WKB Method for Nonlinear Equations of Emden–Fowler Type

Abstract

A method of asymptotic integration based on application of a generalized Prüfer transformation is elaborated for a class of Emden–Fowler type equations, and a relationship with two-scale expansion method is established.

We study the asymptotic behavior at infinity of oscillating solutions to the Emden–Fowler type equations (see [1])

$$u{\kern 1pt} ''(t) + q(t){\text{|}}u(t){{{\text{|}}}^{{\lambda - 1}}}u(t) = 0,$$

(1)

where $q(t) > 0$ for $t \in {{\mathbb{R}}_{ + }}$ and λ > 0 is a parameter. Let

$$\alpha (t): = q{\kern 1pt} '(t) \cdot q{{(t)}^{{ - (\lambda + 5)/(\lambda + 3)}}}.$$

Throughout this paper, we consider the class of Eqs. (1) for which

$$\mathop {\lim \,}\limits_{t \to + \infty } \alpha (t) = 0,\quad \int\limits_{{{\mathbb{R}}_{ + }}} {\text{|}}d\alpha (t){\text{|}} < \infty .$$

(2)

1 REFERENCE EQUATION AND PRÜFER TRANSFORMATION

Let w(x) denote a solution of the reference equation

$$w{\kern 1pt} '{\kern 1pt} '(x)\; + \;{\text{|}}w(x){{{\text{|}}}^{{\lambda - 1}}}w(x) = 0$$

that satisfies the initial conditions $w(0)$ = 0 and $w{\kern 1pt} '(0)$  = 1. Here, w(x) is a periodic function with amplitude A(λ) = ${{\left( {\frac{{\lambda + 1}}{2}} \right)}^{{1/(\lambda + 1)}}}$ and period T(λ) = $4\frac{{A(\lambda )}}{{\lambda \, + \,1}}B\left( {\frac{1}{{\lambda \, + \,1}},\frac{1}{2}} \right);$ moreover,

$$w{\kern 1pt} '{{(x)}^{2}} + \frac{2}{{\lambda + 1}}{\text{|}}w(x){{{\text{|}}}^{{\lambda + 1}}} \equiv 1.$$

(3)

To obtain asymptotic representations of solutions to Eq. (1), we use the generalized Prüfer transformation (cf. [2])

$$u(t) = q{{(t)}^{{ - 1/(\lambda + 3)}}}\rho (t)w(\theta (t)),$$

(4)

$$u{\kern 1pt} '(t) = q{{(t)}^{{1/(\lambda + 3)}}}\rho {{(t)}^{{(\lambda + 1)/2}}}w{\kern 1pt} '(\theta (t)),$$

(5)

where $\rho (t),\theta (t)$ are new action–angle variables. An important role is played by the quantity

$$\begin{gathered} \rho {{(t)}^{{\lambda + 1}}} = q{{(t)}^{{ - 2/(\lambda + 3)}}}u{\kern 1pt} '{{(t)}^{2}} + \\ \, + \frac{2}{{\lambda + 1}}q{{(t)}^{{(\lambda + 1)/(\lambda + 3)}}}{\text{|}}u(t){{{\text{|}}}^{{\lambda + 1}}} = :I[u](t), \\ \end{gathered} $$

(6)

which is an analogue of the Ehrenfest adiabatic invariant (see, e.g., [3]). Note that substitution (4), (5) is nondegenerate, and its Jacobian is given by

$$\frac{{\partial (u,u{\kern 1pt} ')}}{{\partial (\rho ,\theta )}} = - \frac{{\lambda + 1}}{2}\rho {{(t)}^{{(\lambda + 1)/2}}},$$

where $\theta (t)$ and $\rho (t)$ satisfy the system of differential equations

$$\theta {\kern 1pt} '(t) = q{{(t)}^{{2/(\lambda + 3)}}}\rho {{(t)}^{{(\lambda - 1)/2}}} + \frac{1}{{\lambda + 1}}\frac{{q{\kern 1pt} '(t)}}{{q(t)}}w(\theta (t))w{\kern 1pt} '(\theta (t)),$$

$$\frac{{\rho {\kern 1pt} '(t)}}{{\rho (t)}} = - \frac{2}{{(\lambda + 1)(\lambda + 3)}}\frac{{q{\kern 1pt} '(t)}}{{q(t)}}\left( {1 - \frac{{\lambda + 3}}{{\lambda + 1}}{\text{|}}w(\theta (t)){{{\text{|}}}^{{\lambda + 1}}}} \right).$$

In this context, we note that the generalized Liouville transformation

$$\begin{gathered} u(t) = Cq{{(t)}^{{ - 1/(\lambda + 3)}}}{v}(\xi (t)), \\ \xi (t) = {{C}^{{(\lambda - 1)/2}}}\int\limits_0^t q{{(s)}^{{2/(\lambda + 3)}}}ds, \\ \end{gathered} $$

where C > 0, reduces Eq. (1) to the form

$$\begin{gathered} v{\kern 1pt} '{\kern 1pt} '(\xi (t))\; + \;{\text{|}}v(\xi (t)){{{\text{|}}}^{\lambda }}{\text{sgn}}\,v(\xi (t)) \\ \, + {{C}^{{1 - \lambda }}}\{ q{{(t)}^{{ - 3/(\lambda + 3)}}}(q{{(t)}^{{ - 1/(\lambda + 3)}}}){\kern 1pt} '{\kern 1pt} '\} v(\xi (t)) = 0. \\ \end{gathered} $$

If the expression in curly brackets is sufficiently small at infinity in a certain sense, then it is natural to expect that the solution $v(\xi (t))$ of the resulting equation is asymptotically equivalent to the solution $w(\xi (t))$ of the reference equation.

2 ADIABATIC INVARIANT AND ACTION–ANGLE VARIABLES

Statement 1. Under condition (2), for any solution $u(t)$ of Eq. (1), there exists a limit

$$\mathop {\lim }\limits_{t \to + \infty } I[u](t) = :J[u]$$

which is finite or infinite. Moreover, $J[u] > 0$ if $\lambda \leqslant 1,$ and $J[u] < \infty$ if $\lambda \geqslant 1.$

Indeed, if $u(t)$ is a solution of Eq. (1), then

$$dI[u](t) = - \frac{2}{{\lambda + 3}}\alpha (t)d(u(t)u{\kern 1pt} '(t));$$

thus, for arbitrary $t \geqslant s \geqslant 0$, it is true that

$$\begin{gathered} I[u](t) = I[u](s) \\ \, + \frac{2}{{\lambda + 3}}(\alpha (s)u(s)u{\kern 1pt} '(s) - \alpha (t)u(t)u{\kern 1pt} '(t)) \\ \, + \frac{2}{{\lambda + 3}}\int\limits_s^t u(r)u{\kern 1pt} '(r)d\alpha (r). \\ \end{gathered} $$

(7)

In view of the inequality

$${\text{|}}u(t)u{\kern 1pt} '(t){\text{|}} \leqslant A(\lambda )I[u](t{{)}^{{(\lambda + 3)/2(\lambda + 1)}}},$$

(8)

which follows from (6), the boundedness of $I[u](t)$, in view of conditions (2), implies the existence of a limit $J[u] \geqslant 0$. If $J[u] = 0,$ then combining (7) and (8) yields the estimate

$$I[u](t{{)}^{{(\lambda - 1)/2(\lambda + 1)}}} \leqslant \frac{8}{{\lambda + 3}}{{(\lambda + 1)}^{{1/(\lambda + 1)}}}\int\limits_t^\infty {\text{|}}d\alpha (s){\text{|}},$$

(9)

which holds only if $\lambda > 1$ (cf. [4]). Thus, in this case, we have the inequality

$$I[u](t) \leqslant M(\lambda ){{\left( {\int\limits_t^\infty {\text{|}}d\alpha (s){\text{|}}} \right)}^{{2(\lambda + 1)/(\lambda - 1)}}}$$

(10)

with constant M(λ) = (λ + 1)^{2/(λ – 1)}(8/(λ + ${{3))}^{{2(\lambda + 1)/(\lambda - 1)}}}$. If $I[u](t)$ is unbounded as $t \to \infty ,$ then, once again, by virtue of (7) and (8), we have estimate (9); moreover, it is necessary that $\lambda < 1$, and, accordingly, in this case,

$$I[u](t) \geqslant M(\lambda ){{\left( {\int\limits_t^\infty {\text{|}}d\alpha (s){\text{|}}} \right)}^{{2(\lambda + 1)/(\lambda - 1)}}}.$$

(11)

Statement 2. If $0 < J[u] < \infty ,$ then the corresponding solution $u(t)$ of Eq. (1)has the form of (4), (5), where

$$\mathop {\lim \,}\limits_{t \to + \infty } \rho (t) = {{\rho }_{\infty }} = J{{[u]}^{{1/(\lambda + 1)}}},$$

$$\theta (t) = (\rho _{\infty }^{{(\lambda - 1)/2}} + o(1))\int\limits_0^t q{{(s)}^{{2/(\lambda + 3)}}}ds.$$

Indeed, the first limit relation follows directly from Statement 1 and equality (6), while, to prove the second relation, we preliminarily note that, under the assumptions made,

$$\int\limits_0^\infty q{{(t)}^{{2/(\lambda + 3)}}}dt = \infty .$$

Assuming the opposite, in view of (2), we conclude that the integral

$$\int\limits_{}^\infty \alpha (t)q{{(t)}^{{2/(\lambda + 3)}}}dt = \int\limits_{}^\infty d\ln q(t)$$

converges. The last is possible only if there exists $\mathop {\lim \,}\limits_{t \to + \infty } q(t) > 0,$ which is obviously incompatible with the assumption made. From this, by virtue of the differential equation for $\theta (t)$ and the existence of the first integral (3), it follows that $\theta (t) \to \infty$ as $t \to \infty$; moreover, we have the limit relation

$$\begin{gathered} \mathop {\lim }\limits_{t \to + \infty } \theta (t){{\left( {\int\limits_0^t q{{{(s)}}^{{2/(\lambda + 3)}}}ds} \right)}^{{ - 1}}}\, = \,\mathop {\lim }\limits_{t \to + \infty } \theta {\kern 1pt} '(t)q{{(t)}^{{ - 2/(\lambda + 3)}}}\, \\ \, = \mathop {\lim }\limits_{t \to + \infty } \left( {\rho {{{(t)}}^{{(\lambda - 1)/2}}}\, + \,\frac{1}{{\lambda + 1}}\alpha (t)w(\theta (t))w{\kern 1pt} '(\theta (t))} \right)\, = \,\rho _{\infty }^{{(\lambda - 1)/2}}. \\ \end{gathered} $$

Corollary 1. For $\lambda = 1,$ formulas (4) and (5) yield well-known WKB asymptotics for solutions of the linear equation

$$u(t) = q{{(t)}^{{ - 1/4}}}\sin \left( {(1 + o(1))\int\limits_0^t \sqrt {q(s)} ds} \right)(1 + o(1)),$$

$$u'(t) = q{{(t)}^{{1/4}}}\cos \left( {(1 + o(1))\int\limits_0^t \sqrt {q(s)} ds} \right)(1 + o(1)).$$

Statement 3. For an arbitrary ${{\rho }_{\infty }} > 0,$ Eq. (1)has a solution $u(t)$ of the form (4), (5), where

$$\begin{gathered} \mathop {\lim }\limits_{t \to + \infty } \rho (t) = {{\rho }_{\infty }}, \\ \theta (t) = (\rho _{\infty }^{{(\lambda - 1)/2}} + o(1))\int\limits_0^t q{{(s)}^{{2/(\lambda + 3)}}}ds. \\ \end{gathered} $$

(12)

Indeed, by Statement 2, it suffices to show that the functional $J[u]$ takes all possible positive values. For fixed $C > 0$ and $\varepsilon > 0,$ we choose $T > 0$ such that

$${{(C + \varepsilon )}^{{(\lambda + 3)/2(\lambda + 1)}}}\frac{{A(\lambda )}}{{\lambda + 3}}\int\limits_T^\infty {\text{|}}d\alpha (t){\text{|}} < \frac{\varepsilon }{4}.$$

Let $u(t)$ be a solution of Eq. (1) satisfying the condition $I[u](T) = C.$ Then, according to (7) and (8), for $t \geqslant T,$ we have ${\text{|}}I[u](t) - C{\text{|}} < \varepsilon$ and, hence, ${\text{|}}J[u] - C{\text{|}} \leqslant \varepsilon .$ Now we choose ${{C}_{0}},{{C}_{1}}$, and $\varepsilon$ such that C₀ + ε < C < ${{C}_{1}} - \varepsilon ,$ and construct solutions ${{u}_{0}}(t)$ and ${{u}_{1}}(t)$ of Eq. (1) for which $J[{{u}_{0}}] \leqslant {{C}_{0}} + \varepsilon$ and $J[{{u}_{1}}] \geqslant {{C}_{1}} - \varepsilon ,$ respectively. Let ${{u}_{\tau }}(t)$ be the one-parameter family of solutions to Eq. (1) with initial conditions u_τ(0) = $(1 - \tau ){{u}_{0}}(0) + \tau {{u}_{1}}(0)$ and $u_{\tau }^{'}(0) = (1 - \tau )u_{0}^{'}(0) + \tau u_{1}^{'}(0),$ where $\tau \in [0,1].$ By the theorem on continuous dependence of the solution of the Cauchy problem on initial data, $J[{{u}_{\tau }}]$ is a continuous function of the parameter τ and, hence, it takes the given value C on the interval [0, 1]

Along with $I[u](t),$ for the solution $u(t)$ of Eq. (1), we introduce the quantity

$$\begin{gathered} E[u](t): = q{{(t)}^{{ - (\lambda + 1)/(\lambda + 3)}}}I[u](t) \\ = \frac{1}{{q(t)}}u{\kern 1pt} '{{(t)}^{2}} + \frac{2}{{\lambda + 1}}{\text{|}}u(t){{{\text{|}}}^{{\lambda + 1}}}, \\ \end{gathered} $$

whose logarithmic derivative is given by (lnE[u](t))' = $- \frac{{u{\kern 1pt} '{{{(t)}}^{2}}}}{{E[u](t)}}\frac{{q{\kern 1pt} '(t)}}{{q{{{(t)}}^{2}}}},$ and, as a consequence,

$$E[u](t){\text{/}}E[u](s) = \exp \left( { - \int\limits_s^t \frac{{u{\kern 1pt} '{{{(r)}}^{2}}}}{{E[u](r)}}\frac{{dq(r)}}{{q{{{(r)}}^{2}}}}} \right).$$

Let

$${{q}_{ \pm }}(t): = \pm \frac{{q(0)}}{2} + \frac{1}{2}\int\limits_0^t ({\text{|}}q{\kern 1pt} '(s){\text{|}} \pm q{\kern 1pt} '(s))ds$$

so that $q(t) = {{q}_{ + }}(t) - {{q}_{ - }}(t),$ where $q_{ \pm }^{'}(t) \geqslant 0.$ Thus, the following result is true.

Statement 4. For arbitrary $s \leqslant t$, the following two-sided estimate holds:

$$\begin{gathered} E[u](s)\exp \left( { - \int\limits_s^t \frac{{d{{q}_{ + }}(r)}}{{q(r)}}} \right) \leqslant E[u](t) \\ \leqslant E[u](s)\exp \left( {\int\limits_s^t \frac{{d{{q}_{ - }}(r)}}{{q(r)}}} \right). \\ \end{gathered} $$

(13)

3 EQUATIONS WITH QUALIFIED ASYMPTOTICS OF SOLUTIONS

Combining Statements 1–4 with estimates (10) and (11) in the cases $\lambda > 1$ and $\lambda < 1$, respectively, we formulate the conditions under which all solutions of Eq. (1) have the form of (4), (5).

Theorem 1. Suppose that $q(t) \in {{{\text{C}}}^{2}}({{\mathbb{R}}_{ + }})$ and condition (2) is satisfied. If $\lambda < 1,$ then assume additionally that

$$\begin{gathered} q{{(t)}^{{(\lambda + 1)/(\lambda + 3)}}}{{\left( {\int\limits_t^\infty {\text{|}}d\alpha (s){\text{|}}} \right)}^{{2(\lambda + 1)/(1 - \lambda )}}} \\ \, \times \exp \left( {\int\limits_0^t \frac{{d{{q}_{ - }}(s)}}{{q(s)}}} \right) \to 0,\quad t \to \infty , \\ \end{gathered} $$

(14)

while, if $\lambda > 1,$ then assume that

$$\begin{gathered} q{{(t)}^{{ - (\lambda + 1)/(\lambda + 3)}}}{{\left( {\int\limits_t^\infty {\text{|}}d\alpha (s){\text{|}}} \right)}^{{2(\lambda + 1)/(\lambda - 1)}}} \\ \, \times \exp \left( {\int\limits_0^t \frac{{d{{q}_{ + }}(s)}}{{q(s)}}} \right) \to 0,\quad t \to \infty . \\ \end{gathered} $$

(15)

Then any solution $u(t)$ of Eq. (1)has the form of (4), (5), where $\rho (t)$ and $\theta (t)$ satisfy asymptotic relations (12).

In fact, the conditions of Theorem 1 mean some regularity of the behavior of the coefficient $q(t)$ at infinity. If $q(t)$ increases monotonically, then condition (2) in Theorem 1 can be replaced by the assumption that the integral

$${{\widehat S}_{\lambda }}(t): = \int\limits_t^\infty {\text{|}}d(q{\kern 1pt} '(s)q{{(s)}^{{ - 3/2}}}){\text{|}} < \infty ,\quad \lambda \leqslant 1,$$

$${{\widehat S}_{\lambda }}(t): = \int\limits_t^\infty {\text{|}}d(q{\kern 1pt} '(s)q{{(s)}^{{ - (\lambda + 2)/(\lambda + 1)}}}){\text{|}} < \infty ,\quad \lambda \geqslant 1,$$

converges; then the additional constraints (14), (15) hold automatically (see [4]). The class of Eqs. (1) satisfying the condition ${{\widehat S}_{\lambda }}(t) < \infty$ with solutions $u(t)$ representable by asymptotic formulas (4) and (5) can be expanded so that the component ${{q}_{ - }}(t)$ in the corresponding expansion $q(t) = {{q}_{ + }}(t) - {{q}_{ - }}(t)$ is subordinate, in a certain sense, to ${{q}_{ + }}(t).$

Theorem 2. Suppose that ${{\widehat S}_{\lambda }}(t) < \infty ,$ there exists a    monotonically increasing function $\hat {q}(t)$ such that $q(t) \geqslant \hat {q}(t)$ > 0, the ratio $\frac{{q(t)}}{{\hat {q}(t)}}$ is bounded, and

$${{\widehat S}_{\lambda }}{{(t)}^{{2(\lambda + 1)/|\lambda - 1|}}}\exp \left( {\int\limits_0^t \frac{{d{{q}_{ - }}(s)}}{{q(s)}}} \right) \to 0,\quad t \to \infty .$$

(16)

Then any solution $u(t)$ of Eq. (1) has the form of (4), (5), where $\rho (t)$ and $\theta (t)$ satisfy asymptotic relations (12).

For $\lambda < 1,$ the convergence of the integral ${{\widehat S}_{\lambda }}(t)$ obviously implies the existence of a finite limit $\mathop {\lim }\limits_{t \to + \infty } q{\kern 1pt} '(t)q{{(t)}^{{ - 3/2}}}$ = c. Indeed, if $c > 0,$ then $q{\kern 1pt} '(t)q{{(t)}^{{ - 3/2}}}$ > c/2 starting at some sufficiently large ${{t}_{0}},$ which, after integration, immediately leads to a contradiction:

$$2(q{{({{t}_{0}})}^{{ - 1/2}}} - q{{(t)}^{{ - 1/2}}}) > \frac{c}{2}(t - {{t}_{0}}),\quad t \geqslant {{t}_{0}}.$$

If $c < 0,$ then $q{\kern 1pt} '(t)q{{(t)}^{{ - 3/2}}} < \frac{c}{2}$ for $t \geqslant {{t}_{0}}$; therefore,

$$2(q{{({{t}_{0}})}^{{ - 1/2}}} - q{{(t)}^{{ - 1/2}}}) < \frac{c}{2}(t - {{t}_{0}}),\quad t \geqslant {{t}_{0}},$$

whence $q(t) \to 0$ as $t \to \infty$. The last is incompatible with the existence of a monotonically increasing positive function $\hat {q}(t) \leqslant q(t)$; thus, c = 0 and, a fortiori, $\alpha (t) \to 0$ as $t \to \infty ,$ since $q(t)$ is bounded away from zero. Sending $t$ to infinity in the equality

$$\begin{gathered} \alpha (t) - \alpha (0) = \int\limits_0^t q{{(s)}^{{(\lambda - 1)/2(\lambda + 3)}}}d(q{\kern 1pt} '(s)q{{(s)}^{{ - 3/2}}}) \\ \, + \frac{{\lambda - 1}}{{2(\lambda + 3)}}\int\limits_0^t q{\kern 1pt} '{{(s)}^{2}}q{{(s)}^{{ - 2(\lambda + 4)/(\lambda + 3)}}}ds \\ \end{gathered} $$

and taking into account the condition ${{\widehat S}_{\lambda }}(t) < \infty ,$ we conclude that

$$\int\limits_{}^\infty {q{\kern 1pt} '{{{(s)}}^{2}}q{{{(s)}}^{{ - 2(\lambda + 4)/(\lambda + 3)}}}ds < \infty } ;$$

(17)

hence, the assumptions of the theorem ensure that condition (2) holds. Finally, we note that

$$\begin{gathered} \int\limits_t^\infty {\text{|}}d\alpha (s){\text{|}} \leqslant \hat {q}{{(t)}^{{(\lambda - 1)/2(\lambda + 3)}}}{{\widehat S}_{\lambda }}(t) \\ \, + \frac{{1 - \lambda }}{{2(\lambda + 3)}}\int\limits_t^\infty q{\kern 1pt} '{{(s)}^{2}}q{{(s)}^{{ - 2(\lambda + 4)/(\lambda + 3)}}}ds \\ \leqslant 2\hat {q}{{(t)}^{{(\lambda - 1)/2(\lambda + 3)}}}{{\widehat S}_{\lambda }}(t) + \alpha (t) \leqslant 3\hat {q}{{(t)}^{{(\lambda - 1)/2(\lambda + 3)}}}{{\widehat S}_{\lambda }}(t), \\ \end{gathered} $$

since

$$\alpha (t) = - q{{(t)}^{{(\lambda - 1)/2(\lambda + 3)}}}\int\limits_t^\infty d(q'(s)q{{(s)}^{{ - 3/2}}}).$$

Therefore, since the ratio $\frac{{q(t)}}{{\hat {q}(t)}}$ is bounded, condition (16) ensures that (14) holds and, thus, Theorem 1 is applicable in this case.

By analogy with the previous case, for $\lambda > 1$, the convergence of the integral ${{\widehat S}_{\lambda }}(t)$ implies that $\mathop {\lim }\limits_{t \to + \infty } q{\kern 1pt} '(t)q{{(t)}^{{ - (\lambda + 2)/(\lambda + 1)}}}$ = 0. Therefore, since $q(t)$ is bounded away from zero, we have $\alpha (t) \to 0$ as $t \to \infty .$ Passing to the limit as $t \to \infty$ in the equality

$$\begin{gathered} \alpha (t) - \alpha (0) = \int\limits_0^t q{{(s)}^{{(1 - \lambda )/(\lambda + 1)(\lambda + 3)}}}d(q{\kern 1pt} '(s)q{{(s)}^{{ - (\lambda + 2)/(\lambda + 1)}}}) \\ \, + \frac{{1 - \lambda }}{{2(\lambda + 3)}}\int\limits_0^t q{\kern 1pt} '{{(s)}^{2}}q{{(s)}^{{ - 2(\lambda + 4)/(\lambda + 3)}}}ds \\ \end{gathered} $$

with use of the condition ${{\widehat S}_{\lambda }}(t) < \infty ,$ we again obtain (17) and, hence, condition (2) is satisfied. Next, we establish the estimate

$$\begin{gathered} \int\limits_t^\infty {\text{|}}d\alpha (s){\text{|}} \leqslant \hat {q}{{(t)}^{{(1 - \lambda )/(\lambda + 1)(\lambda + 3)}}}{{{\hat {S}}}_{\lambda }}(t) \\ \, + \frac{{\lambda - 1}}{{(\lambda + 1)(\lambda + 3)}}\int\limits_t^\infty q{\kern 1pt} '{{(s)}^{2}}q{{(s)}^{{ - 2(\lambda + 4)/(\lambda + 3)}}}ds \\ \leqslant \,2\hat {q}{{(t)}^{{(1 - \lambda )/(\lambda + 1)(\lambda + 3)}}}{{{\hat {S}}}_{\lambda }}(t)\, + \,\alpha (t)\, \leqslant \,3\hat {q}{{(t)}^{{(1 - \lambda )/(\lambda + 1)(\lambda + 3)}}}{{{\hat {S}}}_{\lambda }}(t), \\ \end{gathered} $$

where

$$\alpha (t) = - q{{(t)}^{{(1 - \lambda )/(\lambda + 1)(\lambda + 3)}}}\int\limits_t^\infty d(q{\kern 1pt} '(s)q{{(s)}^{{ - (\lambda + 2)/(\lambda + 1)}}}).$$

Finally, taking into account the boundedness of the ratio $\frac{{q(t)}}{{\hat {q}(t)}}$ and the equality

$$q(t)\exp \left( {\int\limits_0^t \frac{{d{{q}_{ - }}(s)}}{{q(s)}}} \right) = q(0)\exp \left( {\int\limits_0^t \frac{{d{{q}_{ + }}(s)}}{{q(s)}}} \right),$$

we conclude that, in the case under consideration, when $\lambda > 1,$ condition (16) ensures that (15) holds; thus, under the assumptions made, Theorem 1 is again applicable. The class of coefficients of Eq. (1) that satisfy the conditions of Theorem 2 consists of $q(t)$ such that ${{\widehat S}_{\lambda }}(t) < \infty$ and $\int\limits_{}^\infty {\frac{{d{{q}_{ - }}(s)}}{{q(s)}}} < \infty .$

Somewhat different are the conditions ensuring the asymptotic behavior (4), (5) for solutions of Eq. (1) of the considered type if the coefficient q(t) is monotonically decreasing (see [4]). In this case, condition (2) in Theorem 1 can be replaced by the requirement for the convergence of the integral

$${{\widetilde S}_{\lambda }}(t): = \int\limits_t^\infty {\text{|}}d(q{\kern 1pt} '(s)q{{(s)}^{{ - (\lambda + 2)/(\lambda + 1)}}}){\text{|}} < \infty ,\quad \lambda \leqslant 1,$$

$${{\widetilde S}_{\lambda }}(t): = \int\limits_t^\infty {\text{|}}d(q{\kern 1pt} '(s)q{{(s)}^{{ - 3/2}}}){\text{|}} < \infty ,\quad \lambda \geqslant 1,$$

and, once again, additional constraints (14), (15) hold automatically. The class of Eqs. (1) satisfying the condition ${{\widetilde S}_{\lambda }}(t) < \infty$ with solutions $u(t)$ representable by asymptotic formulas (4), (5) is stable with respect to a small (in a sense) perturbation of the monotonicity property of the coefficient $q(t).$

Theorem 3. Suppose that ${{\widetilde S}_{\lambda }}(t) < \infty ,$ there exists a monotonically decreasing function $\tilde {q}(t)$ such that $\tilde {q}(t) \geqslant q(t) > 0$ and the ratio $\frac{{\tilde {q}(t)}}{{q(t)}}$ is bounded,

$${{\widetilde S}_{\lambda }}{{(t)}^{{2(\lambda + 1)/|\lambda - 1|}}}\exp \left( {\int\limits_0^t \frac{{d{{q}_{ + }}(s)}}{{q(s)}}} \right) \to 0,\quad t \to \infty ,$$

and, additionally, $\mathop {\lim }\limits_{t \to + \infty } q{\kern 1pt} '(t)q{{(t)}^{{ - (\lambda + 2)/(\lambda + 1)}}} = 0$ for $\lambda < 1$ and $\mathop {\lim }\limits_{t \to + \infty } q{\kern 1pt} '(t)q{{(t)}^{{ - 3/2}}} = 0$ for $\lambda > 1.$ Then any solution $u(t)$ of Eq. (1) has the form of (4), (5), where $\rho (t)$ and $\theta (t)$ satisfy asymptotic relations (12).

It is well known (see, e.g., [4]) that the convergence of the integral

$$\int\limits_{}^\infty {{{t}^{{2 - \kappa + \lambda (\kappa - 1)}}}q(t)dt} < \infty ,\quad \kappa = 1,2,$$

ensures the existence of a solution of Eq. (1) with asymptotics

$$u(t) = {{t}^{{\kappa - 1}}}(1 + o(1)),\quad t \to \infty .$$

In view of this, we conclude that the conditions of Theorem 3 are tight in the power scale of coefficients $q(t) = {{t}^{\gamma }}.$ Indeed, for $\lambda \geqslant 1$, the condition ${{\widetilde S}_{\lambda }}(t) < \infty$ holds if $\gamma > - 2,$ but, if $\gamma < - 2$, then the corresponding equation (1) has a solution of the form $u(t) = 1 + o(1),$ $t \to \infty ,$ for which $J[u] = 0$ and asymptotic formulas (4), (5) do not hold. In turn, for $\lambda \leqslant 1$, the condition ${{\widetilde S}_{\lambda }}(t) < \infty$ is satisfied if $\gamma > - (\lambda + 1),$ but, if γ < $- (\lambda + 1)$, then Eq. (1) has a solution with asymptotics $u(t) = t(1 + o(1)),$ $t \to \infty ,$ for which $J[u] = \infty$ and once again representation (4), (5) does not hold. Finally, for λ = 1 and $\gamma = - 2$ in the case of linear equation (1), the WKB formula is also not applicable for describing the behavior of its solution u(t) = $\sqrt t {\text{sin}}(\sqrt 3 t{\text{/}}2)$.

4 RELATIONSHIP WITH THE TWO-SCALE EXPANSION METHOD

Within the framework of a nonlinear modification of the WKB method (see [5, 6]), a solution of the equation

$${{\varepsilon }^{2}}u{\kern 1pt} '{\kern 1pt} '(t) + f(t,u) = 0$$

(18)

with small values of the parameter $\varepsilon$ is sought in the form of an asymptotic series

$$u(t,\varepsilon ) = \sum\limits_{n = 0}^\infty {{\varepsilon }^{n}}{{u}_{n}}(\xi (t){\text{/}}\varepsilon ,t).$$

(19)

This approach is known as the two-scale expansion method, where, in addition to the initial “slow” time t, the “fast” variable $\tau = \xi (t){\text{/}}\varepsilon$ is introduced. By substituting the two-scale expansion (19) into (18) and setting the coefficients of successive powers of $\varepsilon$ to zero, we derive a recurrence system of equations for ${{u}_{n}}(\tau ,t),$ the first two of which are given by

$$\xi '{{(t)}^{2}}\frac{{{{\partial }^{2}}{{u}_{0}}}}{{\partial {{\tau }^{2}}}} + f(t,{{u}_{0}}) = 0,$$

(20)

$$\xi {\kern 1pt} '{{(t)}^{2}}\frac{{{{\partial }^{2}}{{u}_{1}}}}{{\partial {{\tau }^{2}}}} + \frac{{\partial f}}{{\partial u}}(t,{{u}_{0}}){{u}_{1}} = - 2\xi {\kern 1pt} '(t)\frac{{{{\partial }^{2}}{{u}_{0}}}}{{\partial \tau \partial t}} - \xi {\kern 1pt} '{\kern 1pt} '(t)\frac{{\partial {{u}_{0}}}}{{\partial \tau }}.$$

(21)

Equation (20) contains two unknown functions, ${{u}_{0}}(\tau ,t)$ and $\xi (t),$ which can be found only by analyzing Eq. (21) for the first correction term ${{u}_{1}}(\tau ,t).$ When Eq. (20) has a periodic solution ${{u}_{0}}(\tau ,t)$ with a period $T = T(t),$ Eq. (21) has a $T$-periodic solution ${{u}_{1}}(\tau ,t)$ if and only if

$$\int\limits_0^T \frac{{\partial {{u}_{0}}}}{{\partial \tau }}\left( {2\xi {\kern 1pt} '(t)\frac{{{{\partial }^{2}}{{u}_{0}}}}{{\partial \tau \partial t}} + \xi {\kern 1pt} '{\kern 1pt} '(t)\frac{{\partial {{u}_{0}}}}{{\partial \tau }}} \right)d\tau = 0$$

(22)

(see, e.g., [6]). Thus, to determine ${{u}_{0}}(\tau ,t)$ and $\xi (t),$ Eq. (20) has to be supplemented with condition (22). By applying this approach, we can effectively find the leading term ${{u}_{0}}(\xi (t{\text{)/}}\varepsilon ,t)$ of asymptotic expansion (19), which is a nonlinear analogue of the WKB asymptotics. Of course, for the Emden–Fowler type equation (18), the corresponding result agrees with formulas (4), (5).

Proposition 1. If $f(t,u) = q(t){\text{|}}u{{{\text{|}}}^{\lambda }}{\text{sgn}}u,$ where $q(t) > 0,$ for an arbitrary $C > 0$, Eq. (18)has a formal asymptotic solution (19) with the leading term

$$\begin{gathered} {{u}_{0}}(\xi (t){\text{/}}\varepsilon ,t) = Cq{{(t)}^{{ - 1/(\lambda + 3)}}}w(\xi (t){\text{/}}\varepsilon ), \\ \xi (t) = {{C}^{{(\lambda - 1)/2}}}\int\limits_0^t q{{(s)}^{{2/(\lambda + 3)}}}ds, \\ \end{gathered} $$

where $w(x)$ is a solution of the reference equation.

In the case under consideration, a solution of Eq. (20) is sought in the form ${{u}_{0}}(\tau ,t) = a(t)w(\tau ),$ where $a(t) > 0.$ Moreover, $\xi {\kern 1pt} '{{(t)}^{2}} = q(t)a{{(t)}^{{\lambda - 1}}},$ and relation (22) transforms into

$$(a{{(t)}^{2}}\xi {\kern 1pt} '(t)){\kern 1pt} '\int\limits_0^T w{\kern 1pt} '{{(\tau )}^{2}}d\tau = 0,$$

which implies that $a{{(t)}^{2}}\xi {\kern 1pt} '(t) = {\text{const}}$ is a constant parametrizing the family of formal asymptotic solutions. Fixing this constant, we find

$$a(t) = Cq{{(t)}^{{ - 1/(\lambda + 3)}}},\quad \xi {\kern 1pt} '(t) = {{C}^{{(\lambda - 1)/2}}}q{{(t)}^{{2/(\lambda + 3)}}}.$$

Thus, for the leading term of asymptotic expansion (19), we have obtained the closed-form formula ${{u}_{0}}(\xi (t){\text{/}}\varepsilon ,t) = a(t)w(\xi (t){\text{/}}\varepsilon )$, which agrees with (4), (5).

The next perturbative corrections can be determined from a recurrence system of inhomogeneous linear equations for ${{u}_{n}}(\tau ,t)$ in which the slow time t plays the role of a parameter. For applications of the two-scale expansion method, see, for example, [7, 8].