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Register a new userOn Camassa-Holm equation with self-consistent sources and its solutions
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Regarded as the integrable generalization of Camassa-Holm (CH) equation, the CH equation with self-consistent sources (CHESCS) is derived. The Lax representation of the CHESCS is presented. The conservation laws for CHESCS are constructed. The peakon solution, N-soliton, N-cuspon, N-positon and N-negaton solutions of CHESCS are obtained by using Darboux transformation and the method of variation of constants.
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In this paper, we study one of generalized Heisenberg ferromagnet equations with self-consistent sources, namely, the so-called M-CIV equation with self-consistent sources (M-CIVESCS). The Lax representation of the M-CIVESCS is presented. We have shown that the M-CIVESCS and the CH equation with self-consistent sources (CHESCS) is geometrically equivalent each to other. The gauge equivalence between these equations is proved. Soliton (peakon) and pseudo-spherical surfaces induced by these equations are considered. The one peakon solution of the M-CIVESCS is presented.
The soliton solutions of the Camassa-Holm equation are derived by the implementation of the dressing method. The form of the one and two soliton solutions coincides with the form obtained by other methods.
We study the integrability and equivalence of a generalized Heisenberg ferromagnet-type equation (GHFE). The different forms of this equation as well as its reduction are presented. The Lax representation (LR) of the equation is obtained. We observe that the geometrical and gauge equivalent counterpart of the GHFE is the modified Camassa-Holm equation (mCHE) with an arbitrary parameter $kappa$. Finally, the 1-soliton solution of the GHFE is obtained.
An integrable semi-discretization of the Camassa-Holm equation is presented. The keys of its construction are bilinear forms and determinant structure of solutions of the CH equation. Determinant formulas of $N$-soliton solutions of the continuous and semi-discrete Camassa-Holm equations are presented. Based on determinant formulas, we can generate multi-soliton, multi-cuspon and multi-soliton-cuspon solutions. Numerical computations using the integrable semi-discrete Camassa-Holm equation are performed. It is shown that the integrable semi-discrete Camassa-Holm equation gives very accurate numerical results even in the cases of cuspon-cuspon and soliton-cuspon interactions. The numerical computation for an initial value condition, which is not an exact solution, is also presented.
In the present paper, we investigate some geometrical properties of the Camass-Holm equation (CHE). We establish the geometrical equivalence between the CHE and the M-CIV equation using a link with the motion of curves. We also show that these two equations are gauge equivalent each to other.
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