5 A theory for one dimensional asynchronous chemical wave

Based on analytic calculations we identify the local structures which changes the parity of the global traveling wave states on its either sides. We establish the fact that competition of Turing and Hopf modes and multi-stability is not an essential ingredient for the generation of asynchronous one dimensional wave rather, close to a purely Hopf instability boundary and well in the region where the nonlinear Hopf mode exists, such parity inverted nonlinear traveling waves can be sustained by a special type of localized structure which can be nucleated around some local inhomogeneity. One dimensional asynchronous chemical waves (or 1d spirals) were first reported by J.-J.Perraud et al [1] in 1993. An explanation of this phenomenon had been put forward on the basis of bi-stability of Turing and Hopf states and non-variational effects. Historically , in those days when the above mentioned experiment was done, studying droplet of other states inside a global one under uniform conditions was a topic of tremmendous interest. Mainly being influenced by those new observations [2, 3, 4, 5] and some numerically simulated asynchronous wave like structures, the bi-stability of Turing and Hopf modes and their interaction was considered to be the basic underlying principle on which a possible explanation of experimentally observed 1d spirals was given. As a result of considering bi-stability the dynamics was considered not to derive form variational principle [1]. The above mentioned mechanism is till date considered as the basis for such a novel phenomenon of nonlinear waves since any other alternative theory is not present. In the present letter we are going to analytically show that sustenance of parity inverted global traveling wave states requires the presence of a class of asymmetric localized structures nucleated at local inhomogeneities. As a result it will be clearly shown that the *