Dynamics of the main discontinuity

Our task is to describe the profile of the main discontinuity of the field  

$$u(\xi , \eta )={\bf P}_{w}u_{0}(x,y) \equiv \int_{X}w(x;\xi ,\eta ) u_{0} (x,y=\th (x;\xi , \eta ))dx$$ (126)

along the line $\xi =const$.I remind you that

$$u_{0}(x,y)\stackrel{q}{\sim }A_{0}(x)R^{(+)}_{q,\nu }[y-\tau (x)].$$

Applying the operator ${\bf D}^{q+1}{\bf H}^{-\nu }$ we obtain the field

$$U(\xi ,\eta ) \sim \int w(x;\xi , \eta ){\vert\th _{\eta }\vert}^{q+1}A_{0}(x)\delta [ \th _{M}(x)-\tau (x)]dx.$$

Then we fix a line in the plane $(\xi ,\eta)$:

$${d \th _{M} \over dx} =p {\:}(x=x^{\ast }=const, {\:}p=const),$$

which we shall call a pseudoray. For each point M^' belonging to the pseudoray, we apply the expansion of the difference $\th _{M^{'}}(x)-\tau (x)$ into Taylor's series in a neighborhood of the point $x^{\ast }$:

$$\th_{M^{'}}(x)-\tau (x)=\epsilon -\left( {d \over 2} \right) l^{2} +0(l^{2})$$

where $\epsilon=y^{'}-y^{\ast }, {\:}y^{\ast }=\th_{M}(X^{\ast}), {\:}l=x-x^{\ast }$and

$$d={ \left[ {d^{2}\tau \over dx^{2}}-{d^{2}\th _{M} \over dx^{2}} \right]}_{x= x^{\ast }}.$$

The above equations are absolutely the same as those that we have already done for the KO. As a result, we will receive a very similar formula describing the field U(M^'):  

$$U(M^{'}) \sim {\left[ \sqrt{{2\pi \over \vert d\vert}} {\ver... ...0} \right] }_{{\omega}}R^{(\kappa )}_{-{1 \over 2}} (\epsilon )$$ (127)

where $\kappa = {\rm sgn}(d)$.

But here we have some important differences that we have to take into account: Formula (127) describes the profile of the discontinuity along the pseudoray but not along a line $\xi =const$. There is a formalism that connects different profiles of discontinuity. I shall omit this consideration and give you only the final result under $\theta _{\eta}\gt$: 

$$u(\xi ,\eta )\stackrel{q^{'}}{\sim }{({\bf D}^{q+1}{\bf H}^{... ...wA_{0} \right] }_{{\omega}}R^{(+)}_{q^{'},\nu ^{'}}[y-\tau (x)]$$ (128)

where $q^{'}=q+{1 \over 2},{\:}\nu ^{'}=\nu + {\kappa -1 \over 2}$.I consider this formula to be the main result of the developing theory.