# Asymptotic step-like solutions of the singularly perturbed Burgers equation

V. Samoilenko, Yu. Samoilenko, E. Zappale

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V. Samoilenko, Yu. Samoilenko, E. Zappale. Asymptotic step-like solutions of the singularly perturbed Burgers equation. Physics of Fluids, 2023, 35 (6), 10.1063/5.0150685. hal-04992281

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# Asymptotic step-like solutions of the singularly perturbed Burgers equation ✅

V. Samoilenko

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; Yu. Samoilenko

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; E. Zappale

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Physics of Fluids 35, 067106 (2023)

https://doi.org/10.1063/5.0150685

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# Asymptotic step-like solutions of the singularly perturbed Burgers equation

Cite as: Phys. Fluids 35, 067106 (2023); doi: 10.1063/5.0150685

Submitted: 15 March 2023 · Accepted: 12 May 2023 ·

Published Online: 2 June 2023

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V. Samoilenko,¹,ª ID Yu. Samoilenko,²,ᵇ ID and E. Zappale³,ᵇ ID

## AFFILIATIONS

¹Department of Mathematical Physics, Taras Shevchenko National University of Kyiv, vul. Volodymyrs’ka, 60, Kyiv 01033, Ukraine

²Institute Camille Jordan, Claude Bernard Lyon 1 University, Avenue Claude Bernard, 21, Bâtiment Braconnier, Villeurbanne CEDEX, Lyon 69622, France

³Department of Basic and Applied Sciences for Engineering, Sapienza University of Rome, Via Antonio Scarpa, 16, Rome 00161, Italy

ª) Author to whom correspondence should be addressed: valsamyul@gmail.com

ᵇ) Electronic addresses: yuliiasam1976@gmail.com and elvira.zappale@uniroma1.it

## ABSTRACT

This paper deals with a problem of asymptotic step-like solutions of the Burgers equation with variable coefficients and a small parameter. By means of the nonlinear Wentzel–Kramers–Brillouin method, the algorithm of constructing these asymptotic solutions is proposed and statements on justification of the algorithm are proved. The obtained results are illustrated by an example, for which the first asymptotic step-like approximation is explicitly found. The asymptotic solution is global and has a form of the shock wave type function. There are also given graphs of these approximate solutions for certain numerical parameters.

Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0150685

## I. INTRODUCTION

The paper deals with constructing asymptotic solutions of the singularly perturbed Burgers equation with variable coefficients of the following form:

$$ \epsilon u_{xx} = a(x, t, \epsilon)u_t + b(x, t, \epsilon)u u_x, \quad (1) $$

where the functions $a(x, t, \epsilon)$, $b(x, t, \epsilon)$, $(x, t) \in \mathbb{R} \times [0; T]$ are written as the asymptotic (according to Poincaré) series,

$$ \begin{aligned} a(x, t, \epsilon) &= \sum_{k=0}^{N} \epsilon^k a_k(x, t) + O(\epsilon^{N+1}), \\ b(x, t, \epsilon) &= \sum_{k=0}^{N} \epsilon^k b_k(x, t) + O(\epsilon^{N+1}). \end{aligned} \quad (2) $$

Here, $a_0(x, t)$, $b_0(x, t) \neq 0$ for all $(x, t) \in \mathbb{R} \times [0; T]$; and $\epsilon$ is a small parameter.

Equation (1) is a mathematical model of wave processes in medium with small viscosity¹ (or large Reynolds number Re) and variable characteristics, and it is a generalization²⁻⁴ of the Burgers equation with constant coefficients,

$$ u_t + uu_x = \nu u_{xx}, \quad (3) $$

where $u = u(x, t)$ is the velocity of liquid moving, and $\nu$ is the viscosity of liquid. It was proposed as a mathematical model of turbulence.⁵

At present, the equation is used in hydrodynamics, acoustic, gas dynamics,⁶ magnetohydrodynamics,⁷ and other fields of physics in the description of wave processes with viscosity.

Equation (3) has been known since 1906 (see Refs. 8 and 9) and has attracted the attention of researchers, because in the one-dimensional case, the equation is a simple form of Navier–Stokes equation, one of the most important models of hydrodynamics. Recall that the Navier–Stokes system was obtained in the first part of the XIXth century, first by A. Navier and then by O. Cauchy, S. Poisson, A. Saint-Venant, and G. Stokes, and describes moving of a viscous (in the general case, incompressible) liquid (gas) and is based on Newton’s laws. The Navier–Stokes system is nonlinear that is essentially complicate studying its solutions, which are sensitive to the perturbation of the coefficients as well as to the initial (or/and the boundary) conditions.

At the end of the last 30 years of the past century, J. M. Burgers studied Eq. (3) as the simplest model unifying typical nonlinearity and viscosity, which describes the phenomenon of hydrodynamic turbulence.¹⁰⁻¹² If the viscosity is absent (for $\nu = 0$), then Eq. (3) is the invisible Burgers equation,

$$ u_t + uu_x = 0, \quad (4) $$

that is also known as the Hopf equation.¹³

For solution to the equation, the so-called gradient catastrophe takes place.¹⁴ The phenomenon is connected with overturning
(destroying) waves. It is deduced from the formula for the solution to the Cauchy problem for Eq. (4) with initial condition $u(x, 0) = f(x)$, $x \in \mathbb{R}$, that is implicitly written as

$$u(x, t) = f(x - tu(x, t)).$$

In particular, from the note, it can be concluded that if the initial function $f(x)$ is quickly decreasing, then the solution to the corresponding Cauchy problem for Eq. (4) exists only on finite time intervals.¹⁵ The properties are obviously the consequence of the presence of nonlinear term $uu_x$. Even a small perturbation of the form $\nu u_{xx}$, where $\nu > 0$ in Eq. (4), significantly affects the qualitative properties of its solutions.

Model (3) is one of the simplest describing nonlinear effects, in particular, the appearance and evolution of shock waves. Many qualitative properties of Burgers equation [Eq. (3)] can be obtained from formulas for its exact solutions, which are found by reducing nonlinear Eq. (3) to the heat linear equation using the substitution proposed by Hopf¹³ and Cole¹⁶ independently. In such a way, they got formulas for exact solutions of the Burgers equation, which led to analyze the process of appearance shock waves for nonlinear environment with a small viscosity from the initial conditions. In particular, because of viscosity, there is not overturning waves, but the presence of the dissipation induces the appearance of the so-called stationary waves prolonging with a constant velocity.¹⁴,¹⁷ These stationary solutions of Eq. (3) are written as

$$u(x, t) = a \left( 1 - \tanh \left( \frac{a}{2} \frac{x - at}{\nu} + \varphi_0 \right) \right). \quad (5)$$

Here, the real constants $a$ and $\varphi_0$ are arbitrary.

In the case of disappearing viscosity, i.e., if $\nu \to 0$, then the limit value of the solution (5) of Eq. (3) is a discontinuous step-like function, because

$$\lim_{\nu \to 0} u(x, t) = \begin{cases} 0, & x - at > 0, \\ 2a, & x - at < 0. \end{cases} \quad (6)$$

Therefore, there naturally appeared a problem on constructing approximate solutions of Eq. (1) with a small parameter $\varepsilon$ that is analogous to the case of Eq. (3) in their asymptotic behavior (as $\nu \to 0$) and is similar to solutions of the form (5).

This problem is considered in the paper, where using the nonlinear Wentzel-Kramers-Brillouin method (WKB method)¹⁸⁻²¹ for Eq. (1) the asymptotic solution is constructed that is a discontinuous step-like function in limits case (as $\varepsilon \to 0$).²²⁻²⁴

Such a solution is called an *asymptotic step-like solution*.

In the paper, we propose a general algorithm of constructing these solutions, find the first approximation for the asymptotic step-like solution, and prove the theorems on the accuracy with which the main term and the first approximation satisfy Eq. (1). These results are demonstrated on an example.

## II. PRELIMINARY NOTES AND DEFINITIONS

According to the concept of constructing asymptotic solutions of the singularly perturbed differential equations of hydrodynamics with variable coefficients, the searched solution can be written as asymptotic series in the following form:²³,²⁴

$$u(x, t, \varepsilon) = \sum_{j=0}^{\infty} \varepsilon^j [u_j(x, t) + V_j(x, t, \tau)], \quad \tau = \frac{x - \varphi(t)}{\varepsilon}, \quad (7)$$

where $u_j(x, t)$, $V_j(x, t, \tau)$, $j = 0, 1, \dots$, $\varphi = \varphi(t)$ are infinitely differentiable functions that are defined while constructing the asymptotic solution.

Here, the function $\varphi = \varphi(t)$ defines the discontinuity curve.²²,²³

The function

$$U(x, t, \varepsilon) = \sum_{j=0}^{\infty} \varepsilon^j u_j(x, t)$$

is called the regular part of asymptotics (7), and the function

$$V(x, t, \tau, \varepsilon) = \sum_{j=0}^{\infty} \varepsilon^j V_j(x, t, \tau)$$

is called the singular part of the asymptotics.

The regular part is a background function, while the singular part describes qualitative properties of searched asymptotic solution (7).

According to the properties of functions (5), the terms of the singular part of the asymptotics are searched as elements of certain functional space $G$ that is defined as follows.²³,²⁴

The space $G = G(\mathbb{R} \times [0; T] \times \mathbb{R})$ contains functions $f = f(x, t, \tau) \in C^{\infty}(\mathbb{R} \times [0; T] \times \mathbb{R})$, such that the following conditions are fulfilled:

(1) the relation

$$\lim_{\tau \to +\infty} \tau^n \frac{\partial^{p+q+r}}{\partial x^p \partial t^q \partial \tau^r} f(x, t, \tau) = 0, \quad (x, t) \in K$$

is satisfied for all non-negative integers $n, p, q$, and $r$ uniformly with respect to $(x, t)$ on any compact $K \subset \mathbb{R} \times [0; T]$;

(2) there exists an infinitely differentiable function $f^{-}(x, t)$ such that

$$\lim_{\tau \to -\infty} \tau^n \frac{\partial^{p+q+r}}{\partial x^p \partial t^q \partial \tau^r} (f(x, t, \tau) - f^{-}(x, t)) = 0, \quad (x, t) \in K.$$

Remark that according to the notation

$$\tau = \frac{x - \varphi(t)}{\varepsilon}$$

with $\varphi(t) = at$, function (5) belongs to the space $G$. From property (6) for function (5), it follows the appropriateness of using notice of discontinuity curve, which is defined as

$$\Gamma = \{(x, t) \in \mathbb{R} \times [0; T] : x = \varphi(t)\}$$

for solution (7).

By $G_0 = G_0(\mathbb{R} \times [0; T] \times \mathbb{R}) \subset G$, we denote the space of such functions $f = f(x, t, \tau)$ that the condition

$$\lim_{\tau \to -\infty} f(x, t, \tau) = 0$$

holds uniformly with respect to $(x, t) \in K$ for any compact $K \subset \mathbb{R} \times [0; T]$.

## III. ALGORITHM OF CONSTRUCTING THE ASYMPTOTIC SOLUTION

While finding the terms of asymptotic solution (7), we use general approaches and ideas of asymptotic analysis as well as properties
of the space G. For asymptotic series (7), first we determine the terms of the regular part and then we find the terms of the singular part of the asymptotics as well as the function $\varphi = \varphi(t)$ defining the discontinuity curve.

Equations for the regular terms of the asymptotics have the following form:

$$a_0(x, t) \frac{\partial u_0}{\partial t} + b_0(x, t) u_0 \frac{\partial u_0}{\partial x} = 0, \quad (8)$$

$$a_0(x, t) \frac{\partial u_j}{\partial t} + b_0(x, t) \frac{\partial}{\partial x} (u_0 u_j) = f_j(x, t), \quad (9)$$

and they are obtained after substituting solution (7) in (1) using the property (1) of the space G for $V_j(x, t, \tau) \in G, j = 0, 1, \dots$.

The functions $f_j(x, t), j = 1, 2, \dots,$ in (9) are found recurrently. For example,

$$f_1(x, t) = \frac{\partial^2 u_0}{\partial x^2} - a_1(x, t) \frac{\partial u_0}{\partial t} - b_1(x, t) u_0 \frac{\partial u_0}{\partial x}.$$

Since Eq. (8) is quasilinear, and Eq. (9) is linear, their solutions can be found, for example, by means of the method of characteristics. So, we can assume that these solutions are known.

The equations for the singular terms of the asymptotics are written as follows:

$$\frac{\partial^2 V_0}{\partial \tau^2} + a_0(x, t) \varphi'(t) \frac{\partial V_0}{\partial \tau} - b_0(x, t) \left( u_0 \frac{\partial V_0}{\partial \tau} + V_0 \frac{\partial V_0}{\partial \tau} \right) = 0, \quad (10)$$

$$\frac{\partial^2 V_j}{\partial \tau^2} + a_0(x, t) \varphi'(t) \frac{\partial V_j}{\partial \tau} - b_0(x, t) \left( u_0 \frac{\partial V_j}{\partial \tau} + \frac{\partial}{\partial \tau} (V_0 V_j) \right) = F_j(x, t, \tau), \quad (11)$$

and they are found after substituting solution (7) in (1) and taking into account Eqs. (8) and (9).

Here, the functions,

$$F_j(x, t, \tau) = F_j(t, V_0(x, t, \tau), \dots, V_{j-1}(x, t, \tau), u_0(x, t), \dots, u_j(x, t)), \quad j = 1, 2, \dots,$$

are obtained recurrently after finding the regular part of the asymptotics.

Unlike the equations for the regular part of the asymptotics, Eqs. (10) and (11) generally cannot be integrated in a closed form. Additionally, the solutions belong to the functional space G.

Let us consider the problem of the existence of these solutions. The study is done as follows:

1. We assume that the curve $\Gamma$ is known, and Eqs. (10) and (11) are considered on the curve $\Gamma$.

Further, we use the notation

$$v_j = v_j(t, \tau) = V_j(x, t, \tau)|_{x=\varphi(t)}, \quad j = 0, 1, \dots,$$

2. The equation for the main term $v_0(t, \tau)$ is integrated by quadratures, and for the other equations, it is possible to lower their order. As a result, the first order linear inhomogeneous differential equations are deduced, solutions of which are written explicitly. Then, the problem of belonging found solutions to the space G is studied, and the functions, $V_j(x, t, \tau), j = 0, 1, \dots,$ are determined.

3. From the condition that solutions of Eqs. (10) and (11) belong to the space G on the curve $\Gamma$, we also deduce a nonlinear ordinary differential equation for the function $\varphi = \varphi(t)$.

### A. The main term of the singular part of the asymptotics

The function $v_0(t, \tau) = V_0(\varphi(t), t, \tau)$ satisfies the following non-linear partial differential equation:

$$\frac{\partial^2 v_0}{\partial \tau^2} + a_0(\varphi, t) \varphi'(t) \frac{\partial v_0}{\partial \tau} - b_0(\varphi, t) (u_0(\varphi, t) + v_0) \frac{\partial v_0}{\partial \tau} = 0. \quad (12)$$

Since the function $V_0(x, t, \tau)$ belongs to the space G, we find the function $v_0(t, \tau)$ as

$$v_0(t, \tau) = A[1 - \tanh(\beta\tau + c)], \quad (13)$$

where

$$A = A(t) = \frac{a_0(\varphi(t), t)\varphi'(t) - b_0(\varphi(t), t)u_0(\varphi(t), t)}{b_0(\varphi(t), t)}, \quad (14)$$

$$\beta = \beta(t) = \frac{1}{2} A(t) b_0(\varphi(t), t). \quad (15)$$

$c = c(t)$ is a value, which is assumed to be zero.

Next, we show that as the main term of the singular part of the asymptotics $V_0(x, t, \tau)$, we can consider the function $v_0(t, \tau)$ that is given by formula (13).

Considering that the main term of the regular part of the asymptotics is known, we prove the following theorem.

**Theorem 1.** Let condition $a_0(x, t) b_0(x, t) \neq 0$ be fulfilled for all $(x, t) \in \mathbb{R} \times [0; T]$, where the functions $a_0(x, t), b_0(x, t)$ are infinitely differentiable in the variables $(x, t) \in \mathbb{R} \times [0; T]$. Then, for any infinitely differentiable function $x = \varphi(t), t \in [0; T]$, the following function,

$$Y_0(x, t, \varepsilon) = u_0(x, t) + v_0(t, \tau), \quad \tau = \frac{x - \varphi(t)}{\varepsilon}, \quad (16)$$

is the main term of the asymptotic step-like solution of Eq. (1) and satisfies the equation with an asymptotic accuracy $O(1)$ on the set $\mathbb{R} \times [0; T]$. Moreover, as $\tau \to +\infty$, it equates (1) with an asymptotic accuracy $O(\varepsilon)$.

If additionally for function (14) the condition $\frac{dA}{dt} = 0$ holds, then (16) satisfies Eq. (1) with an asymptotic accuracy $O(\varepsilon)$ as $\tau \to -\infty$.

**Proof.** For proving Theorem 1, it is necessary to estimate the residue for (16) in the case of Eq. (1). We find an explicit expression for the residual function by substituting asymptotic solution (16) in Eq. (1) and taking into account the equations for the regular and the singular parts of the asymptotics.

The searched residual function is determined by the formula

$$g_0(x, t, \varepsilon) = \frac{1}{\varepsilon} [a_0(x, t) - a_0(\varphi(t), t)] \varphi'(t) \frac{\partial v_0}{\partial \tau} \\ + \frac{1}{\varepsilon} [u_0(x, t) b_0(x, t) - u_0(\varphi(t), t) b_0(\varphi(t), t)] \varphi'(t) \frac{\partial v_0}{\partial \tau} \\ + \frac{1}{\varepsilon} [b_0(x, t) - b_0(\varphi(t), t)] v_0 \frac{\partial v_0}{\partial \tau} + a_0(x, t) \frac{\partial v_0}{\partial t} \\ + [a_1(x, t) + b_1(x, t)(u_0 + v_0)] \frac{\partial v_0}{\partial \tau} + O(\varepsilon). \quad (17)$$

For all $(x, t)$ belonging to the set
$$ \Omega_{\mu}(\Gamma) = \{(x; t) \in \mathbb{R} \times [0; T] : |x - \varphi(t)| < \mu\}, $$

where $\mu > 0$, we have inequalities

$$ \begin{aligned} & |a_0(x, t) - a_0(\varphi(t), t)| \le C_1|x - \varphi(t)|, \\ & |b_0(x, t) - b_0(\varphi(t), t)| \le C_2|x - \varphi(t)|, \\ & |u_0(x, t)b_0(x, t) - u_0(\varphi(t), t)b_0(\varphi(t), t)| \le C_3|x - \varphi(t)| \end{aligned} $$

with some positive constants $C_k, k = 1, 2, 3$.

Because $\frac{\partial v_0}{\partial \tau}$ is a rapidly decreasing function with respect to the variable $\tau$, the asymptotic equality $g_0(x, t, \varepsilon) = O(1)$ is true.

Since $v_0(x, \tau) \in G$, the function $v_0(x, \tau)$ obviously satisfies Eq. (1) with accuracy $O(\varepsilon)$ as $\tau \to +\infty$. If the value of $A(t)$ in (14) is a constant, then the function $v_0(x, \tau) \in G$ satisfies condition $v_{0t}(x, \tau) \in G_0$. Thus, the function $v_0(x, \tau)$ satisfies Eq. (1) with an asymptotic accuracy $O(\varepsilon)$ as $\tau \to -\infty$.

Theorem 1 is proven.

## B. Higher terms of the singular part of the asymptotics

Let us consider equations for higher terms of the singular part of the asymptotics on the discontinuity curve $\Gamma$. Remind that the functions $v_j = v_j(t, \tau)$, $j \in \mathbb{N}$, belong to the space $G$. We have

$$ \frac{\partial^2 v_j}{\partial \tau^2} + a_0(\varphi, t)\varphi'(t) \frac{\partial v_j}{\partial \tau} - b_0(\varphi, t) \left[ u_0(\varphi, t) \frac{\partial v_j}{\partial \tau} + \frac{\partial}{\partial \tau} (v_0 v_j) \right] = \mathcal{F}_j(t, \tau), \quad (18) $$

where

$$ \mathcal{F}_j(t, \tau) = F_j(t, V_0(x, t, \tau), \dots, V_{j-1}(x, t, \tau), u_0(x, t), \dots, u_j(x, t))|_{x=\varphi(t)} $$

are defined recurrently after finding the functions $u_0(x, t)$, $u_1(x, t)$, ..., $u_j(x, t)$, $V_0(x, t, \tau)$, $V_1(x, t, \tau)$, ..., $V_{j-1}(x, t, \tau)$, $j = 1, \dots, N$. In particular,

$$ \begin{aligned} \mathcal{F}_1(t, \tau) = & a_0(\varphi, t) \frac{\partial v_0}{\partial t} + b_0(\varphi, t) u_{0x}(\varphi, t) v_0 \\ & + [-a_1(\varphi, t) \varphi'(t) + b_1(\varphi, t) u_0(\varphi, t), t] \\ & + b_0(\varphi, t) u_1(\varphi, t)] \frac{\partial v_0}{\partial \tau} + \tau [-a_{0x}(\varphi, t) \varphi'(t) \\ & + b_{0x}(\varphi, t) u_0(\varphi, t) + b_0(\varphi, t) u_{0x}(\varphi, t)] \\ & \times \frac{\partial v_0}{\partial \tau} + [b_1(\varphi, t) + \tau b_{0x}(\varphi, t)] v_0 \frac{\partial v_0}{\partial \tau}. \end{aligned} \quad (19) $$

Since the variable $t$ can be considered as a parameter in the equation, we integrate this equation in the variable $\tau$. We obtain

$$ \frac{\partial v_j}{\partial \tau} = [-a_0(\varphi(t), t)\varphi'(t) + b_0(\varphi(t), t)(u_0(\varphi(t), t) + v_0)] v_j + \Phi_j(t, \tau), \quad (20) $$

where

$$ \Phi_j(t, \tau) = \int_{\tau_0}^{\tau} \mathcal{F}_j(t, \xi) d\xi, \quad j \in \mathbb{N}. \quad (21) $$

Solution of Eq. (20) can be explicitly written in the following form:

$$ v_j(t, \tau) = \left( C_0 \cosh^2(\beta \tau_0) + \int_{\tau_0}^{\tau} \cosh^2(\beta s) \Phi_j(t, s) ds \right) \cosh^{-2}(\beta \tau), \quad (22) $$

where $C_0$ and $\tau_0$ are arbitrary constants, and the value $\beta = \beta(t)$ is defined with formula (15), $j \in \mathbb{N}$.

The following lemma is true.

**Lemma 1.** If $\Phi_j(t, \tau) \in G$, $j \in \mathbb{N}$, then for any $t \in [0; T]$ function (22) is a bounded solution of Eq. (20).

Statement of Lemma 1 concludes from the following inequality:

$$ \left| \cosh^{-2}(\beta \tau) \int_{\tau_0}^{\tau} \cosh^2(\beta s) \Phi_j(t, s) ds \right| \le c_{1j} \cosh^{-2}(\beta \tau) \left( \frac{\sinh(2\beta \tau)}{4\beta} + \frac{\tau}{2} + c_{2j} \right) < \infty, $$

where $c_{1j}$ and $c_{2j}$ are some constants, $j \in \mathbb{N}$.

Lemma 1 is proven.

Let us consider function (22). It is easy to notice the following properties:

(1) The function $v_0(t, \tau)$ is written via $\tanh(\beta \tau)$;

(2) The right-side function (18) depends on $v_k(t, \tau)$, $k = 0, \dots, j-1$, and their derivatives with respect to the variable $\tau$.

So, it is possible to assume that under some conditions on the coefficients of series (2), the functions $\Phi_j(t, \tau)$, $j \in \mathbb{N}$, may have the following form:

$$ \Phi_j(t, \tau) = C_{10}(t) + \sum_{k=1}^{3} C_{1k}(t) \tanh^k(\beta \tau) + \sum_{k=1}^{m} C_{2k}(t) \cosh^{-k}(\beta \tau), \quad (23) $$

where $C_{1k}(t)$, $k = 0, \dots, 3$, $C_{2k}(t)$, $k = 1, \dots, m$, are some values, $m$ is a natural, and $t$ is a parameter.

It is easy to see that under some conditions on the coefficients of expansion in (23), function (22) belongs to the space $G$. Really, for example, let us study the first term of the singular part of the asymptotics on the curve $\Gamma$ for the case of zero background, when the regular part of the asymptotics is trivial $U_N(x, t, \varepsilon) = 0$. Then, we have

$$ \begin{aligned} \Phi_1(t, \tau) = & \alpha_0 + \alpha_1 \tau + \alpha_2 \tau \tanh(\beta \tau) + \alpha_3 \tau \tanh^2(\beta \tau) \\ & + \alpha_4 \ln \cosh(\beta \tau) + \alpha_5 \tanh(\beta \tau) + \alpha_6 \tanh^2(\beta \tau), \end{aligned} \quad (24) $$

where

$$ \alpha_0 = \alpha_0(t) = -A a_1(\varphi(t), t) \varphi'(t) + \frac{A^2}{2} b_1(\varphi(t), t), \quad (25) $$

$$ \alpha_1 = \alpha_1(t) = a_0(\varphi(t), t) \frac{dA}{dt} - \frac{A^2}{2} b_{0x}(\varphi(t), t), \quad (26) $$

$$ \begin{aligned} \alpha_2 = \alpha_2(t) = & -\frac{A}{\beta} a_0(\varphi(t), t) \frac{d\beta}{dt} + A a_{0x}(\varphi(t), t) \varphi'(t) \\ & - A^2 b_{0x}(\varphi(t), t), \end{aligned} \quad (27) $$

$$ \alpha_3 = \alpha_3(t) = \frac{A^2}{2} b_{0x}(\varphi(t), t), \quad (28) $$
$$ \alpha_4 = \alpha_4(t) = -a_0(\varphi(t), t) \frac{d}{dt} \left(\frac{A}{\beta}\right) - \frac{A}{\beta} a_{0x}(\varphi(t), t) \varphi'(t) + \frac{A^2}{\beta} b_{0x}(\varphi(t), t), \quad (29) $$

$$ \alpha_5 = \alpha_5(t) = A a_1(\varphi(t), t) \varphi'(t) - A^2 b_1(\varphi(t), t) + \frac{A^2}{2\beta} b_{0x}(\varphi(t), t), \quad (30) $$

$$ \alpha_6 = \alpha_6(t) = \frac{A^2}{2} b_1(\varphi(t), t). \quad (31) $$

Here, the values A and β are defined by formulas (14) and (15).

Since the function $\Phi_1(t, \tau)$ must be bounded as an element of the space G, it is necessary the following condition

$$ \alpha_1 = \alpha_2 = \alpha_3 = \alpha_4 = 0 \quad (32) $$

takes place.

Using formulas (26)–(29), we write condition (32) in the terms of the coefficients of Eq. (1). From condition $\alpha_3 = 0$, it is deduced equality $b_{0x}(\varphi(t), t) = 0$ that is fulfilled, for example, in the case $b_0(x, t) = b_0(t)$. Under this assumption from condition $\alpha_1 = 0$, we find relation $A'(t) = 0$, which is equivalent to the nonlinear first order ordinary differential equation for determining the discontinuity curve,

$$ \frac{d\varphi}{dt} = \rho \frac{b_0(t)}{a_0(\varphi, t)}, \quad (33) $$

where $\rho \neq 0$ is an arbitrary real.

The function $\varphi = \varphi(t)$ satisfies Eq. (33) with initial condition $\varphi(0) = \varphi_0$ and is defined on some (maximal) interval $(\omega_-, \omega_+) \subset \mathbb{R}$, where the values $\omega_-$, $\omega_+$ depend on initial value $\varphi_0$. In general case, the interval $(\omega_-, \omega_+)$ is finite, although, above all, of considerable interest is the case when the function $\varphi = \varphi(t)$ is given for all $t \ge 0$.

If (33) holds, then the conditions $\alpha_2 = \alpha_4 = 0$ are equivalent to the relation

$$ a_0(\varphi(t), t) b'_0(t) = \rho b_0^2(t) a_{0x}(\varphi(t), t). \quad (34) $$

Thus, if the coefficients $a_0(x, t)$, $b_0(x, t)$ in (2) satisfy conditions $a_0(x, t) = a_0(t)$, $b_0(x, t) = b_0 = \text{const}$, and $\varphi = \varphi(t)$ is a solution of Eq. (33), then the function $\Phi_1(t, \tau)$ belongs to the space G. In addition, according to Lemma 1, the solution $v_1(t, \tau)$ of Eq. (20) is a bounded function that can be analytically obtained from formula (22).

### C. Asymptotic estimate for the first approximation

Let $\tau_0 = 0$ in (22) and conditions (32) be fulfilled. Taking into account representation (24), we find the function $v_1(t, \tau)$ explicitly. We have

$$ v_1(t, \tau) = \frac{\alpha_5}{2\beta} + \left[ c - \frac{\alpha_5}{2\beta} + (\alpha_0 - \alpha_6) \frac{\tau}{2} \right] \cosh^{-2}(\beta\tau) + \frac{\alpha_0 + \alpha_6}{2\beta} \tanh(\beta\tau), \quad (35) $$

where the values $\alpha_0$, $\alpha_5$, and $\alpha_6$ are defined by (25), (30), and (31), and c is a constant of integration.

Obviously, the constructed function $v_1(t, \tau) \in G$. Then, the first asymptotic approximation for the solution of Eq. (1) can be written as

$$ Y_1(x, t, \varepsilon) = v_0(t, \tau) + \varepsilon v_1(t, \tau), \quad \tau = \frac{x - \varphi(t)}{\varepsilon}, \quad (36) $$

where $v_0(t, \tau)$ and $v_1(t, \tau)$ are defined by formulas (13) and (35), respectively.

The property is deduced from the following theorem.

**Theorem 2.** Let the following conditions be fulfilled:

(1) The functions $a_0(x, t)$, $b_0(x, t)$, $a_1(x, t)$, $b_1(x, t)$ are infinitely differentiable with respect to the variables $(x, t) \in \mathbb{R} \times [0; T]$ and $a_0(x, t) \neq 0$, $b_0(x, t) = b_0(t) \neq 0$;

(2) The function $\varphi = \varphi(t)$, $t \in [0; T]$, is a solution of differential Eq. (33);

(3) Relation (34) takes place.

Then, function (36) is the first approximation for the asymptotic step-like solution of Eq. (1) and satisfies Eq. (1) with an asymptotic accuracy $O(\varepsilon)$ on the set $\mathbb{R} \times [0; T]$. Moreover, it satisfies this equation with an asymptotic accuracy $O(\varepsilon^2)$ as $\tau \to +\infty$.

If additionally the condition

$$ \frac{d a_1(\varphi(t), t) \varphi'(t) - b_1(\varphi(t), t) A(t)}{dt} = 0 \quad (37) $$

holds, where the value $A(t)$ is given with formula (14), then function (36) satisfies Eq. (1) with an asymptotic accuracy $O(\varepsilon^2)$ as $\tau \to -\infty$.

**Proof.** As in the case of proving Theorem 1, it is necessary to estimate the residual value for function (36) and Eq. (1). Substituting asymptotic solution (36) in Eq. (1) and taking into account the equations for the singular terms, we get explicit expression for the residual value. Thus, we have to estimate the function

$$ g_1(x, t, \varepsilon) = [a_0(x, t) - a_0(\varphi(t), t)] \frac{\partial v_0}{\partial t} - \frac{1}{\varepsilon} \left[ (a_0(x, t) - a_0(\varphi(t), t) - \varepsilon \tau a_{0x}(\varphi(t), t)) + \varepsilon(a_1(x, t) - a_1(\varphi(t), t)) \right] \varphi' \frac{\partial v_0}{\partial \tau} - [a_0(x, t) - a_0(\varphi(t), t)] \varphi' \frac{\partial v_1}{\partial \tau} + \varepsilon[(a_1(x, t) - a_2(x, t) \varphi')] \frac{\partial v_0}{\partial \tau} + (a_0(x, t) - a_1(x, t) \varphi') \frac{\partial v_1}{\partial \tau} + \frac{1}{\varepsilon} \left[ (b_0(x, t) - b_0(\varphi(t), t) - \varepsilon \tau b_{0x}(\varphi(t), t)) + \varepsilon(b_1(x, t) - b_1(\varphi(t), t)) \right] v_0 \frac{\partial v_0}{\partial \tau} + \varepsilon \left[ b_1(x, t) \frac{\partial}{\partial \tau} (v_0 v_1) + \frac{1}{2} b_2(x, t) \frac{\partial v_0^2}{\partial \tau} + \frac{1}{2} b_0(x, t) \frac{\partial v_1^2}{\partial \tau} \right] + [b_0(x, t) - b_0(\varphi(t), t)] \frac{\partial}{\partial \tau} (v_0 v_1) + O(\varepsilon^2). \quad (38) $$

Consider the first two terms of the residual function. Since the explicit formula for $v_0(t, \tau)$ and condition (2) of Theorem 2 directly deduce inclusion $v_{0t} \in G_0$, for the first term we obtain the asymptotic estimate
$$ \begin{aligned} & \left| [a_0(x, t) - a_0(\varphi(t), t)] \frac{\partial v_0}{\partial t} \right| \\ & \le |a_0(x, t) - a_0(\varphi(t), t)| \left| \frac{\partial v_0}{\partial t} \right| \le M_1 \varepsilon |\tau| \left| \frac{\partial v_0}{\partial t} \right| \le C_1 \varepsilon, \end{aligned} $$

where $M_1 > 0$ and $C_1 > 0$ are certain values.

Analogously, for the second term of the residual function (38), we have inequalities

$$ \begin{aligned} & \left| \frac{1}{\varepsilon} [a(x, t) - a_0(\varphi(t), t) - \varepsilon \tau a_{0x}(\varphi(t), t)] \frac{\partial v_0}{\partial \tau} \right| \\ & \le \frac{1}{\varepsilon} M_2 \varepsilon^2 \tau^2 \left| \frac{\partial v_0}{\partial \tau} \right| \le C_2 \varepsilon, \end{aligned} $$

where $M_2 > 0$ and $C_2 > 0$ are some constants.

Analogously, the other terms of residual function (38) are estimated.

So, we have an asymptotic equality $g_1(x, t, \varepsilon) = O(\varepsilon)$.

Because $v_0(t, \tau)$, $v_1(t, \tau) \in G$, the function $Y_1(x, t, \varepsilon) \in G$ obviously satisfies Eq. (1) with an asymptotic accuracy $O(\varepsilon^2)$ as $\tau \to +\infty$.

If additionally in formula (35) value

$$ \frac{\alpha_0 + \alpha_6}{2\beta} $$

does not depend on variable $t$, i.e., condition (37) holds, then the function $Y_1(x, t, \varepsilon)$ satisfies condition $Y_{1t}(t, \tau) \in G_0$. So, the function $Y_1(x, t, \varepsilon)$ satisfies Eq. (1) with an accuracy $O(\varepsilon^2)$ as $\tau \to -\infty$.

Theorem 2 is proven.

In this way, the higher terms of the singular part of the step-like asymptotics can be constructed, but these formulas are very cumbersome. Therefore, they are not written here because for an adequate description of the solutions of the differential equations with a small perturbation, as a rule, it is sufficient to consider the asymptotic approximation of the first order.25-28

## IV. EXAMPLE

Let us consider the Burgers equation with variable coefficients of the following form:

$$ \varepsilon u_{xx} = (t^2 + 1 + \varepsilon(x^2 + 1)^2)u_t + \left(1 + \varepsilon \frac{(x^2 + 1)^2}{t^2 + 1}\right)u_x \quad (39) $$

and construct the first approximation for its asymptotic solution for the case of zero background.

For this example, the first coefficients of asymptotic series (2) are written as

$$ a_0(x, t) = t^2 + 1, \quad a_1(x, t) = (x^2 + 1)^2, \quad (40) $$

$$ b_0(x, t) = 1, \quad b_1(x, t) = \frac{(x^2 + 1)^2}{t^2 + 1}. \quad (41) $$

Taking into account (40) and (41) and zero background conditions

$$ u_0(x, t) = u_1(x, t) = 0, $$

from (33) we get equation for the function $\varphi = \varphi(t)$ in the form

$$ (t^2 + 1)\varphi' = \rho, \quad \rho \neq 0. $$

The solution of this equation with initial condition $\varphi(0) = 0$ is written as

$$ \varphi(t) = \rho \arctan t $$

and defined for all $t \in \mathbb{R}$.

Then, we find $A = A(t) = \rho, \beta = \rho/2$. You can easily make sure that conditions (34) and (37) with $C_0 = 0$ are fulfilled.

Thus, all conditions of Theorem 1 and Theorem 2 are fulfilled.

The main term of the singular part of the asymptotics for Eq. (39) is given with the following formula:

$$ V_0(x, t, \varepsilon) = 1 - \tanh \left( \rho \frac{x - \rho \arctan t}{2\varepsilon} \right), \quad (42) $$

and the first term of the singular part of the asymptotics is written as

$$ V_1(x, t, \varepsilon) = \left[ c - \rho^2 \frac{(1 + \rho^2 \arctan^2 t)^2}{t^2 + 1} \frac{x - \rho \arctan t}{2\varepsilon} \right] \times \cosh^{-2} \left( \rho \frac{x - \rho \arctan t}{2\varepsilon} \right). \quad (43) $$

It is easy to see that $V_0(x, t, \varepsilon) \in G$, $V_1(x, t, \varepsilon) \in G_0$.

The function

$$ Y_1(x, t, \varepsilon) = 1 - \tanh \left( \frac{x - \rho \arctan t}{2\varepsilon} \right) + \varepsilon \left[ c - \rho^2 \frac{(1 + \rho^2 \arctan^2 t)^2}{t^2 + 1} \frac{x - \rho \arctan t}{2\varepsilon} \right] \times \cosh^{-2} \left( \rho \frac{x - \rho \arctan t}{2\varepsilon} \right) \quad (44) $$

is the first asymptotic step-like approximation for solution of Eq. (39) and, according to Theorem 2, satisfies the equation with an asymptotic accuracy $O(\varepsilon)$. Moreover, the first approximation satisfies the equation with an accuracy $O(\varepsilon^2)$ as $\tau \to \pm\infty$.

Graphs of functions (42), (43), and (44) are given on plots 1-3 for $\rho = 1, c = 0, \varepsilon = 0, 9, \varepsilon = 0, 45$, and $\varepsilon = 0, 15$. These graphs (Figs. 1-3) demonstrate properties of solutions (1) inherent in functions that mathematically describe shock waves. Graphs 1-3 allow us to observe the influence of a small parameter, which consists in the fact that the wave front becomes almost vertical as $\varepsilon$ tends to zero.

## V. CONCLUSIONS AND DISCUSSIONS

The paper deals with the problem on constructing asymptotic step-like solutions of the Burgers equation with variable coefficients and a small parameter. This equation can be considered as a mathematical model of wave processes in media with a small viscosity and variable characteristics and as generalization of the classical Burgers equation. Partial differential equations with variable coefficients are more complicated than those with constant coefficients, but they allow a more accurate description of physical phenomena. Some generalized Burgers equations with variable coefficients have been fruitfully studied in the framework of symmetry analysis (see Refs. 29 and 30 and references therein). As a result, families of their exact solutions were constructed for certain coefficients. Equation (1) has a more general form than in Refs. 29 and 30 and is of great importance for the study of nonlinear waves.
<figure><img src="image_18.png" /><figcaption>FIG. 1. The main term of the asymptotic solution $V_0(x, t, \epsilon)$ as $\epsilon = 0.9$ (on the top), $\epsilon = 0.45$ (in the middle), and $\epsilon = 0.15$ (on the bottom).</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>FIG. 2. The term $\epsilon V_1(x, t, \epsilon)$ as $\epsilon = 0.9$ (on the top), $\epsilon = 0.45$ (in the middle), and $\epsilon = 0.25$ (on the bottom).</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>FIG. 3. The first order asymptotic solution $Y_1(x, t, \epsilon)$ as $\epsilon = 0.9$ (on the top), $\epsilon = 0.45$ (in the middle), and $\epsilon = 0.25$ (on the bottom).</figcaption></figure>

The Burgers equation is known as the simplest mathematical model, which unified typical nonlinearity and viscosity and describes phenomena of hydrodynamical turbulence, blow-up, and overturning (destruction) of waves. When the viscosity tends to zero, the limiting value of the solution is a discontinuous step-like function. Naturally, the problem of finding asymptotic step-like solutions of the Burgers equation with variable coefficients (1) arises. These solutions can be considered as a deformation of the step-like waves of the classical Burgers equation [Eq. (3)].

In this paper, the asymptotic step-like solution of Eq. (1) is constructed by means of the nonlinear WKB method<sup>21,31</sup> since it gives a congruent approximate description of wave processes. This is confirmed by the established accuracy with which the solutions satisfy the considered equation. In addition, the singular part of the constructed asymptotic step-like solution (7) coincides with exact solution (5) in the case of the classical Burgers equation [Eq. (3)]. It follows from formulas (13) and (35). The obtained results are illustrated by an example, for which the first asymptotic step-like approximation is found in an exact form (44). The given formula demonstrate that the wave velocity is variable due to the coefficients of Eq. (39). There are also given graphs of these approximate solutions for certain numerical parameters (Figs. 1–3). In this example, the asymptotic solution, in contrast to the general case, is defined for all values of space and time variables, and has the form of the shock wave type function for sufficiently small parameter $\epsilon$. Analogous to the case of the classical Burgers equation, the constructed asymptotic step-like solution of Eq. (39) tends to discontinuous step-like function as a small parameter tends to zero. Plots 1 and 3 also demonstrate that the main asymptotic term adequately describes the solution of the equation under consideration, while the first asymptotic term provides a complementary approximation.

## ACKNOWLEDGMENTS

A part of this work has been made when the first author was a visitor at Department of Basic and Applied Science for Engineering at Sapienza-University of Rome, through the program “Professori visitatori che si trovino in condizioni di rischio a causa di conflitti.” Hence, he is very grateful for the support and the hospitality. The first and the last author also acknowledge the support of INdAM “Professori visitatori Sportello Ucraina.” The last author is a member of GNAMPA-INdAM, whose support is acknowledged. The second author was supported by Programme PAUSE and the ANR, Project No. ANR-22-PAUK-0038-01 in Claude Bernard Lyon I University, and she is very appreciative and grateful for the support and the hospitality. The authors are also sincerely grateful to Professor Lorenzo Brandolese (Institute Camille Jordan, Claude Bernard Lyon I University) and Professor Volodymyr Matsypura (Taras Shevchenko National University of Kyiv) for discussions and remarks that were useful in preparing the final draft of the manuscript. Finally, we would like to sincerely thank the anonymous referees whose remarks and comments were very useful for us.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.
## Author Contributions

**Valerii Samoilenko:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). **Yuliia Samoilenko:** Conceptualization (lead); Formal analysis (equal); Investigation (equal); Methodology (lead); Supervision (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). **Elvira Zappale:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (supporting); Supervision (equal); Validation (supporting); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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