A non-isospectral linear problem for an integrable 2 + 1 generalization of the non linear Schrödinger equation, which includes dispersive terms of third and fourth order, is presented. The classical symmetries of the Lax pair and the related reductions are carefully studied. We obtain several reductions of the Lax pair that yield in some cases non-isospectral problems in 1 + 1 dimensions.
An integrable two-component nonlinear Schrödinger equation in 2 + 1 dimensions is presented. The singular manifold method is applied in order to obtain a three-component Lax pair. The Lie point symmetries of this Lax pair are calculated in terms of nine arbitrary functions and one arbitrary constant that yield a non-trivial infinite-dimensional Lie algebra. The main non-trivial similarity reductions associated to these symmetries are identified. The spectral parameter of the reduced spectral problem appears as a consequence of one of the symmetries.
Microstructure and defect development in the gas tungsten arc weld process is influenced by the solidification and melt-pool dynamics. Melt-pool geometrical parameters which depend mainly on heat input have profound influence on the dendrite growth velocity and growth pattern in the melt pool. Temperature magnitude and history during the process directly determine the molten pool dimensions and surface integrity. However, due to the transient nature and small size of the molten pool, the temperature gradient and the molten pool size are very challenging to measure and control. The proposed research aims to establish a methodology for characterizing direct energy deposited metals by linking processing variables to the resulting microstructure and subsequent material properties. Secondary Dendrite Arm Spacing (SDAS) optical metallographic measurements of equiaxed solidified IN-738LC gas tungsten arc welds were conducted to find a new expression that links the cooling rate that is imposed on the welding during solidification, and the resultant scale of the grain substructure.
The Lie algebra of the symmetry group of the (n + 1)-dimensional generalization of the dispersionless Kadomtsev-Petviashvili (dKP) equation is obtained and identified as a semi-direct sum of a finite dimensional simple Lie algebra and an infinite dimensional nilpotent subalgebra. Group transformation properties of solutions under the subalgebra sl(2, R) are presented. Known explicit analytic solutions in the literature are shown to be actually group-invariant solutions corresponding to certain specific infinitesimal generators of the symmetry group.
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