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MATHEMATICAL PHYSICS
G. M. ZASLAVSKY, S. S. MOISEEV, R. Z. SAGDEEV
ASYMPTOTIC METHOD FOR SOLVING A FOURTH-ORDER DIFFERENTIAL EQUATION WITH TWO SMALL PARAMETERS IN THE HYDRODYNAMIC THEORY OF STABILITY
(Presented by Academician M. A. Leontovich, 11 V 1964)
In the problem of natural oscillations in media with slowly varying parameters, the method of geometrical optics (the WKB approximation), well developed for second-order differential equations, is often applied. Often, however, in various applications the problem of natural oscillations leads to the need to study differential equations of higher order. A classical example is the Orr—Sommerfeld equation, well known in the theory of hydrodynamic stability \((^{1})\), containing a small parameter at the fourth derivative. We shall be interested, in the case of a weakly inhomogeneous medium, in the differential equation
\[ \alpha \beta^{2}\varphi^{\mathrm{IV}}-\beta U_{2}(x,k,\omega)\varphi''+U_{1}(x,k,\omega)\varphi=0, \tag{1} \]
where \(k,\ \omega\) are, respectively, the wave vector and the frequency of the wave; \(\beta\) is a small parameter of “quasiclassicality,” taking into account the weak inhomogeneity in \(x\); \(\alpha\) is a small parameter associated with the particular formulation of the problem (in the Orr—Sommerfeld equation, for example, \(\alpha\) is proportional to the viscosity); \(U_{1}, U_{2}\sim 1\) except for small regions near points where \(U_{1}\) and \(U_{2}\) vanish. We note that, as will be seen below, the WKB method breaks down not only near points where \(U_{1}=0\) (near such points, for two of the solutions of (1), the wave vector becomes small), but also near points where \(U_{2}=0\). In this connection there arises the problem, taking into account the indicated features, of obtaining rules for finding the eigenfrequencies (“quantization rules”) for finite solutions of (1).
For convenience we choose a specific form of \(U_{1}, U_{2}\), as shown in Fig. 1. Far from \(A, B, O_{1}, O_{2}\) (turning points), solutions of (1) are sought in the form of an asymptotic series in the small parameter \(\sqrt{\beta}\) and have the form:
\[ \varphi_{1,2}=\frac{\mathrm{const}}{\sqrt{q_{1,2}}}\exp \frac{1}{\sqrt{\beta}}\int^{x} q_{1,2}(x)\,dx, \tag{2} \]
\[ \varphi_{3,4}=\frac{\mathrm{const}}{\sqrt{q_{3,4}^{5}}}\exp \frac{1}{\sqrt{\beta}}\int^{x} q_{3,4}(x)\,dx, \tag{3} \]
where*
\[ q_{1,2}=\pm\sqrt{\frac{U_{2}}{2\alpha}-\sqrt{\frac{U_{2}^{2}}{4\alpha^{2}}-\frac{U_{1}}{\alpha}}}, \quad q_{3,4}=\pm\sqrt{\frac{U_{2}}{2\alpha}+\sqrt{\frac{U_{2}^{2}}{4\alpha^{2}}-\frac{U_{1}}{\alpha}}}. \tag{4} \]
\[ \overline{\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}} \]
* Here and below, for convenience of notation, the pre-exponential factor valid for \(x>\sqrt{\alpha}\) is written out.
To obtain the solution near the points where \(U_2=0\), we set \(U_2=Ux\) \((U\sim 1)\), \(x=\beta y\). This leads to the equation
\[ \frac{\alpha}{\beta^2}\varphi^{\mathrm{IV}}-U_y\varphi''+U_1\varphi=0. \tag{5} \]
It is not difficult to see that near the point where \(U_2=0\), there exist two points at which, respectively, \(q_1=q_3\), \(q_2=q_4\) (points of “intersection” of the solutions). Near the indicated points, a strict separation of the “normal” solutions (2) and (3) from one another is, generally speaking, impossible—they “transform” into one another.
The physical picture of the solution of the problem depends essentially on the magnitude of the parameter \(\alpha/\beta^2\). If \(\alpha/\beta^2\ll 1\), then, as is seen from (4), the distance between the points of “intersection” of the solutions is small compared with the wavelength of the intersecting solutions. We note that only this case has been investigated in the study of Poiseuille flow \((^1)\), and also in connection with other physical problems \((^2)\). In this case the connection between the solutions \(\varphi_{1,2}\) and \(\varphi_{3,4}\) is effected only in the next order in \(\alpha/\beta^2\)* (weak coupling). In the case under consideration the finite solutions correspond to the following “quantization rules” \((^4)\):
\[ \int_{O_1}^{O_2}\sqrt{\frac{U_2}{\alpha\beta}}\,dx = \left(n+\frac{1}{2}\right)\pi; \qquad \int_{O_2}^{A}\sqrt{\frac{U_1}{\beta U_2}}\,dx = \left(n+\frac{1}{2}\right)\pi . \tag{6} \]
A qualitatively new picture arises when \(\alpha/\beta^2\gg 1\). In this case, around each point of “intersection” of solutions, one can always distinguish such a region in the complex \(x\)-plane where the “quasiclassical” approximation (2), (3) is valid for the solution of (5). Obviously, near such points a solution of type (2) can already in the zeroth approximation “transform” into a solution of type (3), and conversely (strong coupling).
Fig. 1 Fig. 2
We obtain the solutions of equation (5) by applying the Laplace method:
\[ \varphi(y) = \int \frac{1}{t^2} \exp\left\{ yt-\frac{\alpha}{\beta^2}\frac{t^3}{3U} +\frac{1}{t}\frac{U_1}{U} \right\}\,dt, \tag{7} \]
where the integral is taken in the plane of the complex variable \(t\) along a contour at whose ends the function
\[ \exp\left\{ yt-\frac{\alpha}{\beta^2}\frac{t^3}{3U} +\frac{1}{t}\frac{U_1}{U} \right\} \]
vanishes.
The solution (7), like equation (5) itself, is valid in the region \(y<1/\beta\). For
\[ 1<y<1/\beta . \tag{8} \]
to compute (7) one may use the saddle-point method and obtain the following four linearly independent solutions:
\[ \varphi_i(y)\sim \sqrt{ \frac{\pi}{ y\left( \dfrac{U_1}{U}\dfrac{1}{q_i^3} -\dfrac{\alpha}{\beta^2}\dfrac{\bar q_i}{U} \right)} } \frac{1}{q_i^2} \exp\int^y q_i(y)\,dy \qquad (i=1,2,3,4), \tag{9} \]
\[ \text{* With the exception of a certain region of the complex \(x\)-plane (see (3)).} \]
where
\[ \begin{aligned} \bar q_{1,2}&=\pm \sqrt{\frac{\beta^{2}}{\alpha}\frac{U}{2}}\, \sqrt{\,y-\sqrt{y^{2}-\frac{\alpha}{\beta^{2}}\frac{4U_{1}}{U^{2}}}\,},\\ \bar q_{3,4}&=\pm \sqrt{\frac{\beta^{2}}{\alpha}\frac{U}{2}}\, \sqrt{\,y+\sqrt{y^{2}-\frac{\alpha}{\beta^{2}}\frac{4U_{1}}{U^{2}}}\,}. \end{aligned} \tag{10} \]
Using (10), we obtain the solutions (9) in the form
\[ \varphi_{1,2}\sim (\bar q_i)^{-1/4}\exp\int^y \bar q_i(y)\,dy,\qquad \varphi_{3,4}\sim (\bar q_i)^{-5/4}\exp\int^y \bar q_i(y)\,dy, \tag{11} \]
which respectively pass into (2) and (3). The “intersection” points of the solutions correspond to
\[ y_0\equiv ia=\pm \sqrt{\frac{\alpha}{\beta^{2}}\frac{4U_1}{U^{2}}}. \tag{12} \]
It follows from (12) and (8) that, when \(\alpha/\beta^2\gg 1\), in accordance with what was said above, the points \(y_0\) can always be surrounded by a region where one may use the solution (9), which passes into the “quasiclassical” solutions (2) and (3).
We note that in the immediate vicinity of the points \(y_0\), the contours in the \(t\)-plane corresponding to the linearly independent solutions (5) coalesce, and an additional study of the character of the solutions at the points \(y_0\) is required. We shall not, however, be interested in this question, since for obtaining the quasiclassical “quantization rules” it is sufficient to know the rules for going around the points \(y_0\) in the complex \(x\)-plane.
We use the representation:
\[ \pm\sqrt{\,y-\sqrt{y^2+a^2}\,} =\pm\frac{1}{\sqrt2}\left(\sqrt{y+ia}-\sqrt{y-ia}\right) \qquad (y>0), \]
\[ =\pm\frac{i}{\sqrt2}\left(\sqrt{|y|+ia}-\sqrt{|y|-ia}\right) \qquad (y<0); \]
\[ \pm\sqrt{\,y+\sqrt{y^2+a^2}\,} =\pm\frac{1}{\sqrt2}\left(\sqrt{y+ia}+\sqrt{y-ia}\right) \qquad (y>0), \]
\[ =\pm\frac{i}{\sqrt2}\left(\sqrt{|y|+ia}+\sqrt{|y|-ia}\right) \qquad (y<0). \]
After this, the solutions (2), (3) for (5) are written in the form:
\[ \varphi_{1,2}=(w_1-w_2)^{-1/2} \exp\left\{\pm i\int^y (w_1(y)-w_2(y))\,dy\right\} \qquad (y>0), \]
\[ =(w_1-w_2)^{-1/2} \exp\left\{\pm\int^y (w_1(|y|)-w_2(|y|))\,dy\right\} \qquad (y<0); \tag{13} \]
\[ \varphi_{3,4}=(w_1+w_2)^{-5/2} \exp\left\{\pm\int^y (w_1(y)+w_2(y))\,dy\right\} \qquad (y>0), \]
\[ =(w_1+w_2)^{-1/2} \exp\left\{\pm i\int^y (w_1(|y|)+w_2(|y|))\,dy\right\} \qquad (y<0), \]
where
\[ w_1=\sqrt{\frac{\beta^2 U}{2\alpha}(y-ia)},\qquad w_2=\sqrt{\frac{\beta^2 U}{2\alpha}(y+ia)}. \]
Using formulas (13), one can construct the picture of the level lines of \(w_1,w_2\) for each of the intersection points of the solutions separately (Fig. 2)*. The rules for stitching the solutions (13) near the point \(O_1\) are obtained in the following way. From (13) it is seen that one can go around separately in the complex \(y\)-plane about the points \(a_1=ia\) and \(a_2=-ia\). When going around \(a_1\), the pairs of solutions \((\varphi_1,\varphi_4)\) and \((\varphi_2,\varphi_3)\), and when going around \(a_2\), the pairs \((\varphi_1,\varphi_3)\) and \((\varphi_2,\varphi_4)\), behave independently. Denote by \(A_i,B_i,C_i,D_i\) the systems of coefficients of the solution for \(\varphi_1,\varphi_2,\varphi_3,\varphi_4\), respectively, near the lines numbered \(i\), issuing from
* The meaning of all notations and letters not specified in the text is clear from Fig. 2.
points \(a_1, a_2\). Using, in going around each point separately, rules of the type (5), after simultaneously going around \(a_1, a_2\) we obtain:
\[ \begin{gathered} A_2=A_1+iD_1,\quad B_2=B_1+iD_1,\quad C_2=iA_1+iB_1+C_1-D_1,\quad D_2=D_1,\\ A_3=A_2+iC_2,\quad B_3=B_2+iC_2,\quad C_3=C_2,\quad D_3=iA_2+iB_2-C_2+D_2,\\ A'_1=A_3+iD_3,\quad B'_1=B_3+iD_3,\quad C'_1=iA_3+iB_3+C_3-D_3,\quad D'_1=D_3, \tag{14}\\ A'_1=-B_1,\quad B'_1=-A_1,\quad C'_1=-D_1,\quad D'_1=-C_1. \end{gathered} \]
To the left of the point \(A\) we write an arbitrary solution tending to zero as \(-\infty\):
\[ \varphi= |w_1-w_2|^{-1/2}\exp\left\{-i\int_A^y w_1(y)\,dy+i\int_A^y w_2(y)\,dy\right\}+ \]
\[ {}+D\,|w_1+w_2|^{-5/2}\exp\left\{\int^y w_1(y)\,dy+\int^y w_2(y)\,dy\right\}. \tag{15} \]
Using (13)—(15) and requiring finiteness of the solution as \(+\infty\), we obtain the following “quantization rules”:
\[ \Phi_1+\Phi_2+\Phi_3=\left(n+\frac12\right)\pi,\qquad \Phi_1=i\int_{L_1}p_1\,dz-i\int_{L_2}p_2\,dz, \]
\[ \Phi_2=\int_{L_3}p_1\,dz+\int_{L_4}p_2\,dz,\qquad \Phi_3=-i\int_{L'_1}p_1\,dz+i\int_{L'_2}p_2\,dz, \tag{16} \]
\[ p_1=\sqrt{\frac{\beta^2}{2\alpha}\left(U_2-\sqrt{4U_1\alpha/\beta^2}\right)},\qquad p_2=\sqrt{\frac{\beta^2}{2\alpha}\left(U_2+\sqrt{4U_1\alpha/\beta^2}\right)}. \]
Here the contours \(L_1, L_2\), beginning at the point \(A\), go along the real axis and end: the contour \(L_1\) at the point \(a_1\), and \(L_2\) at the point \(a_2\). The contours \(L_3, L_4\) begin respectively at the points \(a_1, a_2\), then descend to the real axis, go along it, and end respectively at the points \(b_1, b_2\). The contours \(L'_1, L'_2\) are analogous to the contours \(L_1, L_2\), with \(a_1, a_2\) replaced by \(b_1, b_2\) and \(A\) by \(B\). The expressions for \(p_1, p_2\) in the vicinity of the points \(O_1, O_2\) pass respectively into \(w_1, w_2\).
It is not difficult to verify that the quantities \(\Phi_1,\Phi_2,\Phi_3\) defined in (18) are purely real. Consider, for example, \(\Phi_2\). Taking into account that on the real axis \(L_3\) and \(L_4\) coincide, we have: \(\int_{O_1}^{O_2}(p_1+p_2)\,dy\) is purely real. Moreover,
\[ \int_{a_2}^{O_1}\sqrt{z+ia}\,dz=(ia)^{3/2},\qquad \int_{a_1}^{O_1}\sqrt{z-ia}\,dz=(-ia)^{3/2}. \]
From this the assertion made immediately follows.
We express our gratitude to A. A. Galeev, I. B. Khriplovich, and V. N. Oraevsky for valuable discussions.
Novosibirsk State University
Received
23 IV 1964
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