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The Hydrodynamic-type Equations and the Solitary Solutions

Sergiy LYASHKO1, Valerii SAMOILENKO1, Yuliia SAMOILENKO2,3 and Ihor GAPYAK1

1 Taras Shevchenko National University of Kyiv, Ukraine

2 National Technical University of Ukraine Igor Sikorsky Kyiv Polytechnic Institute, Ukraine

3 Claude Bernard Lyon 1 University, France

This chapter analyzes the problem of solitary waves for hydrodynamic-type equations. It mainly focuses on the Korteweg–de Vries (KdV) equation with a small singular perturbation. It should be noted that in the study of various problems associated with the singularly perturbed KdV and Kdv-like equations, linear and nonlinear Wentzel–Kramers–Brillouin (WKB) methods are used. A description of the main idea of this technique is given, and its effectiveness for constructing asymptotic soliton-like solutions for KdV and KdV-like equations with variable coefficients and a singular perturbation is provided.

1.1. Introduction

Nonlinear hydrodynamic-type equations are associated with important models of modern physics and applied mathematics (Whitham 1974; Ablowitz 2011). These equations arise in the study of various wave processes and phenomena in hydrodynamics and many other fields of natural science. They are quite difficult to analyze because of their nonlinear nature and the fundamental impossibility of their integration in a closed form. Therefore, it is natural to study either their particular solutions, for example, solutions in the form of a traveling wave, or such equations, using numerical analysis.

These equations include the Boussinesq equation, the Korteweg–de Vries equation (the KdV equation), the modified Korteweg–de Vries equations, the Burgers’ equation and the Benjamin–Bona–Mahony equation (Bullough and Caudry 1980; Lamb 1980; Dodd et al. 1982; Newell 1985). The equations have the traveling wave solution u = f(x − at).

The Boussinesq equation:

and the Korteweg–de Vries equation:

appeared in the 19th century as a result of scientific discussions by many outstanding scientists about the propagation of long waves in rectangular channels and the search for mathematical models to describe the solitary waves that were observed by John Scott Russell in 1834 (Russell 1844).

Equations [1.1] and [1.2] have solutions that describe the waves with profile Acosh−2 (c(x − vt)). A characteristic feature of waves is localization in space and time, as well as the dependence of the propagation velocity on their amplitude (Bullough and Caudry 1980).

The Burgers’ equation (Burgers 1939)

is one of the simple models describing the evolution of shock waves. It is deduced from the well-known Navier–Stokes system in the one-dimensional case.

The Benjamin–Bona–Mahony equation or regularized long wave equation (Peregrin 1966)

was proposed in 1966 because of a search for an alternative to the KdV equation for describing long waves on the surface of a liquid. Although equation [1.4] has one-soliton solutions (Benjamin et al. 1972), it does not have m-soliton (m ≥ 2) solutions. Moreover, [1.4] is not an integrable system because it only has three conservation laws.

The integrability of the Korteweg–de Vries (KdV) equation and the KdV-like equations makes it possible to use powerful algebro-geometric approaches (Blackmore et al. 2011) and methods of the Hamiltonian analysis (Faddeev and Takhtajan 1987) to study the properties of the equations and construct a wide set of their exact solutions.

However, the consideration of wave processes in media with variable characteristics and, in particular, with a small dispersion, requires the study of nonlinear equations with variable coefficients and small perturbations. The presence of variable coefficients in nonlinear equations significantly complicates the researching of the corresponding systems. As a result, either the methods of asymptotic analysis are the only mathematical tool for studying such systems or such equations must be studied using numerical analysis (Lyashko 1991, 1995).

1.2. The Korteweg–de Vries equation and the soliton solutions

Most of the above-mentioned equations are models of fluid motion. The Korteweg–de Vries equation is used to mathematically describe the dynamics of solitary waves on a liquid surface. Later, in the middle of the 20th century, it was found that equation [1.2] and the KdV-like equations arise while studying many various phenomena and processes, in particular, in plasma, solid body theory, optic, biology, telecommunication systems, etc. Equation [1.2] is currently one of the fundamental equations of modern physics. If we consider this equation as an example, then we can find many different characteristic phenomena and properties that are inherent in hydrodynamic-type equations.

An in-depth study of the Korteweg–de Vries equation began in the late 1960s after the publication of the paper by Zabusky and Kruskal (1965). The authors pointed out the connection between the equation and the Fermi–Pasta–Ulam problem. Zabusky and Kruskal found that the Korteweg–de Vries equation possesses solutions that have the property of retaining their waveform after collision with waves of the same nature. Such solutions are called soliton solutions or solitons (Zabusky and Kruskal 1965).

The amazing discovery drew great attention from both mathematicians and physicists to the Korteweg–de Vries equation. This attention increased significantly after the creation of a new technique called the inverse scattering transform, which has been successfully applied to the study of many integrable systems (Gardner et al. 1967). Subsequently, the inverse scattering transform became a powerful tool for studying numerous nonlinear systems of modern theoretical and mathematical physics.

Many monographs are devoted to different mathematical aspects of the theory of nonlinear integrable systems, including the Korteweg–de Vries equation. In particular, one-, two-, multi-soliton solutions, finite-gap solutions, etc. have been constructed for the KdV equation. For this equation and its generalizations, problems on existence, uniqueness and other properties (smoothness, exponential decay, etc.) of its solutions, and solutions to the Cauchy problem for various classes of initial functions were also studied (Sjoberg 1970; Hruslov 1976; Kato 1979; Faminskii 1990).

It has been found that properties of the solutions to the KdV equation essentially depend on the initial conditions (Pokhozhayev 2010; Turbal et al. 2015). In addition to soliton, periodic, finite-gap and shock wave solutions, the KdV equation has solutions with some singularities on the curves (Arkadyev et al. 1984), as well as solutions that are destroyed roughly or destroyed according to the gradient catastrophe scenario (Pokhozhayev 2010).

1.3. The Korteweg–de Vries equation with a small perturbation

The discovery of new classes of solutions to the KdV equation had a great influence on the development of the perturbation theory of nonlinear systems. Such research began with the study of the perturbed Korteweg–de Vries equation by asymptotic methods. In the paper by Miura and Kruskal (1974), the authors generalized the linear Wentzel–Kramers–Brillouin (WKB) method for the nonlinear equations and constructed the leading term of the asymptotic series for the quasi-periodic solution to the KdV equation with a small dispersion:

where δ is a small parameter. Thus, the nonlinear WKB method was created. It has become a powerful mathematical tool for studying nonlinear systems of the hydrodynamic-type (Maslov 1982; Samoilenko and Samoilenko 2008, 2019; Lyashko et al. 2021) with a singular perturbation.

By means of the Bogoliubov M.M. averaging method (Bogoliubov and Mitropolsky 1961), Flaschka et al. (1980) studied the modulation of nonlinear waves, which are described by the Korteweg–de Vries equation. Although the equation under consideration does not contain a small parameter δ explicitly, the dependence of its finite-gap solutions on the small parameter manifests through a small deformation of the associated Riemann surfaces.

Furthermore, Lax and Levermore (1983a, 1983b, 1983c) considered the problem on the limit of the solution to the Cauchy problem for the singularly perturbed Korteweg–de Vries equation [1.5], as a small parameter tends to zero. While solving the problem they used the inverse scattering transform. In particular, the authors constructed eigenfunctions for the Lax operator with a small perturbation by the linear WKB method. Subsequently, this problem under other initial conditions was considered in many papers and studied mainly by numerical methods.

Asymptotic methods turned out to be an effective means of investigating other problems for the Korteweg–de Vries...