Qiao Equation Research Articles

Singularity formation for the high-order generalized Fokas–Olver–Rosenau–Qiao equation

Singularity formation for the high-order generalized Fokas–Olver–Rosenau–Qiao equation

2-Peakon Solutions and Non-uniqueness of the Fokas–Olver–Rosenau–Qiao Equation with Higher-Order Nonlinearity

2-Peakon Solutions and Non-uniqueness of the Fokas–Olver–Rosenau–Qiao Equation with Higher-Order Nonlinearity

Wave breaking phenomenon for the generalized FORQ equation

In this paper, we consider the generalized Fokas–Olver–Rosenau–Qiao equation (also called the ab-family of equation). The local well-posedness and blow-up scenario are investigated. Sufficient conditions on the initial data to guarantee wave breaking are established for b = 2. For b = 2 and 1/6 < a ≤ 1/3, we also show that if the initial momentum density is positive in a compact support and the length of the support is small enough, then wave breaking occurs.

Non-uniform continuity of the Fokas–Olver–Rosenau–Qiao equation in Besov spaces

In this paper, we consider the solution map of the Cauchy problem to the Fokas–Olver–Rosenau–Qiao equation on the real line and prove that the solution map of this problem is not uniformly continuous on the initial data in Besov spaces. Our result extends the previous results in Himonas and Mantzavinos (Nonlinear Anal 95:499–529, 2014) and Li et al. (J Math Fluid Mech 22:50, 2020).

Well-posedness and continuity properties of the new shallow-water model with cubic nonlinearity

This paper is devoted to the new shallow-water model (also called the modified Camassa–Holm–Novikov equation) with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the Fokas–Olver–Rosenau–Qiao equation and the Novikov equation as two special cases. It is shown that the Cauchy problem of the modified Camassa–Holm–Novikov equation for the periodic and the nonperiodic case is well-posed in Sobolev spaces in the sense of Hadamard, that is, the data-to-solution map is continuous. However, the solution map is not uniformly continuous.

Blow-up phenomena for the generalized FORQ/MCH equation

In this paper, we are concerned with the generalized FORQ/MCH equation, which includes the celebrated Fokas–Olver–Rosenau–Qiao equation (or also called the modified Camassa–Holm equation). Firstly, a priori estimates in the transport equation theory and 1-D Morse-type estimates were applied to derive a blow-up criterion. Then, we exploit the characteristic ordinary differential equation to construct a conservative property, which leads to the precise blow-up scenario. Finally, making use of the fine structure and conservation laws, we present a blow-up result for the strong solutions with respect to the initial data.

GROUP-INVARIANT SOLUTIONS, NON-GROUP-INVARIANT SOLUTIONS AND CONSERVATION LAWS OF QIAO EQUATION

This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators.

Blow-up phenomena and peakons for the b-family of FORQ/MCH equations

This paper is devoted to studying the modified b-family of equations with cubic nonlinearity, called the b-family of FORQ/MCH equations, which includes the cubic Camassa–Holm equation (also called Fokas–Olver–Rosenau–Qiao equation) as a special case. We first study the local well-posedness for the Cauchy problem of the equation, and then make good use of fine structure of the equation, we derive the precise blow-up scenario and a new blow-up result with respect to initial data. Finally, peakon solutions are derived.

Non-uniqueness for the Fokas–Olver–Rosenau–Qiao equation

For the Fokas–Olver–Rosenau–Qiao equation (FORQ), on both the line and the circle, it is proved that there exist initial data in the Sobolev spaces H s , with s < 3 / 2 , which have nonunique solutions. This phenomenon is demonstrated by constructing a 2-peakon solution with an asymmetric peakon–antipeakon initial profile. The peakon and antipeakon eventually collide, and the solution at the collision time coincides with a second solitary peakon solution. Using the time-reversibility of the equation, this scenario proves nonuniqueness.

On the Cauchy problem of generalized Fokas–Olver–Resenau–Qiao equation

ABSTRACTIn this paper, we study the Cauchy problem of the generalized Fokas–Olver–Resenau–Qiao equation. Firstly, by means of transport equation and Littlewood–Paley theory, we obtain the local well-posedness of the equation in the nonhomogeneous Besov space with , . Secondly, we consider the local well-posedness in the critical space and show the continuality of the data-to-solution mapping. Thirdly, two blow-up criterion are addressed.

On the curvature blow-up phenomena for the Fokas–Qiao–Xia–Li equation

We study the blow-up phenomena of the strong solutions for the Fokas–Qiao–Xia–Li (FQXL) equation with quadratic and cubic nonlinearities, which include the celebrated Camassa–Holm equation and the Fokas–Olver–Rosenau–Qiao (FORQ) equation as its special case. The FQXL equation models the unidirectional propagation of the shallow water waves over a flat bottom and describes the physics more accurately than its celebrated counterpart, the Gardner equation. Two kinds of conditions on the initial data, including cases where the initial momentum density is not of one sign, are formulated to guarantee finite time blow-up for the strong solutions for this equation. The first condition relies on the sign persistence property of the momentum density m. The second condition is based on the full use of the conservation laws of the equation, especially the conservation quantity H2[u0].

Exponential rational function method for solving nonlinear equations arising in various physical models

In this paper, we explore new applications of the exponential rational function method to the scalar Qiao equation and the Kuramoto-Sivashinsky equation. The obtained solutions may be of significance for the explanation of some practical physical problems. The method is very suitable, easy and effective handling of the solution process of nonlinear equations.

Invariant analysis of nonlinear time fractional Qiao equation

In this study, Lie’s invariant analysis method is applied to nonlinear time fractional Qiao equation with the Riemann–Liouville derivative. Fractional derivative value \(\alpha \) is considered for three cases and each cases symmetry generators are given. Using these generators group-invariant solutions in explicit form are obtained.

Reciprocal Transformations of Two Camassa–Holm Type Equations**Supported by the Global Change Research Program of China under Grant No. 2015CB953904, National Natural Science Foundation of China under Grant Nos.11275072, 11375090 and 11435005, Research Fund for the Doctoral Program of Higher Education of China (No. 20120076110024), The Network Information Physics Calculation of basic research innovation research group of China under Grant No.

The relation between the Camassa–Holm equation and the Olver–Rosenau–Qiao equation is obtained, and we connect a new Camassa–Holm type equation proposed by Qiao etc. with the first negative flow of the KdV hierarchy by a reciprocal transformation.

The Cauchy problem and blow-up phenomena of a new integrable two-component Camassa–Holm system

This paper considers the Cauchy problem and blow-up phenomena of a new integrable two-component Camassa–Holm system, which is a natural extension of the Fokas–Olver–Rosenau–Qiao equation. Firstly, the local well-posedness of the system in the critical Besov space B2,112(R)×B2,112(R) is investigated, and it is shown that the data-to-solution mapping is Hölder continuous. Then, a blow-up criteria for the Cauchy problem in the critical Besov space is derived. Moreover, with conditions on the initial data, a new blow-up criteria is obtained by virtue of the blow-up criteria at hand and the conservative property of m and n along the characteristics. Finally, a global existence result for the strong solution is established.

Blow-up phenomena for an integrable two-component Camassa–Holm system with cubic nonlinearity and peakon solutions

This paper is devoted to an integrable two-component Camassa–Holm system with cubic nonlinearity, which includes the cubic Camassa–Holm equation (also called the Fokas–Olver–Rosenau–Qiao equation) as a special case. The one peaked solitons (peakons) and two peakon solutions are described in an explicit formula. Then, the local well-posedness for the Cauchy problem of the system is studied. Moreover, we target at the precise blow-up scenario for strong solutions to the system, and establish a new blow-up result with respect to the initial data.

Exact traveling soliton solutions for the scalar Qiao equation

We investigate exact traveling wave solutions of the scalar Qiao equation proposed by Li and Qiao (2010 J. Math. Phys. 51 042703) using the generalized auxiliary equation method. This equation is known to have bi-Hamiltonian structure and Lax pair, which imply integrability of the equation for a fixed value of k. Symmetries of the scalar Qiao equation and its solutions are also considered. The obtained solutions include kink and antikink solitons, bright and dark solitons, singular solutions and exponential solutions. This method could be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics. This work could be also relevant for numerical studies of the scalar Qiao equation.

Hölder Continuity for the Fokas–Olver–Rosenau–Qiao Equation

It has been shown that the Cauchy problem for the Fokas–Olver–Rosenau–Qiao equation is well-posed for initial data \(u_0\in H^s\), \(s>5/2\), with its data-to-solution map \(u_0\mapsto u\) being continuous but not uniformly continuous. This work further investigates the continuity properties of the solution map and shows that it is Holder continuous in the \(H^r\) topology when \(0\le r<s\). The Holder exponent is given explicitly and depends on both \(s\) and \(r\).

The Cauchy problem for the Fokas–Olver–Rosenau–Qiao equation

It is shown that the initial value problem for the Fokas–Olver–Rosenau–Qiao equation (FORQ) is well-posed in Sobolev spaces Hs, s>5/2, in the sense of Hadamard. Furthermore, it is proved that the dependence on initial data is sharp, i.e. the data-to-solution map is continuous but not uniformly continuous. Also, peakon travelling wave solutions are derived on both the circle and the line and are used to prove that the solution map is not uniformly continuous in Hs for s<3/2.

The dual modified Korteweg-de Vries–Fokas–Qiao equation: Geometry and local analysis

We study a bi-hamiltonian equation with cubic nonlinearity shown to appear in the theory of water waves by Fokas, derived by Qiao using the two-dimensional Euler equation, and also known to arise as the dual of the modified Korteweg-de Vries equation thanks to work by Fokas, Fuchssteiner, Olver, and Rosenau. We present a quadratic pseudo-potential, we compute infinite sequences of local and nonlocal conservation laws, and we construct an infinite-dimensional Lie algebra of symmetries which contains a semi-direct sum of the $sl(2,\mathbb {R})$sl(2,R)-loop algebra and the centerless Virasoro algebra. As an application we prove a theorem on the existence of smooth solutions, and we construct some explicit examples. Moreover, we consider the Cauchy problem and we prove existence and uniqueness of weak solutions in the Sobolev space $H^{q+2}({\mathbb R})$Hq+2(R), q &amp;gt; 1/2.