In contrast with the soliton equations, the evolution of the eigenfunctions in the Lax representation of soliton equation with self-consistent sources ͑SESCS͒ possesses singularity. We present a general method to treat the singularity to determine the evolution of scattering data. The AKNS hierarchy with self-consistent sources, the MKdV hierarchy with self-consistent sources, the nonlinear Schrödinger equation hierarchy with self-consistent sources, the Kaup-Newell hierarchy with selfconsistent sources and the derivative nonlinear Schrödinger equation hierarchy with self-consistent sources are integrated directly by using the inverse scattering method. The N soliton solutions for some SESCS are presented. It is shown that the insertion of a source may cause the variation of the velocity of soliton. This approach can be applied to all other (1ϩ1)-dimensional soliton hierarchies.

A method is proposed to construct a new extended KP hierarchy, which includes two types of KP equation with self-consistent sources and admits reductions to k-constrained KP hierarchy and to Gelfand-Dickey hierarchy with sources. It provides a general way to construct soliton equations with sources and their Lax representations.Comment: Published in Phys. Lett. A, 13 page

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Two binary (integral type) Darboux transformations for the KdV hierarchy with self-consistent sources are proposed. In contrast with the Darboux transformation for the KdV hierarchy, one of the two binary Darboux transformations provides non auto-Bäcklund transformation between two n-th KdV equations with self-consistent sources with different degrees. The formula for the m-times repeated binary Darboux transformations are presented. This enables us to construct the N-soliton solution for the KdV hierarchy with self-consistent sources.

In addition to fluids, the concept of sediment‐derived melts infiltrating the fore‐arc mantle during subduction initiation has been proposed based on studies of modern subduction zones and ophiolite mélange. However, outcrops that contain the products of such melts are rare, especially in conjunction with boninite. New U‐Pb zircon dating reveals that the Darutso volcanic rocks (DVRs) within ophiolitic mélange in the Beila area, central Tibet, crystallized at ∼164–162 Ma. This is the first recognition of Jurassic volcanic rocks in the middle section of the Bangong‐Nujiang Suture Zone. Geochemically, the DVRs are high‐Mg andesites with moderate SiO2 (59.03–63.62 wt %) and high MgO (3.74–6.53 wt %), Cr (up to 395 ppm), and Mg# (50.3–67.9). They also have high Th contents, (La/Sm)N ratios, and (87Sr/86Sr)i values (0.7085–0.7147); low Ba/Th, U/Th, and Sr/Y ratios; and negative values of εNd(t) (−8.7 to −9.8) and zircon εHf(t) (−7.4 to −9.9). The εNd(t) values of the DVRs overlap those of regional sediments. Detailed analyses of these geochemical characteristics indicate that the DVRs were derived from partial melting of subducted sediments and subsequent interaction with overlying mantle peridotite in a shallow and hot setting. In combination with the regional geology, in particular adjacent ophiolites that contain MORB‐like and boninite mafic lavas, these rocks collectively recorded the evolution of a fore‐arc setting during the initiation of the northward subduction of the south branch of the Bangong‐Nujiang Ocean. Therefore, the results provide direct evidence for sediment melting during subduction initiation and constrain the Jurassic tectonic evolution of the Lhasa terrane.

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