Abstract
We present a new approach to find travelling wave solutions for the Korteweg-de Vries type equations, which allows extending the class of known soliton solutions. Also we propose method for studying the multisoliton solutions of the Korteweg-de Vries type equations.
1. Introduction
In recent years the investigation of separated waves plays an important role in many applied scientific fields. Travelling wave solutions can describe various phenomena in fluid mechanics, hydrodynamics, optics, plasma physics, solid state physics, biology, meteorology, and other fields. Pay attention that separated waves often occur at the boundaries of dynamic environments with different physical characteristics, such as “water-air” (in this case we consider the shallow water equations), and the limits of stratified fluids, on the verge of “gas-vacuum” (in [1] it is considered thin gas disks that rotate in a gravitational field) and on the border of crust and mantle (Moho surface).
Many models were proposed to describe the physical phenomena of separated waves existence and a variety of methods were proposed to construct the exact and approximate solutions to nonlinear equations. It is well known the KdV equations describe the unidirectional propagation of shallow water waves and a number of generalizations; for example, (is given by [2]), , (is given by [3]), (is given by [4]), (equation Kuramoto-Sivashinsky, KdV [5]), (the Kawahara equation [6]), , (KdV-Burgers-Kuramoto [7]).
The KdV equations extended to several physical problems such as long internal waves in a density stratified ocean and acoustic waves on a crystal lattice. In recent years many efficient methods of finding the traveling wave solutions were developed such as Infinite Series method [8], Backlund transformation method [9], Darboux transformation [10], tanh method [11, 12], extended tanh function method [4], modified and extended tanh function method [13], the generalized hyperbolic function [14], the variable separation method, first integral method, and exp-function method [15]. A number of papers were devoted to the problems of the asymptotic solutions of the Korteweg-de Vries equation [16].
In this paper, we propose a new technique of finding the PDE’s traveling wave solutions which is based on the T-transformations.
2. New Solution of KdV Equation Based on the T-Representation
Let us consider a general nonlinear differential equation in the formwhere is dependent variable, , are independent variables, and is a polynomial function concerning indicated variables. According to well known travelling wave approach we unite the independent variables and into one particular wave variable , where is speed of the wave. Then (1) is converted to ODE. But then we have problem which is how to find separated wave solution.
Let be function, determining the point of maximum wave disturbance. Define a function that describes the shape of the wave in the form , where is sufficiently smooth function, which satisfies conditions
Obviously that function can describe a positive perturbation of any shape; is amplitude parameter; is a parameter that determines the location of disturbance. In the simplest case function is measure defined on the set of intervals in .
Let be a function, . Let us consider the solution which we are looking for in the next form:
It is easy to build some kind of solution (3) that would be general for the system of differential equations in partial derivatives:
We expand the solution in the formwhere , is function of measures defined on the set of intervals , (, ), are amplitude functions, is a parameter that determines the location of disturbance, and are functions that determine the trajectory of perturbation.
Representation (3) makes it possible to find new approach to research the multisoliton solution. In the case of small or large parameter we can investigate some infinitesimal properties of travelling wave solution. Let us consider an example, the well known Korteweg-de Vries equation in the formWe can write the derivatives of the function :Substituting derivatives into (5) we obtain the equation
Let . Then, based on the properties of functions from (8) we obtainTherefore we can formulate proposition.
Proposition 1. All travelling wave solutions of the KdV equation have constant amplitude.
From (8) we obtainEquation (10) defines a general condition for functions and at representation (3). Formulate the following statement.
Proposition 2. Infinitely wide soliton (2) is a trivial solution of the KdV equation.
This property follows from the analysis of the boundaries of the form
Proposition 3. Infinitely narrow soliton (2) is a solution of the KdV.
This proposition follows from the limitsThe last limit follows from these considerations:
It is obvious that similar properties have the solutions of other KdV-type equations that admit a trivial solution. At the same time, special KdV with variable coefficients species (given by [15]) does not have the appropriate properties.
Let . Then (10) can be expressed in the form
Let , . Then we obtain
Let . Тhen
Let . ThereforeWe obtain linear equation. The partial solution can be obtained in the formThe general solution is .
Therefore and .
We obtain the conditionUnder the condition of (2), we have .
Finally we can define the function as solution of Koshi problem:The well known solution of KdV equation can be shown in the formwhere , are parameters.
Let us prove that (22) is a partial case of (3). Let . Then from (10) we obtain . Problem (21) can be shown in the formTaking into account (22) and (2), we get .
Therefore .
Let . Then
Let . It is easy to prove that (24) is solution of (23). Differentiating we obtain . Substituting (24) into second part of (23), we get
The proof is completed.
3. Method for Studying the Interaction of Solitons
Let us consider the double-soliton solution. Let , , exact solutions of the KdV equation. Obviously, the function is a solution of the KdV equation in the region . Therefore, we consider the following generalization. Let the parameters determining the amplitudes be functions of time, .
Substituting the sum into (6), we obtain
Let . If and from (26) yield
then from (27) we get
Writing the obvious initial conditions and , we get the Cauchy problem that can be solved with known velocities , . Thus, we find exact solutions of the KdV equation in the region .
Let , , . For the well known KdV equation solution the rate is equal to the double amplitude. Then from (27) we obtain the Cauchy problem:
In Figure 1 we show a graphical illustration of appropriate solutions.
Obviously, the proposed solution in the form is approximate and the KdV equation is satisfied only in the area . For further clarification, consider new functions in the forms and , where , are some parameters and inserts to the KdV equation. Let . To simplify record superscript amplitude functions will lower. Using a similar approach, if we getwhere .
Similarly and :
Hence we obtain the system of equations:
Writing the obvious initial conditions, and , we get new Cauchy problem and can build the exact solution in the area .
Besides, we can consider the other laws of wave motion. Suppose
For this case we construct similar Cauchy problem for the amplitude functions and specification. An example of corresponding solutions is shown in Figure 5.
The -soliton solutions for any can be always constructed in a similar way (Figure 6).
4. Numerical Results
Let us consider some numerical results for special case of the functions and . Let , , , and . These parameters satisfied (10). In Figure 1 we plot 2D graphs of the function , in Figure 2 , in Figure 3 classical soliton solution (22), and in Figure 4 amplitude functions, solution of system (29).
Let us consider an example of soliton interaction. According to approach given in Section 3, define the functions , . Let and , , . Solving problem (29) with initial conditions , , , and , we can construct function .
In case (33) we get a similar system of equations and for initial conditions , , , and , see Figure 7.
5. Conclusions
In this paper, we propose a new technique of finding the PDE’s traveling wave solutions; this technique is based on the T-transformations. Using T-representation method we find a new class of KdV solution and prove that well known solution (22) is a partial case of representation (3).
The proposed method can be applied to find solutions of other differential equations in partial derivatives in the form of solitary waves and can be useful for the investigation of multisoliton solutions. In order to obtain the resultant equation for the amplitudes of perturbations representations (1) or their combinations should be used. We get the resulting equation only for the maximum of perturbation and ignore the exact wave profiles. Wave profile can be found in the simplest cases, in particular for one-soliton dynamics. This approach is effective for equations of shallow water and others.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.