A direct and systematic algorithm is proposed to find one-dimensional optimal system for the group invariant solutions, which is attributed to the classification of its corresponding one-dimensional Lie algebra. Since the method is based on different values of all the invariants, the process itself can both guarantee the comprehensiveness and demonstrate the inequivalence of the optimal system, with no further proof. To leave the algorithm clear, we illustrate each stage with a couple of well-known examples: the Korteweg-de Vries equation and the heat equation. Finally, we apply our method to the Novikov equation and use the found optimal system to investigate the corresponding invariant solutions.

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A direct algorithm of one-dimensional optimal system for the group invariant solutions

The Cauchy problem for the integrable Novikov equation

In this paper we consider the Cauchy problem for the Novikov equation. We prove that the Cauchy problem for the Novikov equation is not locally well-posed in the Sobolev spaces with in the sense that its solutions do not depend uniformly continuously on the initial data. Since the Cauchy problem for the Novikov equation is locally well-posed in with s > 3/2 in the sense of Hadamard, our result implies that s = 3/2 is the critical Sobolev index for well-posedness. We also present two blow-up results of strong solution to the Cauchy problem for the Novikov equation in with s > 3/2.

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The Cauchy problem for the Novikov equation

A 4-parameter polynomial family of equations generalizing the Camassa-Holm and Novikov equations that describe breaking waves is introduced. A classification of low-order conservation laws, peaked travelling wave solutions, and Lie symmetries is presented for this family. These classifications pick out a 1-parameter equation that has several interesting features: it reduces to the Camassa-Holm and Novikov equations when the polynomial has degree two and three; it has a conserved H1 norm and it possesses N-peakon solutions when the polynomial has any degree; and it exhibits wave-breaking for certain solutions describing collisions between peakons and anti-peakons in the case N = 2.

A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations

On the Cauchy problem for the modified Novikov equation with peakon solutions

Blow-up phenomenon for the integrable Novikov equation

This paper deals with the asymptotic behavior of strong solutions to the 3D Navier–Stokes equations with a nonlinear damping term | u | β − 1 u ( β ≥ 3 ) . First, we establish an upper bound for the difference between the solution of our equation and the heat equation in L 2 space. Then, we optimize the upper bound of decay for the solutions and obtain their algebraic lower bound by using Fourier Splitting method.

Asymptotic behavior of strong solutions to the 3D Navier–Stokes equations with a nonlinear damping term

This paper deals with a new integrable two-component Camassa–Holm (CH2) system which describes the motion of fluid. We are concerned with discovering the wave breaking mechanism for the Cauchy problem. Analogous to McKean’s theorem (McKean (1998) [19] ) for the Camassa–Holm equation, we establish the sufficient condition on the initial data to guarantee the blow-up of the solution. It turns out that our blow-up theorem is not depend on the initial energy but depend on the sign of initial data y 0 ( x ) and ρ 0 ( x ) .

Zaihong Jiang

Papers

Blow-up phenomenon for the integrable Novikov equation

Asymptotic behavior of strong solutions to the 3D Navier–Stokes equations with a nonlinear damping term

Wave breaking of an integrable Camassa–Holm system with two components

Optimal time decay of the Boltzmann equation with frictional force

Analytical properties for the fifth-order b-family Novikov model