In this article we continue to develop the theory of generating symmetries for integrable equations. A technique computation using Maple is presented. The based on standard symmetry method. By it find KdV, Camassa–Holm, mKdV, sine-Gordon, Boussinesq, associated Degasperis-Procesi and Novikov

We study non-linear integrable partial differential equations naturally arising as bi-Hamiltonian Euler equations related to the looped cotangent Virasoro algebra. This infinite-dimensional Lie algebra (constructed in [16]) is a generalization of the classical Virasoro algebra to the case of two space variables. Two main examples of integrable equations we obtain are quite well known. We show t...

A simple algoritm for the inverse scattering approach to the Camassa-Holm equation is presented.

This paper shows that the Camassa-Holm (CH) spectral problem yields two different integrable hierarchies of nonlinear evolution equations (NLEEs), one is of negative order CH hierachy while the other one is of positive order CH hierarchy. The two CH hierarchies possess the zero curvature representations through solving a key matrix equation. We find that the well-known CH equation is included i...

We propose a novel technique for analyzing the long-time asymptotics of integrable wave equations in the case where the underlying isospectral problem has purely discrete spectrum. To this end we introduce a natural coupling problem for entire functions which serves as a replacement for the usual Riemann–Hilbert approach, which does not work for these kind of problems. As a prototypical example...

A generalization of the negative Camassa-Holm hierarchy to 2 + 1 dimensions is presented under the name CHH(2+1). Several hodograph transformations are applied in order to transform the hierarchy into a system of coupled CBS (Calogero-Bogoyavlenskii-Schiff) equations in 2 + 1 dimensions that pass the Painlevé test. A non-isospectral Lax pair for CHH(2+1) is obtained through the above mentioned ...

Considered in this paper is the generalized Camassa–Holm–Novikov equation with high order nonlinearity, which unifies Camassa–Holm and Novikov equations as special cases. We show that solution map of not uniformly continuous on initial data Besov spaces $$B_{p, r}^s(\mathbb {R})$$ $$s>\max \{1+\frac{1}{p}, \frac{3}{2}\}$$ , $$1\le p, r< \infty $$ well critical space $$B_{2, 1}^{3/2}(\mathbb {R}...

In this paper, we prove the convergence of the vortex blob method for a family of nonlinear evolutionary partial differential equations (PDEs), the so-called b-equation. This kind of PDEs, including the Camassa-Holm equation and the Degasperis-Procesi equation, has many applications in diverse scientific fields. Our convergence analysis also provides a proof for the existence of the global weak...