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This paper considers the whole hierarchy of bi-Hamiltonian integrable equations associated to each of the Short-Pulse (SP) equation and the Sine-Gordon (SG) equation. We prove that the transformation that relates the SP equation with the SG equation also serves to establish the correspondence between their flows and Hamiltonian conservation laws in respective hierarchy.
We first derive an integrable deformed hierarchy of short pulse equation and their Lax representation. Then we concentrated on the solution of integrable deformed short pulse equation (IDSPE). By proposing a generalized reciprocal transformation, we
Using the determinant representation of gauge transformation operator, we have shown that the general form of $tau$ function of the $q$-KP hierarchy is a q-deformed generalized Wronskian, which includes the q-deformed Wronskian as a special case. On
Under three relations connecting the field variables of Toda flows and that of KdV flows, we present three new sequences of combination of the equations in the Toda hierarchy which have the KdV hierarchy as a continuous limit. The relation between th
Critical points of semiclassical expansions of solutions to the dispersionful Toda hierarchy are considered and a double scaling limit method of regularization is formulated. The analogues of the critical points characterized by the strong conditions
An iterative algorithm for determining a class of solutions of the dispersionful 2-Toda hierarchy characterized by string equations is developed. This class includes the solution which underlies the large N-limit of the Hermitian matrix model in the