<p>This paper contains several technical refinements of our previously obtained results on the monodromy parametrisation of small-<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation> asymptotics of solutions <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$u(\tau )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of the degenerate third Painlevé equation, <Equation ID="Equ131"> <EquationSource Format="TEX">$ u^{\prime \prime }(\tau ) \! = \! \frac{(u^{\prime }(\tau ))^{2}}{u(\tau )} \! - \! \frac{u^{\prime }(\tau )}{\tau } \! + \! \frac{1}{\tau } \! \left( -8 \varepsilon (u(\tau ))^{2} \! + \! 2ab \right) \! + \! \frac{b^{2}}{u(\tau )}, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msup> <mi>u</mi> <mo>″</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="-0.166667em" /> <mo>=</mo> <mspace width="-0.166667em" /> <mfrac> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>u</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> <mspace width="-0.166667em" /> <mo>-</mo> <mspace width="-0.166667em" /> <mfrac> <mrow> <msup> <mi>u</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mi>τ</mi> </mfrac> <mspace width="-0.166667em" /> <mo>+</mo> <mspace width="-0.166667em" /> <mfrac> <mn>1</mn> <mi>τ</mi> </mfrac> <mspace width="-0.166667em" /> <mfenced close=")" open="("> <mo>-</mo> <mn>8</mn> <mi>ε</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mspace width="-0.166667em" /> <mo>+</mo> <mspace width="-0.166667em" /> <mn>2</mn> <mi>a</mi> <mi>b</mi> </mfenced> <mspace width="-0.166667em" /> <mo>+</mo> <mspace width="-0.166667em" /> <mfrac> <msup> <mi>b</mi> <mn>2</mn> </msup> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varepsilon \! = \! \pm 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mspace width="-0.166667em" /> <mo>=</mo> <mspace width="-0.166667em" /> <mo>±</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varepsilon b \! > \! 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mi>b</mi> <mspace width="-0.166667em" /> <mo>></mo> <mspace width="-0.166667em" /> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$a \! \in \! \mathbb {C},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mspace width="-0.166667em" /> <mo>∈</mo> <mspace width="-0.166667em" /> <mi mathvariant="double-struck">C</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and of its associated <i>mole function</i>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varphi (\tau )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, which satisfies <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\varphi ^{\prime }(\tau ) \! = \! \tfrac{2a}{\tau }\! + \! \tfrac{b}{u(\tau )}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>φ</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="-0.166667em" /> <mo>=</mo> <mspace width="-0.166667em" /> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>a</mi> </mrow> <mi>τ</mi> </mfrac> </mstyle> <mspace width="-0.166667em" /> <mo>+</mo> <mspace width="-0.166667em" /> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>b</mi> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mstyle> </mrow> </math></EquationSource> </InlineEquation>. We also describe three families of three-real-parameter solutions <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$u(\tau )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> which have infinite sequences of zeros converging to the origin of the complex <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>-plane. Furthermore, for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$a=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, a numerical visualisation of the formulae connecting the asymptotics as <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\tau \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\tau \rightarrow +\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo stretchy="false">→</mo> <mo>+</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> of solutions <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$u(\tau )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\varphi (\tau )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> having logarithmic behaviour as <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\tau \rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is given. Bibliography: 24 titles.</p>

ASYMPTOTICS OF SOLUTIONS OF THE DEGENERATE THIRD PAINLEVÉ EQUATION IN THE NEIGHBOURHOOD OF THE REGULAR SINGULAR POINT: THE ISOMONODROMY DEFORMATION APPROACH

期刊论文
发表日期：

2026年5月19日

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A. V. Kitaev,
A. Vartanian

摘要

This paper contains several technical refinements of our previously obtained results on the monodromy parametrisation of small- $\tau $ $τ$ asymptotics of solutions $u(\tau )$ $u (τ)$ of the degenerate third Painlevé equation, $ u^{\prime \prime }(\tau ) \! = \! \frac{(u^{\prime }(\tau ))^{2}}{u(\tau )} \! - \! \frac{u^{\prime }(\tau )}{\tau } \! + \! \frac{1}{\tau } \! \left( -8 \varepsilon (u(\tau ))^{2} \! + \! 2ab \right) \! + \! \frac{b^{2}}{u(\tau )}, $ $u^{″} (τ) = \frac{{(u^{'} (τ))}^{2}}{u (τ)} - \frac{u^{'} (τ)}{τ} + \frac{1}{τ} (- 8 ε {(u (τ))}^{2} + 2 a b) + \frac{b^{2}}{u (τ)},$ where $\varepsilon \! = \! \pm 1$ $ε = \pm 1$ , $\varepsilon b \! > \! 0$ $ε b > 0$ , $a \! \in \! \mathbb {C},$ $a \in C,$ and of its associated mole function, $\varphi (\tau )$ $φ (τ)$ , which satisfies $\varphi ^{\prime }(\tau ) \! = \! \tfrac{2a}{\tau }\! + \! \tfrac{b}{u(\tau )}$ $φ^{'} (τ) = \frac{2 a}{τ} + \frac{b}{u (τ)}$ . We also describe three families of three-real-parameter solutions $u(\tau )$ $u (τ)$ which have infinite sequences of zeros converging to the origin of the complex $\tau $ $τ$ -plane. Furthermore, for $a=0$ $a = 0$ , a numerical visualisation of the formulae connecting the asymptotics as $\tau \rightarrow 0$ $τ \to 0$ and $\tau \rightarrow +\infty $ $τ \to + \infty$ of solutions $u(\tau )$ $u (τ)$ and $\varphi (\tau )$ $φ (τ)$ having logarithmic behaviour as $\tau \rightarrow 0$ $τ \to 0$ is given. Bibliography: 24 titles.