This paper is concerned with the generalized fractional Benjamin-Bona-Mahony (gfBBM) equation
which models the propagation of small amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. The equation is derived in erbayelastic ; hae as a special case of the generalized fractional Camassa-Holm equation by using an asymptotic expansion technique. Here is a positive integer and the operator denotes the Riesz potential of order , for any
. The operator can be defined via Fourier transform by
where is the Fourier transform of a function .
The effects of the relation between the nonlinearity and the dispersion on the dynamics of solutions has been the focus of many studies. The problem mostly handled by fixing the dispersion and increasing the nonlinearity. Studies for the more physically significant case with lower dispersion has become popular only in the recent years. The well-known Korteweg-de Vries (KdV) and Benjamin-Bona-Mohany (BBM) equations are investigated by using corresponding fractional forms such as
fractional KdV (fKdV) equation and
fractional BBM (fBBM) equation. These equations have been intensively studied in the past years for in terms of global well-posedness, stability and blow-up etc. We refer fonseca ; albert ; pava for a more detailed discussion and review. The research on the more delicate case has been increased in the last few years. For , Linares et. al. linares proved the local well-posedness for the Cauchy problem and they have also investigated the solitary wave solutions in terms of existence and stability in linares2015 . The stability and linear instability results for a general nonlinearity are obtained by Pava pava . In kleinfBBM the blow-up and the global existence problems are handled and solitary wave solutions for the fKdV are constructed numerically. Duran used efficient numerical methods to investigate the solitary wave solution of the fKdV equation in aduran .
The gfBBM equation contains the fractional terms of both the fKdV and the fBBM equations, but unlike them, the gfBBM equation models a physical phenomena. The effects of these terms on the solutions when occurring together, such as well-posedness of the Cauchy problem, existence of solitary waves and the nature of solutions in time is therefore a curious problem. The aim of the current study is to investigate the dynamics of the gfBBM equation with a general power type nonlinear term.
The paper is organized as follows: In Section 2, we prove the local existence and uniqueness of the Cauchy problem for the gfBBM equation together with the initial condition
by the help of deriving a suitable energy estimate. In Section 3, we derive the conserved quantities for the gfBBM equation. The existence-nonexistence results for solitary wave solutions are given in Section 4. We use Pohozaev type identites to show the non-existence of solitary wave solutions and the results offranklenzmann are applied for the existence of positive solitary waves for certain values of and the wave speed . Section 5 is devoted to numerical investigation of the solutions; we construct the solitary wave solutions numerically by using Petviashvili method and propose a Fourier pseudo-spectral method for the time evolution of the solutions. We perform numerical experiments for several values of and to investigate the effects of dispersion and nonlinearity.
Throughout this study, is the usual Lebesgue space with the norm for . is the Sobolev space with the norm
for and denotes the generic constant. Here, the Fourier transform and its inverse for the given function defined as
denotes the Bessel potentials of order with .
2 Local Existence and Uniqueness
In this section, we obtain the local well-posedness of the gfBBM equation. The main result of the Section is as follows:
Proof: Applying the operator to the eq. (1.1), multiplying both sides of the equation by and then integrating on the whole line, we have
Using the fractional Leibniz rule kenig , the term is written as
where is the remainder. Then, the eq. (2.2) becomes
By integration by parts and the Hölder’s inequality, the first term of RHS is estimated as
Here, we have used the following Sobolev imbeddings
where implies . Note that with is an algebra. Therefore ensures , as the above condition guarantees that is an algebra. The estimation of the second term in RHS of the eq. (2) is given as
where . Here the Sobolev imbeddings
The estimation of last term of the eq. (2),
By a standard compactness method as in linares , we get the existence of a solution where .
We now prove the uniqueness of the corresponding solution. Let and be two solutions of the eq. (1.1) then satisfies the following equation
where . Multiplying both sides of the equation by , integrating on the whole line we have
Using integration by parts, we get
By the help of Hölder inequality and Sobolev imbedding theorem, the RHS of the eq. (2.8) can be estimated as
for all . Using Gronwall’s lemma, we get
Finally we conclude uniqueness result of Theorem (2.1) by choosing .
3 Conserved Quantities
In this section, we derive the conserved quantities of the gfBBM eq. (1.1) for the smooth enough solutions which tend to 0 as . The equation is rewritten in the conservative like form
Therefore, the first conserved integral is
Multiplying the eq. (1.1) by and integrating on the whole line, we have
Thanks to the Plancherel theorem, one can write
Rewriting the eq. (3) in the form
allows us to write a second conserved integral
for any . Therefore, the Cauchy problem for the eq. (1.1) admits a unique global weak solution in .
4 Solitary Wave Solutions
To find the localized solitary wave solutions of the eq. (1.1), we use the ansatz with
which leads to the ordinary differential equation
Here denotes the derivative with respect to . Integrating the above equation, we have
The following theorem shows the non-existence of the nontrivial solutions of the eq. (4.15) for some values of and .
Assume that one of the following cases
is satisfied. Then, the eq. (4.15) does not admit any nontrivial solution .
By using the Plancherel’s formula
On the other hand, multiplying the eq. (4.15) by and integrating over , we have
Later, the equality
and the proof of the theorem follows directly from the positivity of the left hand integral.
Combining the results of Theorem 4.1 and the condition which ensures that the Hamiltonian (3.14) is well-posed for we conclude that in order to have a non-trivial solution we must have and or .
On the other hand if we assume that the solution is positive and then the eq. (4.16) gives a contradiction while the RHS of the equation is positive and the LHS is negative. Therefore we are able to say that the eq. (4.15) has no positive solitary wave solutions unless .
In order to show the existence and uniqueness of the solitary wave solutions, we recall the results of Frank and Lenzmann franklenzmann .
converts the eq. (4.15) into the eq. (4.21). In Proposition and Theorem of franklenzmann , Frank and Lenzmann prove the existence and uniqueness of the positive ground state solutions of the eq. (4.21) when and holds, where the critical exponent is defined as
Therefore, the eq. (4.15) has a unique positive ground state solution for and . That is why in the following section, we choose the parameters and satisfying these conditions to obtain positive solitary wave solutions, numerically.
5 Numerical results for gfBBM Equation
In this section, we discuss the numerical solutions of the gfBBM equation. We first construct the solitary wave solutions by using the Petviashvili’s iteration method and then we investigate time evolution of the constructed solutions by a Fourier pseudo-spectral method.
5.1 Numerical Generation of Solitary Waves
Since we do not know the analytical solitary wave solutions of the eq. (1.1) for any , we use the Petviashvili’s iteration method pelinovsky ; petviashvili ; yang ; uyen to construct the solitary wave solution numerically. The solitary wave solution of the gfBBM equation satisfies the eq. (4.15). Applying the Fourier transform to the eq. (4.15) yields
We propose standard iterative algorithm in the form
where is used instead of for simplicity. The condition guarantees the non-resonance condition for any . The main idea in the Petviashvili method is to add a stabilizing factor into the fixed-point iteration in order to prevent the iterated solution to converge to zero solution or diverge. Then, the Petviashvili method for the gfBBM eq. is given by
with stabilizing factor
for some parameter . Pelinovsky et. al. in pelinovsky showed that the fastest convergence occurs when . Therefore to reduce the CPU time, we use for the rest of the paper. The Fourier pseudo-spectral method is employed to implement the scheme (5.25). The MATLAB functions “fft” and “ifft” compute the discrete Fourier transform and its inverse for any function by using efficient Fast Fourier Transform. We note that the Petviashvili iteration method can also be used for approximating the periodic waves uyen .
A solitary wave solution of the equation
satisfies the ODE
The eq. (5.27) has the solution
for in kleinfBBM . By the convenient scaling, the exact solitary wave solution of gfBBM eq. can be written as
for and .
In the first numerical experiment, we test our scheme by comparing the numerical result with the exact solution. The space interval and number of grid points are chosen as and , respectively. The overall iterative process is controlled by the error,
between two consecutive iterations defined with the number of iterations, the stabilization factor error
and the residual error
In Figure , we present the difference between the obtained numerical and exact solitary wave solution and the variation of three different errors with the number of iterations in semi-log scale. As it is seen from the Figure , our proposed numerical scheme captures the solution remarkably well.
In order to understand the effects of the fractional dispersion, we illustrate the solitary wave profiles generated by Petviashvili’s iteration method for various values of and for in Figure . We observe that the solution becomes more and more peaked with decreasing values of .
Since we do not know the exact solitary wave solution for different values of , we cannot compare the numerical solution with the exact solution. Therefore, the iteration, stabilization factor and the residual errors are depicted in Figure , respectively, for , and . These results show that the solitary wave solution generated by Petviashvili’s method converges rapidly to the accurate solution.
In order to understand the effects of the nonlinearity, we present the solitary wave profiles generated by Petviashvili’s iteration method for various nonlinearities with fixed in Figure . This numerical result agrees well with the fact that the wave steepens with increasing nonlinear effects.
In Figure , we illustrate the variation of the amplitude with the speed parameter for various values of and the fixed and various values of and fixed nonlinearity .
5.2 The Fourier-pseudospectral method for gfBBM Equation
In this subsection, we propose a Fourier pseudo-spectral method for the time evolution of numerically generated solitary waves. Since the fractional derivative in the gfBBM equation is defined by a Fourier multiplier, the Fourier spectral method will be the most appropriate method for investigating the evolution of the solution in time. We assume that has periodic boundary condition on the truncated domain .
In order to use the MATLAB functions “fft” and “ifft”, the spatial period is normalized from finite interval to using the transformation . In this case, the eq. (1.1) becomes
The interval is divided into equal subintervals with grid spacing , where the integer is even. The spatial grid points are given by , . The time interval is divided into equal subintervals with time step . The temporal grip points are shown by , . The discrete Fourier transform of the sequence , i.e.
gives the corresponding Fourier coefficients. Likewise, can be recovered from the Fourier coefficients by the inversion formula for the discrete Fourier transform (5.32), as follows:
Here denotes the discrete Fourier transform and its inverse. Applying the discrete Fourier transform to the eq. (5.31), we get the ordinary differential equation given by
We then use the fourth order Runge-Kutta method to solve the resulting ODE in time. Finally, we find the approximate solution by using the inverse discrete Fourier transform.
We first show that the numerical solution captures the exact solution (5.29) for well enough. We use the initial data (5.29) with , , . The problem is solved in the space interval up to . We set the number of grid points as . Figure illustrates variations of the conserved quantities and with time and shows that they are preserved by the numerical scheme.
Figure shows the wave profile calculated by the Fourier pseudo-spectral method with time step at and . To observe the wave profile more clear, we focus on the space interval . We also show the change in the conserved quantities and in Figure . It is seen from the figure, the conserved quantities remain constant in time and this behavior provides a valuable check on the numerical results.