Periodic Blow-up Solutions Research Articles

Bifurcations and Exact Traveling Wave Solutions of the Celebrated Green–Naghdi Equations

In this paper, we study the bifurcations and exact traveling wave solutions of the celebrated Green–Naghdi equations by using the qualitative theory of differential equations and the bifurcation theory of dynamical systems. We obtain all possible phase portraits of bifurcations of the system under various conditions about the parameters associated with the planar dynamical system. Then we show the existence of traveling wave solutions including solitary wave solutions, blow-up solutions, periodic wave solutions and periodic blow-up solutions, and give their exact explicit expressions. These results can help to understand the dynamical behavior of the traveling wave solutions of the system.

The explicit periodic wave solutions and their limit forms for a generalized b-equation

In this paper, for b ∈ (−∞,∞) and b ≠ −1,−2, we investigate the explicit periodic wave solutions for the generalized b-equation ut + 2kux − uxxt + (1+b)u2ux = buxuxx + uuxxx, which contains the generalized Camassa-Holm equation and the generalized Degasperis-Procesi equation. Firstly, via the methods of dynamical system and elliptic integral we obtain two types of explicit periodic wave solutions with a parametric variable α. One of them is made of two elliptic smooth periodic wave solutions. The other is composed of four elliptic periodic blow-up solutions. Secondly we show that there exist four special values for α. When α tends to these special values, these above solutions have limits. From the limit forms we get other three types of nonlinear wave solutions, hyperbolic smooth solitary wave solution, hyperbolic single blow-up solution, trigonometric periodic blow-up solution. Some previous results are extended. For b = −1 or b = −2, we guess that the equation does not have any one of above solutions.

Periodic wave solutions and their limits for the modified KdV-KP equations

In this paper, we use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the modified KdV-KP equations. Some explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain solitary wave solutions, periodic wave solutions, kink wave solutions and unbounded solutions.

Periodic wave solutions and their limits for the ZK–BBM equation

In this paper, we use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the ZK–BBM equation. Some explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain solitary wave solutions, periodic wave solutions, kink wave solutions, unbounded solutions and blow-up wave solutions.

Application of the bifurcation method to the modified Boussinesq equation

In this paper, we investigate the modified Boussinesq equation utt uxx # uxxxx 3(u 2 )xx + 3(u 2 ux)x = 0. Firstly, we give a property of the solutions of the equation, that is, if 1 + u(x,t) is a solution, so is 1 u(x,t). Secondly, by using the bifurcation method of dynamical sys- tems we obtain some explicit expressions of solutions for the equation, which include kink-shaped solutions, blow-up solutions, periodic blow-up solutions and solitary wave solutions. Some previous results are extended.

Exact Traveling Wave Solutions of Explicit Type, Implicit Type, and Parametric Type for K(m, n) Equation

By using the integral bifurcation method, we study the nonlinear K(m, n) equation for all possible values of m and n. Some new exact traveling wave solutions of explicit type, implicit type, and parametric type are obtained. These exact solutions include peculiar compacton solutions, singular periodic wave solutions, compacton‐like periodic wave solutions, periodic blowup solutions, smooth soliton solutions, and kink and antikink wave solutions. The great parts of them are different from the results in existing references. In order to show their dynamic profiles intuitively, the solutions of K(n, n), K(2n − 1, n), K(3n − 2, n), K(4n − 3, n), and K(m, 1) equations are chosen to illustrate with the concrete features.

Application of the bifurcation method to the Whitham–Broer–Kaup-Like equations

In this paper, we investigate the Whitham–Broer–Kaup-Like equations. Some explicit expressions of solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain kink-shaped solutions, blow-up solutions, periodic blow-up solutions and solitary wave solutions. Some previous results are extended.

The Application of Bifurcation Method to Klein-Gordon-Zakharov Equations

Bifurcation method of dynamical systems is employed to study the Klein-Gordon-Zakharov equations. Under some parameter conditions, some explicit expressions of solutions for the equation are obtained. These solutions contain solitary wave solutions, blow-up solutions, periodic solutions, periodic blow-up solutions and kink-shaped solutions. Key Words: Klein-Gordon-Zakharov equations; Exact solutions; Bifurcation method

EXPLICIT PERIODIC WAVE SOLUTIONS AND THEIR BIFURCATIONS FOR GENERALIZED CAMASSA–HOLM EQUATION

In this paper, through qualitative analysis and integration, we study the explicit periodic wave solutions and their bifurcations for the generalized Camassa–Holm equation [Formula: see text] When the parameter k satisfies k < 3/8 and the constant wave speed c satisfies [Formula: see text], we obtain two types of explicit periodic wave solutions, elliptic smooth periodic wave solution and elliptic periodic blow-up solutions. These solutions include a bifurcation parameter α which has four bifurcation values αi(i = 1, 2, 3, 4). When α tends to the bifurcation values, the elliptic periodic wave solutions become three types of other solutions, the hyperbolic smooth solitary wave solution, the hyperbolic blow-up solution and the trigonometric periodic blow-up solution. Especially, a new bifurcation phenomenon is found, that is, the periodic blow-up solution can become a smooth solitary wave solution when α varies. When k > 3/8, we guess that there is no other explicit solution except the explicit periodic blow-up solution.

New exact solutions for the classical Drinfel’d–Sokolov–Wilson equation

In this paper, we investigate the classical Drinfel’d–Sokolov–Wilson equation (DSWE) u t + pvv t = 0 , v t + ruv x + su x v + qv xxx = 0 , where p, q, r, s are some nonzero parameters. Some explicit expressions of solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain solitary wave solutions, blow-up solutions, periodic solutions, periodic blow-up solutions and kink-shaped solutions. Some previous results are extended.

Periodic blow-up solutions and their limit forms for the generalized Camassa–Holm equation

In this paper, we consider the generalized Camassa–Holm equation u t + 2 ku x - u xxt + au 2 u x = 2 u x u xx + uu xxx . Under substitution ξ = x - ct , some new explicit periodic wave solutions and their limit forms are presented through some special phase orbits. These periodic wave solutions tend to infinity on ξ - u plane periodically. Thus we call them periodic blow-up solutions. To our knowledge, such periodic blow-up solutions have not been found in any other equations.

Lower bounds for the $L^2$ minimal periodic blow-up solutions of critical nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation with critical power $iu_t=-\Delta u-|u|^{4/N}u$, $t\geq 0$ and $x\in\mathbb T^N$ (the space-periodic case) in $H^1$. We consider a blow-up solution with minimal mass. We obtain in this context an optimal lower bound for the blow-up rate (that is, for $ | {\nabla u(t)}| _{L^2}$), and we observe that this lower bound equals the blow-up rate (which is explicitly known) of the minimal blow-up solutions in $\mathbb R^N$.