A New Auto-Bäcklund Transformation of the KdV Equation with General Variable Coefficients and Its Application

First, by improving some key steps in the homogeneous balance method, a new auto-Bäcklund transformation (BT) to the KdV equation with general variable coefficients is derived. The new auto-BT in this paper does not require the coefficients of the equation to be linearly dependent. Then, based on the new auto-BT in which there is only one quadratic homogeneity equation to be solved, an exact soliton-like solution containing 2-solitary wave is given.

Potential‐field sounding using Euler’s homogeneity equation and Zidarov bubbling

Potential‐field (gravity) data are transformed into a physical‐property (density) distribution in a lower half‐space, constrained solely by assumed upper bounds on physical‐property contrast and data error. A two‐step process is involved. The data are first transformed to an equivalent set of line (2-D case) or point (3-D case) sources, using Euler’s homogeneity equation evaluated iteratively on the largest residual data value. Then, mass is converted to a volume‐density product, constrained to an upper density bound, by “bubbling,” which exploits circular or radial expansion to redistribute density without changing the associated gravity field. The method can be developed for gravity or magnetic data in two or three dimensions. The results can provide a beginning for interpretation of potential‐field data where few independent constraints exist, or more likely, can be used to develop models and confirm or extend interpretation of other geophysical data sets.