Long-time behavior of the solution of the Cauchy problem for the third-order Airy equation

Abstract

The long-time asymptotic behavior of the solution of the Cauchy problem for the evolutionary third-order Airy equation describing wave propagation in dispersive physical media is derived by using the auxiliary parameter method. For the solution in the form of the convolution of the initial data and the Airy function, an asymptotic Erdélyi expansion in inverse powers of the cube root of the time variable with the coefficients depending on a self-similar variable and the logarithm of time is obtained. To refine the asymptotics, a family of special function classes for its coefficients is introduced. It is pointed out how the used method is connected with the geometrical optics approach and how the obtained result can be applied to nonlinear third-order PDEs.

Keywords

Third-order Airy equation linearized KdV equation Cauchy problem Airy function convolution auxiliary parameter method long-time asymptotics Poincaré and Erdélyi expansions self-similarity

1. Introduction

In the present paper, we study the long-time behavior of the solution of the Cauchy problem for the third-order Airy equation: $\begin{matrix} (1.1) & \frac{\partial u}{\partial t} + \frac{\partial^{3} u}{\partial x^{3}} = 0, u (x, 0) = f (x), t ⩾ 0, x \in R, \end{matrix}$ where f is a locally Lebesgue integrable function satisfying the asymptotic relations $\begin{matrix} (1.2) & f (x) = \sum_{n = 0}^{\infty} f_{n}^{\pm} x^{- n}, x \to \pm \infty, \end{matrix}$ in the sense of Poincaré. The partial differential equation (1.1) is used in the theory of wave propagation in dispersive physical media [4] and it can be obtained from the linearized Korteweg–de Vries equation [15] $\begin{matrix} \frac{\partial v}{\partial t} + c \frac{\partial v}{\partial s} + \frac{\partial^{3} v}{\partial s^{3}} = 0 \end{matrix}$ by the change $s = x + c t$ , $v (s, t) = u (x, t)$ . The solution of problem (1.1) is given by the convolution $\begin{matrix} (1.3) & u (x, t) = \frac{1}{\sqrt[3]{3 t}} \int_{- \infty}^{+ \infty} f (y) Ai (\frac{x - y}{\sqrt[3]{3 t}}) d y, \end{matrix}$ with $\begin{matrix} (1.4) & Ai (x) = \frac{1}{π} \int_{0}^{+ \infty} cos (\frac{θ^{3}}{3} + x θ) d θ \end{matrix}$ being the Airy function, and is understood in this case in the sense of the limit ${lim}_{R \to + \infty} \int_{- \infty}^{R} \dots d y$ of the standard Lebesgue integral.

The investigation of the behavior of solution (1.3) is of undoubted interest for the asymptotic analysis of nonlinear dispersive equations, e.g., the KdV equation and its generalizations [3,9,11,12], in particular, for applying the matching method to initial-boundary-value problems, because finding the asymptotics of coefficients of expansions at infinity is one of the main difficulties of the method [8]. Also note that since function (1.4) introduced by G. Airy and the Airy transform (convolution (1.3)) have important applications in wave optics [2] and quantum mechanics [14], in astrophysics [10], and modern laser technologies [13], the technique of the present paper may be useful in these areas as well. There are significant results on asymptotic expansions of integral transforms, e.g., [18] including the Airy transform of the form $\int_{0}^{+ \infty} Ai (s t) f (t) d t$ as $s \to \infty$ . In this connection, it is also worth noting that the investigation of the convolution with the Airy function is necessary for obtaining the asymptotics of eigenfunctions of the Schrödinger operator [7].

The asymptotic behavior of the integral in formula (1.3) can be easily found for large values of t and bounded values of x if $f (x)$ exponentially tends to zero as $x \to \infty$ ; in this case, it suffices to expand the Airy function $Ai (\frac{x - y}{\sqrt[3]{3 t}})$ into Taylor’s series in $y t^{- 1 / 3}$ . However, for the more general conditions (1.2) the problem becomes much more difficult.

2. Auxiliary parameter method

Following the auxiliary parameter method suggested in [5] and applied in [16] to the solution of the heat equation, we represent function (1.3) in the form $\begin{matrix} (2.1) & u (x, t) = U_{1}^{-} (x, t) + U_{0}^{-} (x, t) + U_{0}^{+} (x, t) + U_{1}^{+} (x, t), \end{matrix}$ where $\begin{array}{l} (2.2) & \begin{array}{l} U_{1}^{-} (x, t) = \int_{- \infty}^{- σ} \dots, U_{0}^{-} (x, t) = \int_{- σ}^{0} \dots, U_{0}^{+} (x, t) = \int_{0}^{σ} \dots, \\ U_{1}^{+} (x, t) = \int_{σ}^{+ \infty} \dots, σ = {(| x |^{3} + t)}^{p / 3}, 0 < p < 1, \end{array} \end{array}$ p is an auxiliary arbitrary parameter and dots denote the integrand in the right-hand side of (1.3) together with the factor ${(3 t)}^{- 1 / 3} d y$ . To find the asymptotics of the integral $U_{1}^{-} (x, t)$ as $σ \to \infty$ , we make the change $y = θ \sqrt[3]{3 t}$ . Using condition (1.2) as $x \to - \infty$ and setting $\begin{matrix} (2.3) & μ = \frac{σ}{\sqrt[3]{3 t}}, η = \frac{x}{\sqrt[3]{3 t}}, \end{matrix}$ we obtain $\begin{matrix} (2.4) & \begin{matrix} U_{1}^{-} (x, t) & = \int_{- \infty}^{- μ} f (θ \sqrt[3]{3 t}) Ai (η - θ) d θ \\ = \sum_{n = 0}^{N - 1} f_{n}^{-} {(3 t)}^{- n / 3} \int_{- \infty}^{- μ} θ^{- n} Ai (η - θ) d θ + \int_{- \infty}^{- μ} R_{N} (θ \sqrt[3]{3 t}) Ai (η - θ) d θ, \end{matrix} \end{matrix}$ where $R_{N} (s) = O (| s |^{- N})$ as $s \to - \infty$ by the meaning of Poincaré’s expansion. From the last relation we have the following estimate of the remainder: $\begin{matrix} (2.5) & \int_{- \infty}^{- μ} R_{N} (θ \sqrt[3]{3 t}) Ai (η - θ) d θ = O (σ^{- N + 1}), σ \to \infty . \end{matrix}$

Let us find the dependence of the integral $\int_{- \infty}^{- μ} θ^{- n} Ai (η - θ) d θ$ on the value of μ. For this purpose, it is convenient to specify the set of independent variables $T_{α} = {(x, t) : x \in R, t ⩾ | x |^{α}, α > 3 p}$ and to note that $\begin{matrix} μ = \frac{{(| x |^{3} + t)}^{p / 3}}{\sqrt[3]{3 t}} ⩽ \frac{{(t^{3 / α} + t)}^{p / 3}}{\sqrt[3]{3 t}} = \{\begin{matrix} O (t^{p / α - 1 / 3}), & 3 p < α ⩽ 3, \\ O (t^{(p - 1) / 3}), & α > 3, \end{matrix} \end{matrix}$ and, therefore, $μ \to 0$ as $σ \to \infty$ for $(x, t) \in T_{α}$ . In addition, the obvious inequality $σ^{3 / p} ⩽ t^{3 / α} + t$ implies that $t ⩾ 2^{- α / 3} σ^{α / p}$ for $3 p < α ⩽ 3$ and $t ⩾ 2^{- 1} σ^{3 / p}$ for $α > 3$ ; it follows that $\begin{matrix} (2.6) & μ = O (σ^{- γ}), σ \to \infty, (x, t) \in T_{α}, where γ = min {\frac{1}{p} - 1, \frac{α}{3 p} - 1} > 0 . \end{matrix}$ Then for $n = 0$ and any $N ⩾ 1$ , using Taylor’s expansion of $Ai (η + θ^{'})$ in $θ^{'}$ and the asymptotic estimate (2.6), we have $\begin{array}{l} \int_{- \infty}^{- μ} Ai (η - θ) d θ & = \int_{0}^{+ \infty} Ai (η + θ^{'}) d θ^{'} - \int_{0}^{μ} Ai (η + θ^{'}) d θ^{'} \\ = \int_{η}^{+ \infty} Ai (η^{'}) d η^{'} - \sum_{r = 1}^{N} \frac{μ^{r}}{r!} {Ai}^{(r - 1)} (η) + O (σ^{- γ N}), \\ (2.7) & σ \to \infty, (x, t) \in T_{α} . \end{array}$ For $n ⩾ 1$ , subtracting singularities as $θ \to - 0$ , we obtain $\begin{matrix} \int_{- \infty}^{- μ} θ^{- n} Ai (η - θ) d θ & = \int_{- \infty}^{- 1} θ^{- n} Ai (η - θ) d θ + \int_{- 1}^{- μ} Ψ_{n} (θ, η) d θ \\ + \sum_{r = 0}^{n - 1} \frac{{(- 1)}^{r}}{r!} {Ai}^{(r)} (η) \int_{- 1}^{- μ} θ^{r - n} d θ, \end{matrix}$ where $\begin{matrix} (2.8) & Ψ_{n} (θ, η) = θ^{- n} [Ai (η - θ) - \sum_{r = 0}^{n - 1} \frac{{(- θ)}^{r}}{r!} {Ai}^{(r)} (η)] \end{matrix}$ and the sum in the square brackets is a partial sum of the Taylor series for $Ai (η - θ)$ in variable θ. Thus, we have $\begin{matrix} (2.9) & \begin{matrix} \int_{- \infty}^{- μ} θ^{- n} Ai (η - θ) d θ & = \int_{- \infty}^{- 1} θ^{- n} Ai (η - θ) d θ + {(- 1)}^{n - 1} \frac{{Ai}^{(n - 1)} (η)}{(n - 1)!} ln μ \\ - \sum_{r = 0}^{n - 2} μ^{r - n + 1} \frac{{(- 1)}^{n} {Ai}^{(r)} (η)}{r! (r - n + 1)} - \int_{- μ}^{0} Ψ_{n} (θ, η) d θ + B_{n}^{-} (η), \end{matrix} \end{matrix}$ where $\begin{matrix} (2.10) & B_{n}^{-} (η) = \sum_{r = 0}^{n - 2} \frac{{(- 1)}^{n} {Ai}^{(r)} (η)}{r! (r - n + 1)} + \int_{- 1}^{0} Ψ_{n} (θ, η) d θ, \end{matrix}$ for $n = 1$ we put $B_{1}^{-} (η) = \int_{- 1}^{0} θ^{- 1} [Ai (η - θ) - Ai (η)] d θ$ .

Lemma 1.
For each $n ⩾ 1$ the following asymptotic estimates hold: $\begin{matrix} (2.11) & \int_{- μ}^{0} Ψ_{n} (θ, η) d θ = \sum_{r = 1}^{N} μ^{r} q_{r} {Ai}^{(n - 1 + r)} (η) + O (σ^{- γ N}), σ \to \infty, (x, t) \in T_{α}, \end{matrix}$ where $N ⩾ 1$ and $q_{r}$ are some constants, $\begin{matrix} (2.12) & \int_{- 1}^{0} Ψ_{n} (θ, η) d θ = O (| η |^{- 1 / 4 + n / 2}), \int_{0}^{1} Ψ_{n} (θ, η) d θ = O (| η |^{- 1 / 4 + n / 2}), η \to \infty, \end{matrix}$
Proof.
From definition (2.8) we conclude that $Ψ_{n} (θ, η)$ has no singularities as $θ \to - 0$ ; whence we easily obtain estimate (2.11) for any $N ⩾ 1$ . Using formula (1.4), let us write function (2.8) in the form $\begin{matrix} Ψ_{n} (θ, η) = \frac{1}{2 π θ^{n}} Re \int_{- \infty}^{+ \infty} [exp (- i θ z) - \sum_{r = 0}^{n - 1} \frac{{(- i θ z)}^{r}}{r!}] exp {i (\frac{z^{3}}{3} + η z)} d z . \end{matrix}$ Then we get $\begin{matrix} \int_{- 1}^{0} Ψ_{n} (θ, η) d θ = \frac{1}{2 π} Re \int_{- \infty}^{+ \infty} \int_{- 1}^{0} θ^{- n} [exp (- i θ z) - \sum_{r = 0}^{n - 1} \frac{{(- i θ z)}^{r}}{r!}] d θ exp {i (\frac{z^{3}}{3} + η z)} d z . \end{matrix}$ Integrating by parts, we find the following estimate: $\begin{matrix} \int_{- 1}^{0} θ^{- n} [exp (- i θ z) - \sum_{r = 0}^{n - 1} \frac{{(- i θ z)}^{r}}{r!}] d θ = O (z^{n}), z \to \infty . \end{matrix}$ As $η \to + \infty$ by the saddle-point method we have a negligible contribution in the asymptotics of $\int_{- 1}^{0} Ψ_{n} (θ, η) d θ$ due to the superexponential factor $exp (- 2 η^{3 / 2} / 3)$ . As $η \to - \infty$ it is sufficient to use the stationary-phase approximation with two stationary points $z_{1, 2} = \pm \sqrt{- η}$ exactly by the scheme of calculation of the asymptotics of the Airy function in the form $\begin{matrix} Ai (η) = \frac{1}{2 π} Re \int_{- \infty}^{+ \infty} exp {i (\frac{z^{3}}{3} + η z)} d z, \end{matrix}$ which gives the first estimate (2.12). The proof of the second estimate (2.12) is analogous. The lemma is proved. □

Substituting relations (2.7) for $n = 0$ and (2.9) for $n ⩾ 1$ into the right-hand side of (2.4), by formulas (2.5) and (2.11), we find $\begin{matrix} (2.13) & \begin{matrix} U_{1}^{-} (x, t) & = f_{0}^{-} \int_{η}^{+ \infty} Ai (θ) d θ + \sum_{n = 1}^{N - 1} f_{n}^{-} {(3 t)}^{- n / 3} [\int_{- \infty}^{- 1} θ^{- n} Ai (η - θ) d θ + B_{n}^{-} (η)] \\ + V_{1, N}^{-} (μ, η, t) + O (σ^{- γ N}), σ \to \infty, \end{matrix} \end{matrix}$ where $V_{1, N}^{-} (μ, η, t) = \sum_{r_{s}^{2} + q_{s}^{2} \neq 0} v_{1, s}^{-} {Ai}^{(m_{s})} (η) t^{l_{s}} μ^{r_{s}} {ln}^{q_{s}} μ$ is a finite sum with $v_{1, s}^{-}$ being constant coefficients and $m_{s}$ , $l_{s}$ , $r_{s}$ , $q_{s}$ being some integers, the positive number γ is defined in (2.6); the form of the square brackets in (2.13) is due to the first and the fifth summands in the right-hand side of (2.9); the terms with $μ^{r_{s}} {ln}^{q_{s}} μ$ in $V_{1, N}^{-} (μ, η, t)$ are due to the second and the third summands in the right-hand side of (2.9), and fourth summand rewritten by formula (2.11).

Now, let us study the integral $U_{0}^{-} (x, t) = \frac{1}{\sqrt[3]{3 t}} \int_{- σ}^{0} f (y) Ai (\frac{x - y}{\sqrt[3]{3 t}}) d y$ on the set $T_{α}$ . Since for $| y | ⩽ σ$ there holds the estimate $\frac{y}{\sqrt[3]{3 t}} = O (σ^{- γ})$ , we can use Taylor’s formula. Then, for any $N ⩾ 1$ we find $\begin{matrix} U_{0}^{-} (x, t) = \sum_{n = 0}^{N - 1} t^{- (n + 1) / 3} \frac{{(- 1)}^{n} 3^{- (n + 1) / 3}}{n!} {Ai}^{(n)} (η) \int_{- σ}^{0} y^{n} f (y) d y + O (σ^{- γ N}), σ \to \infty . \end{matrix}$ Let us transform the integral $\int_{- σ}^{0} y^{n} f (y) d y$ as follows: $\begin{array}{l} \int_{- σ}^{0} y^{n} f (y) d y & = \int_{- 1}^{0} y^{n} f (y) d y + \int_{- σ}^{- 1} (f_{0}^{-} y^{n} + \dots + f_{n}^{-} + \frac{f_{n + 1}^{-}}{y}) d y \\ + \int_{- σ}^{- 1} y^{n} [f (y) - f_{0}^{-} - \dots - \frac{f_{n}^{-}}{y^{n}} - \frac{f_{n + 1}^{-}}{y^{n + 1}}] d y \\ = \int_{- 1}^{0} y^{n} f (y) d y + \int_{- \infty}^{- 1} Φ_{n}^{-} (y) d y - \int_{- \infty}^{- σ} Φ_{n}^{-} (y) d y \\ (2.14) & - f_{n + 1}^{-} ln σ + P_{n} (σ), \end{array}$ where $\begin{matrix} Φ_{n}^{-} (y) = y^{n} [f (y) - \sum_{m = 0}^{n + 1} \frac{f_{m}^{-}}{y^{m}}] . \end{matrix}$ From relations (1.2) we obtain estimates $| Φ_{n}^{-} (y) | ⩽ C_{n} y^{- 2}$ for $y ⩽ - 1$ and $\begin{matrix} \int_{- \infty}^{- σ} Φ_{n}^{-} (y) d y = \sum_{m = 1}^{N - 1} φ_{n, m} σ^{- m} + O (σ^{- N}), σ \to \infty, \end{matrix}$ where $φ_{n, m}$ are some constants. Taking into account these estimates and substituting $σ = μ \sqrt[3]{3 t}$ into (2.14), we obtain $\begin{matrix} \int_{- σ}^{0} y^{n} f (y) d y = I_{n} - \frac{f_{n + 1}^{-}}{3} ln t - f_{n + 1}^{-} ln μ + \sum_{r \neq 0} a_{n, r} μ^{r} t^{r / 3} + O (σ^{- N}), σ \to \infty, \end{matrix}$ where $I_{n}$ and $a_{n, r}$ are constants. Then, we have $\begin{matrix} (2.15) & U_{0}^{-} (x, t) = \sum_{n = 0}^{N - 1} t^{- (n + 1) / 3} {Ai}^{(n)} (η) [b_{n}^{-} + {\tilde{b}}_{n}^{-} ln t] + V_{0, N}^{-} (μ, η, t) + O (σ^{- γ N}), σ \to \infty, \end{matrix}$ where $V_{0, N}^{-} (μ, η, t) = \sum_{r_{s}^{2} + q_{s}^{2} \neq 0} v_{0, s}^{-} {Ai}^{(m_{s})} (η) t^{l_{s}} μ^{r_{s}} {ln}^{q_{s}} μ$ is a finite sum with $b_{n}^{-}$ , ${\tilde{b}}_{n}^{-}$ , and $v_{0, s}^{-}$ being constant coefficients, $m_{s}$ , $l_{s}$ , $r_{s}$ , $q_{s}$ being some integers, and the positive number γ is defined in (2.6).

Analogously to formulas (2.13) and (2.15), we find for any $N ⩾ 2$ $\begin{matrix} (2.16) & \begin{matrix} U_{1}^{+} (x, t) & = f_{0}^{+} \int_{- \infty}^{η} Ai (θ) d θ + \sum_{n = 1}^{N - 1} f_{n}^{+} {(3 t)}^{- n / 3} [\int_{1}^{+ \infty} θ^{- n} Ai (η - θ) d θ + B_{n}^{+} (η)] \\ + V_{1, N}^{+} (μ, η, t) + O (σ^{- γ N}), σ \to \infty, \end{matrix} \end{matrix}$ where $V_{1, N}^{+} (μ, η, t) = \sum_{r_{s}^{2} + q_{s}^{2} \neq 0} G_{1, s} (η) t^{l_{s}} μ^{r_{s}} {ln}^{q_{s}} μ$ and the smooth functions $\begin{matrix} B_{n}^{+} (η) = \sum_{r = 0}^{n - 2} \frac{{(- 1)}^{r} {Ai}^{(r)} (η)}{r! (r - n + 1)} + \int_{0}^{1} Ψ_{n} (θ, η) d θ \end{matrix}$ are constructed similarly to functions (2.10), $\begin{matrix} (2.17) & U_{0}^{+} (x, t) = \sum_{n = 0}^{N - 1} t^{- (n + 1) / 3} {Ai}^{(n)} (η) [b_{n}^{+} + {\tilde{b}}_{n}^{+} ln t] + V_{0, N}^{+} (μ, η, t) + O (σ^{- γ N}), σ \to \infty, \end{matrix}$ $b_{n}^{+}$ and ${\tilde{b}}_{n}^{+}$ are some constants, $V_{0, N}^{+} (μ, η, t) = \sum_{r_{s}^{2} + q_{s}^{2} \neq 0} G_{0, s} (η) t^{l_{s}} μ^{r_{s}} {ln}^{q_{s}} μ$ , $G_{0, s} (η)$ and $G_{1, s} (η)$ are some smooth functions. Lemma 2.
For $(x, t) \in T_{α}$ and each $N ⩾ 2$ the following estimate holds: $\begin{matrix} (2.18) & V_{0, N}^{-} (μ, η, t) + V_{1, N}^{-} (μ, η, t) + V_{0, N}^{+} (μ, η, t) + V_{1, N}^{+} (μ, η, t) = O (σ^{- γ N}), σ \to \infty, \end{matrix}$ where the positive number γ is defined in ( 2.6 ).
Proof.
By relations (2.2), (2.3), and the above expressions for the functions $V_{i, N}^{\pm} (μ, η, t)$ we obtain the following formula: $\begin{matrix} V_{0, N}^{-} (μ, η, t) + V_{1, N}^{-} (μ, η, t) + V_{0, N}^{+} (μ, η, t) + V_{1, N}^{+} (μ, η, t) = \sum_{r_{s}^{2} + q_{s}^{2} \neq 0} v_{s} (η) t^{l_{s}} {(ln t)}^{m_{s}} Λ^{p r_{s}} {(ln Λ)}^{p q_{s}}, \end{matrix}$ with $Λ = {(| x |^{3} + t)}^{1 / 3} \to \infty$ , $v_{s} (η)$ being some smooth functions of slow growth, $l_{s}$ , $m_{s}$ , $r_{s}$ , $q_{s}$ being integers, and p being an arbitrary parameter ( $0 < p_{1} < p < p_{2} < 1$ for positive $p_{1}$ and $p_{2}$ ). Then the statement of the lemma directly follows from A.R. Danilin’s proposition [5, Lemma 4.4]. However, for the convenience of the reader, here we explain this key moment of the auxiliary parameter method. Let, for example, $v (x, t) = a (η, t) μ^{q} + O (σ^{- γ N})$ , $q \neq 0$ , where $v (x, t)$ is the difference between the function $u (x, t)$ and the sum of all expressions in the right-hand sides of formulas (2.13), (2.15), (2.16), and (2.17) except for $a (η, t) μ^{q}$ . Consequently, taking into account the definitions of σ and μ, we get $v (x, t) = b (η, t) Λ^{q p} + O (Λ^{- γ p N})$ as $Λ = {(| x |^{3} + t)}^{1 / 3} \to \infty$ . Then for any $p_{1}$ and $p_{2}$ ( $0 < p_{1} < p_{2} < 1$ ), we obtain $b (η, t) Λ^{q p_{1}} - b (η, t) Λ^{q p_{2}} = O (Λ^{- γ p_{1} N})$ . By virtue of the arbitrariness of $p_{1}$ and $p_{2}$ , it follows that $v (x, t) = O (σ^{- γ N})$ . Thus, we arrive at relation (2.18), successively “eliminating” terms with any nonzero powers of μ or $ln μ$ , because all $V_{i, N}^{\pm} (μ, η, t)$ are finite sums. □
Lemma 3.
The following formulas hold: $\begin{matrix} (2.19) & \begin{matrix} \int_{η}^{+ \infty} Ai (θ) d θ = - \int_{0}^{+ \infty} \frac{1}{π θ} sin (\frac{θ^{3}}{3} + η θ) d θ + \frac{1}{2}, \\ \int_{- \infty}^{η} Ai (θ) d θ = \int_{0}^{+ \infty} \frac{1}{π θ} sin (\frac{θ^{3}}{3} + η θ) d θ + \frac{1}{2} . \end{matrix} \end{matrix}$
Proof.
From the explicit form the Airy function (1.4), it is easy to see that $\begin{matrix} \int_{η}^{+ \infty} Ai (θ) d θ = - \int_{0}^{+ \infty} \frac{1}{π θ} sin (\frac{θ^{3}}{3} + η θ) d θ + const . \end{matrix}$ In addition, for $η > 0$ we have $\begin{matrix} \int_{0}^{+ \infty} \frac{1}{θ} sin (\frac{θ^{3}}{3} + η θ) d θ & = \int_{0}^{η^{- 1 / 2}} + \int_{η^{- 1 / 2}}^{+ \infty} \\ = \int_{0}^{η^{1 / 2}} \frac{1}{z} sin (\frac{z^{3}}{3 η^{3}} + z) d z - \int_{η^{- 1 / 2}}^{+ \infty} \frac{1}{θ (θ^{2} + η)} d cos (\frac{θ^{3}}{3} + η θ) . \end{matrix}$ As $η \to + \infty$ the first integral on the right-hand side of this relation tends to the limit value of the sine integral $Si (+ \infty) = \int_{0}^{+ \infty} \frac{sin z}{z} d z = \frac{π}{2}$ , since $z^{3} / η^{3} ⩽ η^{- 3 / 2}$ on the whole integration interval. Integrating by parts, we see that $\begin{matrix} \int_{η^{- 1 / 2}}^{+ \infty} \frac{1}{θ (θ^{2} + η)} d cos (\frac{θ^{3}}{3} + η θ) = O (η^{- 1 / 2}), η \to + \infty; \end{matrix}$ whence we get the first formula (2.19). Taking into account the well-known identity $\int_{- \infty}^{+ \infty} Ai (θ) d θ = 1$ , we obtain the second formula (2.19). The lemma is proved. □

3. Asymptotics of the solution

Now, substituting expressions (2.13), (2.15), (2.16), and (2.17) into (2.1), using estimate (2.18) and formulas (2.19), for any $N ⩾ 2$ we obtain $\begin{matrix} (3.1) & \begin{matrix} u (x, t) & = (f_{0}^{+} - f_{0}^{-}) \int_{0}^{+ \infty} \frac{1}{π θ} sin (\frac{θ^{3}}{3} + \frac{x θ}{\sqrt[3]{3 t}}) d θ + \frac{f_{0}^{+} + f_{0}^{-}}{2} \\ + \sum_{n = 1}^{N - 1} t^{- n / 3} (F_{n} (η) + {\tilde{F}}_{n} (η) ln t) + O (σ^{- γ N}), σ \to \infty . \end{matrix} \end{matrix}$ Substituting expansion (3.1) for the solution $u (x, t)$ into equation (1.1), we find $\begin{matrix} (3.2) & F_{n}^{‴} - η F_{n}^{'} - n F_{n} = - 3 {\tilde{F}}_{n}, {\tilde{F}}_{n}^{‴} - η {\tilde{F}}_{n}^{'} - n {\tilde{F}}_{n} = 0 . \end{matrix}$ Since the coefficients $F_{n} (η)$ and ${\tilde{F}}_{n} (η)$ are linear combinations of smooth bounded functions, derivatives of the Airy function up to the $(n - 1)$ th order, inclusively, and the integrals $\int_{- 1}^{0} Ψ_{n} (θ, η) d θ$ , $\int_{0}^{1} Ψ_{n} (θ, η) d θ$ , from estimates (2.12) and the asymptotics of the Airy function [1] $\begin{array}{l} Ai (x) = \frac{x^{- 1 / 4}}{2 π} exp (- \frac{2}{3} x^{3 / 2}) \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} Γ (\frac{3 n + 1}{2})}{3^{2 n} (2 n)!} x^{- 3 n / 2}, x \to + \infty, \\ Ai (x) = \frac{| x |^{- 1 / 4}}{π} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{[n / 2]} Γ (\frac{3 n + 1}{2})}{3^{2 n} (2 n)!} sin (\frac{2}{3} | x |^{3 / 2} + \frac{π}{4} {(- 1)}^{n}) | x |^{- 3 n / 2}, x \to - \infty, \end{array}$ it follows that $F_{n} (η)$ and ${\tilde{F}}_{n} (η)$ cannot grow faster than $| η |^{- 1 / 4 + n / 2}$ as $η \to \infty$ ; analogously, their lth order derivatives cannot grow faster than $| η |^{- 1 / 4 + (n + l) / 2}$ as $η \to \infty$ . Consequently, these functions belong to the class $A_{1 / 4 - n / 2}$ , where $\begin{matrix} A_{m} = {F (η) \in C^{\infty} (R) : \forall l ⩾ 0 sup_{η \in R} | {(1 + | η |)}^{m - l / 2} \frac{d^{l} F (η)}{d η^{l}} | < \infty}, \end{matrix}$ the nth term in (3.1) cannot grow faster than $ln t | η |^{- 1 / 4} | \frac{x}{t} |^{n / 2}$ , and the series $\sum_{n = 1}^{\infty} t^{- n / 3} (F_{n} (η) + {\tilde{F}}_{n} (η) ln t)$ keeps its asymptotic character for $t > | x |^{1 + δ}$ with any $δ > 0$ in the sense that its nth term has the increasingly well estimate $O (ln t | η |^{- 1 / 4} t^{- δ n / (2 + 2 δ)})$ . Thus, we arrive at the following statement.

Theorem 1.

If for a locally Lebesgue integrable function $f : R \to R$ the conditions $\begin{matrix} f (x) = \sum_{n = 0}^{\infty} f_{n}^{\pm} x^{- n}, x \to \pm \infty, \end{matrix}$ are fulfilled, then for the solution of the Cauchy problem ( 1.1 ) there holds the asymptotic formula $\begin{matrix} u (x, t) & = (f_{0}^{+} - f_{0}^{-}) \int_{0}^{+ \infty} \frac{1}{π θ} sin (\frac{θ^{3}}{3} + \frac{x θ}{\sqrt[3]{3 t}}) d θ + \frac{f_{0}^{+} + f_{0}^{-}}{2} \\ + \sum_{n = 1}^{N - 1} t^{- n / 3} (F_{n} (η) + {\tilde{F}}_{n} (η) ln t) + O (σ^{- γ N}) \end{matrix}$ as $| x | + t \to \infty$ for $t > | x |^{1 + δ}$ with arbitrary natural $N ⩾ 2$ and any positive δ, where the functions $F_{n} (η)$ and ${\tilde{F}}_{n} (η)$ of the self-similar variable $η = \frac{x}{\sqrt[3]{3 t}}$ belong to the class $A_{1 / 4 - n / 2}$ and satisfy the system of differential equations ( 3.2 ), the positive number γ is defined in ( 2.6 ).

Remark 1.

According to formula (3.1), the asymptotic series in the theorem should be understood in the sense of Erdélyi [6] with the asymptotic sequence ${{(| x |^{3} + t)}^{- γ n}}_{n = 1}^{\infty}$ , $γ > 0$ .

Remark 2.

The change of the integration variable $θ = z {(3 t)}^{1 / 3}$ in the leading term of expansion (3.1) gives an expression containing the integral $\int_{- \infty}^{+ \infty} z^{- 1} exp (i S (x, t, z)) d z$ , where the phase $S (x, t, z) = t z^{3} + x z$ satisfies the eikonal equation $S_{t} = {(S_{x})}^{3}$ , establishing a connection between the above analysis and the geometrical optics approach of [4] to problem (1.1).

Remark 3.

The obtained result can be applied to studying asymptotics of solutions of more general equations of the form $u_{t} = Q (x, u, u_{x}, u_{x x}) + ε u_{x x x}$ , e.g., the nonlinear Korteweg–de Vries equation, by the matching method in the same way as the long-time asymptotics of solutions of the heat equation [16] is applied to the Cauchy problem for a quasi-linear parabolic equation [17]. In this connection, it should be emphasized that in singularly perturbed problems the knowledge of infinite asymptotic series for the coefficients of expansions allows one to construct approximations of solutions with any required accuracy [8].

Remark 4.

From the theoretical point of view, the present investigation provides a good example of using the auxiliary parameter method in the asymptotic analysis by showing the general scheme of application to a wide class of integral transforms.

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