For implementation of soliton resonance approach to target detection, we have to deal with inhomogeneous KdV equation. We will provide a qualitative theoretical description that will guide our numerical algorithms. We shall start with the problem formulation. Consider inhomogeneous 1D KdV equation of the form

where h (x, t) is a forcing function. We will search for the solution with zero initial and absorbing boundary conditions and consider two extreme cases: (a) small times and (b) large times.

For the small times the Equation (12) can be simplified by dropping the nonlinear term:

Let the forcing function be, where is the solution of the equation

Then the particular solution of the Equation (13) can be written as

The result could be verified by substitution of the solution (15) into the Equation (13). From the physical viewpoint, the Equation (15) represents a typical linear resonance when the solution of the homogeneous equation is fortified by the external input that has a similar profile.

Now we consider inhomogeneous version of the KdV equation for large time intervals assuming that the external force could depend not only on the solution of the homogeneous equation, but also on its space derivatives as. The reason for such generalization is the following. Since we do not know in advance the microstructure of the target as a “thick” point, we have to be sure that regardless of a different “internal” configuration, the target effect on the global property of the KdV solutions will be qualitatively the same as long as the target is localized in space. The generalized form of the external force in the Equation (12) allows us to consider fundamentally different intrinsic shapes. Let the external force be. Then the solution of the Equation (12) will have the following form

where the soliton velocity is

If the external force is, the solution of the Equation (12) gives us the soliton

where the soliton velocity is defined as

.(19)

Finally taking the external force in the form we obtain the soliton

propagating with the velocity.

In all three cases, the shapes of the forcing function are topologically the same although the “details” of these shapes are different. This demonstrates that all the targets that are localized within a small interval (compare to the length of the whole trajectory) will cause a similar global effect in terms of the KdV solution.

As follows from the Equations (16), (18) and (20), the width of the soliton increases with the decrease of the value of and with the increase of the value b. The motion of the target sets up the amplitude of the soliton. One can see that two variables characterizing the shape of the soliton as the solution of KdV equation, namely, U and a, i.e. speed and amplitude, are connected only by one equation, and therefore one of variables, for instance U, can be set up arbitrarily. That opens up an opportunity to consider targets having variable speeds: indeed, when the speed of the target is changing, the KdV equation “accommodates” it by changing the amplitude of the running soliton.

One of significant advantages of the proposed approach is that the corresponding KdV equation is supposed to be solved only subject to initial conditions, while there are no boundary conditions imposed upon the solution. This property simplifies both theoretical and practical aspect of the methodology of solution. The theory of PDE without boundary conditions is equipped with the theorem of existence of a unique solution; while for PDE with boundary conditions similar theorems do not exist. In addition to that, more closed form analytical solutions are available for PDE without boundary conditions. However when numerical approaches are applied, the following obstacle occurs: we should introduce artificial boundary conditions. These conditions are supposed to be formulated so that they do not affect the solution within the area of interest, i.e. they must provide no reflection. In literature such a boundary condition is called an absorbing one. These conditions are used in our approach.

Although 1D KdV equation is of fundamental theoretical importance, its practical importance is limited and 2D case could be effectively exploited for moving target detection. However if we turn from 1D KdV equation to 2D KdV equations, we can conclude that the response to a thick-point-external-force is a solitary wave that is represented by a thick line rather than a thick point, and the whole idea of target detection fails. The only way the target moving on 2D plane can be detected using 1D KdV equation is by decomposing the target motion on two projections and detecting each projection by the corresponding 1D KdV equation.

Suppose that the motion under consideration takes place on a plane, and 2D image of this motion is presented by a 2D forcing function h(x, y, t). Then two 1D KdV equations can be introduced as

and

where the forcing functions in the Equations (22) and (23) are found as integrals.

Letters h and l define the sides of the 2D area.

We will start from 1D case when the target movement is described by the equation. Here x₀ is the initial position of the target, V and w are the target speed and acceleration. The forcing function for this type of target movement can be modeled as

where the parameter d allows us to adjust the width of the target image. As 1D example we consider a simple case of target movement. Let the target move according to the equation. The solution u (x, t) of KdV equation for this case is shown in Figure 1. The soliton propagates along x-axis during 20 seconds interval with constant velocity. As follows from this plot, the soliton propagates along x-axis during 10 seconds interval with contrast velocity.

To estimate the target velocity we define the soliton movement by locating its maxima during target detection time and use the fitting line to compute the trajectory characteristics. The soliton position as a function of time and the fitting line are shown by the red and blue colors in Figure 2. The fitting line is defined by the equation x = −3 + 10.2t. Therefore, the soliton velocity is about 10.2 m/sec.

The accelerating targets are also well detected by this method. Now we will describe the applications of soliton

resonance method to the detection of a moving target in 2D. In the next section we consider targets moving in a straight line and along a curved trajectory.

Let the target trajectory be given by the equations and. The forcing function implementing this target movement is

We consider examples of target detection with soliton resonance technique for a target moving in a straight line with constant acceleration and a target moving along a curved trajectory. We compute the forcing functions h(x, t) and h (y, t) using the Equations (26) and (24), solve inhomogeneous KdV Equations (22)-(23), find soliton trajectories along x and y directions and, finally, obtain the target trajectory in 2D.

For the first example we will assume that the target trajectory is. Writing the forcing function as

and solving the system of two KdV equations we obtain two solitons propagating in x and y directions. The solitons’ trajectories are shown in Figure 3 by the red curves. The equations for fitting curves, shown by dashed blue lines in Figure 3, are x = −2 + 0.49t² and y = −5.6 + 0.95t², Therefore the solitons propagate along x any y-axes with the accelerations w_x = 0.99 m/sec² and w_y=1.9 m/sec², that is the accelerations of the solitons are the same as the input accelerations w_x and w_y.

Combining red curves from Figure 3 we obtain the simulated target trajectory in (x − y) plane. This trajectory is shown by the dashed red line in Figure 4. The blue line is the real trajectory of the target.

In the second example we will assume that the trajectory of the target is given by the equations x = t²/2 and y = 6 t. Writing the forcing function as

and numerically solving KdV equations we find solitons’ trajectories. They are shown in Figure 5 by the red lines.

The fitting curves (the dashed blue lines in Figure 5) are defined by the equations x = −0.6 + 0.49 t² and y = −4 + 6t.

Therefore the characteristics of the moving target are w_x = 0.98 m/sec² and V_y = 6 m/sec. The target trajectory is constructed with the use of red curves in Figure 5. The target trajectory simulated by the soliton resonance method is given in Figure 6 by the red color. The real target trajectory is shown by the dashed blue curve.

The change of 1D solitons’ amplitudes in time allows us to model a change of the 2D amplitude of the target image. The result is shown in Figure 7. The dashed green line shows the growth of the simulated target amplitude for 12 seconds interval for the unchanged amplitude of the input target signal.