presented.

This paper studies low-regularity solutions of the periodic general Degasperis–Procesi equation with an initial value. The existence and the uniqueness of solutions are proved. The results are illustrated by considering the periodic peakons of the periodic general Degasperis–Procesi equation.

In this paper, we investigate the global existence of the higher-order Camassa–Holm equation in the case of k = 2 . We prove the local well-posedness of this equation and find a conservation law. Then a global existence result is obtained.

Traveling wave solutions for a generalized sinh-cosh-Gordon equation are studied. The equation is transformed into an auxiliary partial differential equation without any hyperbolic functions. By using the theory of planar dynamical system, the existence of different kinds of traveling wave solutions of the auxiliary equation is obtained, including smooth solitary wave, periodic wave, kink and antikink wave solutions. Some explicit expressions of the blow-up solution, kink-like solution, antikink-like solution and periodic wave solution to the generalized sinh-cosh-Gordon equation are given. Planar portraits of the solutions are shown.

Considered herein is the orbital stability of floating periodic peakons for the Camassa–Holm (CH) equation, which describes one-dimensional surface waves at a free surface of shallow water under the influence of gravity. The floating periodic peakons shift up or down according to the change of the parameter. The result shows that the floating periodic peakons are orbitally stable and their stability is independent of the parameter in the CH equation.

In this paper, we employ the bifurcation theory of planar dynamical systems to investigate the traveling wave solutions of a 2-component of the Degasperis-Procesi equation. The expressions for smooth soliton, kink and antikink solutions are obtained., Comment: 16 pages, 18 figures

In this paper, we employ the bifurcation theory of planar dynamical systems to investigate the travelling-wave solutions to a dual equation of the Kaup–Boussinesq system. The expressions for smooth solitary-wave solutions are obtained.

In this paper, we classify the travelling wave solutions to the nonlinear dispersive KdV equation (called K(2, 2) equation). The parameter region is specified and the parameter dependence of its solitary waves is described. Besides the previously known compacton solutions, the equation is shown to admit more new solutions such as cuspons, peakons, loopons, stumpons and fractal-like waves. Furthermore, by the qualitative results, we give some new explicit travelling wave solutions.

Considered herein is the linear stability of compacton solutions to the nonlinear dispersive Ostrovsky equation, which is also a modification of the K ( m , n ) equation. We show that the equation does not pass the Painleve test for integrability and has compacton solutions which are different from conventional forms. Furthermore, compactons are proved to be weak solutions. A new equation similar to the nonlinear dispersive Ostrovsky equation is derived from Lagrangian, and some important conservation laws as well as the Hamiltonian structure are given. Finally, the stability of the compacton solutions to the similar equations is studied via using linear stability analysis. The result shows that the linear stability analysis it follows that, unlike compactons to the K ( m , n ) equation, the weak-compacton solutions are linear stable under certain conditions.

This paper investigates the problem of boundary control of the viscous Fornberg–Whitham equation. The existence and uniqueness of weak solution to the viscous Fornberg–Whitham equation are proved in a short interval. Moreover, we prove the Fornberg–Whitham equation global exponential stability in L 2 , H 1 , H 2 , H 3 under the given boundary condition.

We apply the theory of Weierstrass elliptic function to study exact solutions of the generalized Benjamin-Bona-Mahony equation. By using the theory of Weierstrass elliptic integration, we get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. This method is effective to find exact solutions of many other similar equations which have arbitrary-order nonlinearity.

This paper aims to present a proof of the existence of the attractor for the one-dimensional viscous Fornberg–Whitham equation. In this paper, the global existence of solution to the viscous Fornberg–Whitham equation in L 2 under the periodic boundary conditions is studied. By using the time estimate of the Fornberg–Whitham equation, we get the compact and bounded absorbing set and the existence of the global attractor for the viscous Fornberg–Whitham equation.

This paper investigates the boundary control problem of the viscosity Degasperis–Procesi equation in [0, 1]. Results of the existence of a solution to the viscosity Degasperis–Procesi equation in a short time interval under the boundary condition, are given in the paper. It proves the global exponential stability under the given boundary condition in L 2 , H 1 , H 2 , H 3 space.

This paper studies the problem of optimal control of the viscous Camassa–Holm equation. The existence and uniqueness of weak solution to the viscous Camassa–Holm equation are proved in a short interval. According to variational method, optimal control theories and distributed parameter system control theories, we can deduce that the norm of solution is related to the control item and initial value in the special Hilbert space. The optimal control of the viscous Camassa–Holm equation under boundary condition is given and the existence of optimal solution to the viscous Camassa–Holm equation is proved.

In this paper, we first introduced improved projective Riccati method by means of two simplified Riccati equations. Applying the improved method, we consider the general types of KdV and KdV–Burgers equations with nonlinear terms of any order. As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions are obtained. Some of them are found for the first time.

In this paper the dynamical behaviors of a dispersive shallow water equation with viscosity, viscosity Degasperis–Procesi equation, are investigated. The existence of global solution to viscosity Degasperis–Procesi equation in L 2 under the periodical boundary condition is studied and the existence of the global attractor of semi-group to solution on viscosity Degasperis–Procesi equation in H 2 is obtained.

This paper aims at presenting a proof of the existence of the attractor for the two-dimensional weakly damped KdV equations in belt field. By using the Bourgain spaces and the orthogonal projection, we prove the existence of the attractor for the reduced equation which is obtained from the two-dimensional weakly damped KdV equation in belt field. By using the time estimate of the blow-up of the KdV equation, we get the compact and bounded aborting set and prove the existence of the attractor for the two-dimensional weakly damped KdV equation in belt field.

In this paper, the generalized Ostrovsky equation is introduced. Using a direct and effective method, some new solitary solutions to the generalized Ostrovsky equation, such as compacton solutions, multi-compacton solutions and compact-like kink solutions can be obtained. The homogenous balance (HB) method is used to obtain the Backlund transformation. And some new solitary solutions, particularly new double symmetric peakon solutions, are given by the transformation.

We establish the local well-posedness for the peakon b-family of equations which includes both the Camassa-Holm equation and Degasperis-Procesi equation as special cases. We show that the blow-up phenomena occur in finite time for certain initial profiles. Moreover, we obtain a global existence result.

By variational iteration method, we obtain new solitary solutions for non-linear dispersive equations. Particularly, shock-peakon solutions in K(2,2) equation and shock-compacton solutions in K(3,3) equation are found by this simple method. These two types of solutions are new solitary wave solutions which have the shapes of shock solutions and compacton solutions (or peakon solutions).

In this paper, we study the generalized coupled Hirota–Satsuma KdV system by using the two new improved projective Riccati equations method. As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and combined formal solitary wave solutions and combined formal periodic wave solutions are obtained.

Two generalizations of the Camassa–Holm system associated with the singular analysis are proposed for Painleve integrability properties and the extensions of already known analytic solitons. A remarkable feature of the physical model is that it has peakon solution which has peak form. An alternative WTC test which allowed the identifying of such models directly if formulated in terms of inserting a formed ansatz into these models. For the two models have Painleve property, Painleve–Backlund systems can be constructed through the expansion of solitons about the singularity manifold. By the implementations of Maple, plentiful new type solitonic structures and some kink waves, which are affected by the variation of energy, are explored. If the energy is infinite in finite time, there will be a collapse in soliton systems by direct numerical simulations. Particularly, there are two collapses coexisting in our regular solitons, which occurred around its central regions. Simulation shows that in the bottom of periodic waves arises the non-zero parts of compactons and anti-compactons. We also get floating solitary waves whose amplitude is infinite. In contrary to which a finite-amplitude blow-up soliton is obtained. Periodic blow-ups are found too. Special kinks which have periodic cuspons are derived.

In this Letter, after introducing some potentials for a coupled KdV equation which is derived from the two layer fluid model, the nonlocal Lie–Backlund transformation is obtained. Using the Lie–Backlund transformation theorem to the trivial zero solution, the single soliton solution is also found.

The analysis qualitative methods of planar dynamical systems are used to study the peaked solitary wave solutions for Degasperis–Procesi equation with the dispersion term. By using the phase portrait bifurcation of traveling wave system, periodic wave solutions and solitary wave solutions are constructed in two different ways, and their convergence is showed when g varies. The general explicit expression of peaked solitary wave solutions is obtained under some parameter conditions.

Some special solutions of the Toda lattice model with a transversal degree of freedom are obtained. With the aid of Mathematica and Wu elimination method, more explicit solitary wave solutions, including compacton solutions, multiple compacton solutions, peakon solutions, as well as periodic solutions are found in this paper.

In this paper, we introduce the concept of nonlinear intensity and study the new type solitary wave solutions of nonlinear intensity Klein–Gordon equation. Applying the improved generalized projective Riccati method, abundant exact solitary wave solutions are obtained. By using ansatzes, nonsymmetrical compacton, multi-compacton solutions, pattern solitary wave solutions and singular periodic wave solutions are found. Then, we discuss the changes of compacton solutions under various nonlinear intensity parameters and get the compacton solutions of higher dimensions nonlinear intensity Klein–Gordon equation.

This paper investigates the existence of solution on the weak dissipative Camassa-Holm equation, and studies the dynamical behaviors of the solution for this equation. An interesting result is given that the solution for the Cauchy problem on weak dissipative Camassa-Holm equation is local under certain conditions. Moreover, discussions on solitary solution to weak dissipation Camassa-Holm equation are given.

We study a sort of nonlinear reaction diffusion equation based on the modified Korteweg–de Vries (mKdV) equation with a higher order singularly perturbing term as the Kuramoto–Sivashinsky (KS) equation, called mKdV–KS equation. Special attention is paid to the question of the existence of solitary wave solutions. Based on the analogue between solitary wave solution and homoclinic orbits of the associated ordinary differential equations, from geometric singular perturbation point of view, we prove that solitary wave persists when the perturbation parameter is suitably small. This argument does not require an explicit expression for the original mKdV solitary wave solution.

We introduce a fifth-order K ( m , n ,1) equation with nonlinear dispersion to obtain multi-compacton solutions by Adomian decomposition method. Using the homogeneous balance (HB) method, we derive a Backlund transformation of a special equation K (2,2,1) to determine some solitary solutions of the equation. To study the stability of multi-compacton solutions in K ( m , n ,1) and to obtain some conservation laws, we present a similar fifth-order equation derived from Lagrangian. We finally show the linear stability of all obtained multi-compacton solutions.

In this paper, we consider generalized Camassa–Holm equations and the generalized weakly dissipative Camassa–Holm equations and derive some new exact peaked solitary wave solutions. For m=3, where m is representative of the strength of the nonlinearity, we give two types new exact traveling wave solutions of the generalized weakly dissipative Camassa–Holm equations.

We study the limiting behavior of smooth periodic waves for the Degasperis–Procesi equation as the parameters trend to some special values. Using an improved qualitative method, the existence of smooth periodic waves is considered. For some limiting values of parameters of smooth periodic waves, one obtains peaked solitary waves.

This paper studies the dynamical behavior of a weakly damped forced Korteweg–de Vries (KdV) equation in a wavelet basis and introduces the wavelet approximate inertial manifold and wavelet Galerkin solution of weakly damped forced KdV equation. The results includes theorems that for the KdV equation the wavelet approximate inertial manifold (WAIM) exists and sets up the wavelet Galerkin method of the equation. Error estimates are given by using the WAIM.

This paper sets up the approximate inertial manifold(AIM) in the nonselfadjoint nonlinear evolutionary equation and finds AIMs which are explicitly defined in the weakly damped forced KdV equation (WDF KdV).

This paper studies the wave-breaking criterion for the generalized weakly dissipative two-component Hunter-Saxton system in the periodic setting. We get local well-posedness for the generalized weakly dissipative two-component Hunter-Saxton system. We study a wave-breaking criterion for solutions and results of wave-breaking solutions with certain initial profiles.