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Two-dimensional inverse scattering for quasi-linear biharmonic operator

  • * Corresponding author: Jaakko Kultima

    * Corresponding author: Jaakko Kultima 
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  • The subject of this work concerns the classical direct and inverse scattering problems for quasi-linear perturbations of the two-dimensional biharmonic operator. The quasi-linear perturbations of the first and zero order might be complex-valued and singular. We show the existence of the scattering solutions to the direct scattering problem in the Sobolev space $ W^1_{\infty}( \mathbb{{R}}^2) $. Then the inverse scattering problem can be formulated as follows: does the knowledge of the far field pattern uniquely determine the unknown coefficients for given differential operator? It turns out that the answer to this classical question is affirmative for quasi-linear perturbations of the biharmonic operator. Moreover, we present a numerical method for the reconstruction of unknown coefficients, which from the practical point of view can be thought of as recovery of the coefficients from fixed energy measurements.

    Mathematics Subject Classification: Primary: 35P25, 65M32; Secondary: 35R30, 35J91.


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  • Figure 1.  The potentials $ \beta $ for Examples 1 (top left), 2 (top right), 3 (bottom left) and 4 (bottom right). In Example 1 we have only potential $ V $ which is a characteristic function of an ellipse. In Example 2 $ V $ is the characteristic function of an L-shaped domain and $ \overrightarrow{W} = (0,\varphi_2)\sin(|u|) $ has one component, where $ \varphi_2 $ is a smooth bump function in a circular domain. In Example 3 both components of $ \overrightarrow{W} = (\varphi_1\frac{|u|^2}{1+|u|^2},\varphi_2|u|^2) $ are multiplied by smooth bump functions $ \varphi_1 $ and $ \varphi_2 $ supported in ellipses located at the top and bottom right in the figure, respectively. The coefficient $ \varphi_3 $ of potential $ V $ is also a smooth bump function supported in an ellipse, located in the middle-left side of the figure. In example 4 all coefficients are smooth bump functions (see also Figure 6), but their supports are intersecting

    Figure 2.  The scattered fields for Examples 1 (top left), 2 (top right), 3 (bottom left) and 4 (bottom right) with $ k = 25 $. The locations of the supports of the potentials are presented in black. Here the incident field is travelling from the left to the right

    Figure 3.  Example 1. Left: The unknown target $ \beta $. Right: The numerical reconstruction $ \beta_ \mathrm{num} $

    Figure 4.  Example 2. Left: The unknown target $ \beta $. Right: The numerical reconstruction $ \beta_ \mathrm{num} $. This example shows recovery of corners and recovery of a shape with piece-wise smooth boundary

    Figure 5.  Example 3. Left: The unknown target $ \beta $. Right: The numerical reconstruction $ \beta_ \mathrm{num} $. We see that weak potentials are quite difficult to detect while stronger potentials are clearly visible in comparison, as is expected

    Figure 6.  Example 4. Top left $ V(x,1) $, top middle $ W_1(x,1) $ and top right $ W_2(x,1) $. Bottom left: The unknown target $ \beta $. Bottom right: The numerical reconstruction $ \beta_ \mathrm{num} $. In this example the supports of potentials $ V $, $ W_1 $ and $ W_2 $ overlap. We can not distinguish these functions from each other

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