Bifurcations and exact traveling wave solutions for the regularized Schamel equation

Qiue Cai; Kaixuan Tan; Jiang Li

doi:10.1515/math-2021-0136

Article Open Access

Bifurcations and exact traveling wave solutions for the regularized Schamel equation

Qiue Cai , Kaixuan Tan and Jiang Li

Published/Copyright: December 31, 2021

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$Open Mathematics$

From the journal Open Mathematics Volume 19 Issue 1

Abstract

In the present paper, we focus on studying the bifurcations and the traveling wave solutions (TWSs) for the regularized Schamel equation. Based on the bifurcation method of a dynamical system, a complete phase portrait analysis is given in various parameter conditions and some novel TWSs with the same energy of the Hamiltonian system are discovered. Various significant results on exact expressions of TWSs, including solitary waves, periodic waves, cusp waves, weak kink waves, loop solitons, compactons in different conditions are obtained.

Keywords: traveling wave solutions; phase portraits; bifurcation method; regularized Schamel equation

MSC 2010: 34C11; 34C23; 34C37

1 Introduction

The KdV equation, which has been found to possess stable steady-state solutions with remarkable properties, is usually used for describing the long waves. The form of the KdV equation [1,2] is defined by

(1) u t + u x + u u x + u x x x = 0 ,

where the terms u u x and u x x x represent the nonlinear steepening and dispersion, respectively. Recent analyses have shown that the KdV equation has wide applications as a model in shallow water, acoustic waves and heat pulses in anharmonic crystals, electron plasma waves in a cylindrical plasma, magneto-sonic waves in a magnetized plasma, ion acoustic waves and so on. The authors [3] pointed out certain difficulties associated with the KdV equation that arise spuriously, being irrelevant to the original problem. So they proposed a regularized equation, i.e., BBM equation:

(2) u t + u x + u u x − u x x t = 0 ,

as a more suitable posed model for long waves.

Originally, in order to describe the propagation of weakly nonlinear ion acoustic waves, Schamel [4,5] derived the following equation:

(3) u t + ( u + ∣ u ∣ 3 / 2 ) x + u x x x = 0 ,

which is now usually referred to as the Schamel equation. Compared to the classical KdV equation, the Schamel equation possesses a stronger nonlinearity. Carrying out much of the same regularization process leads to the regularized Schamel equation, which is defined by

(4) u t + ( u + ∣ u ∣ 3 / 2 ) x − u x x t = 0 .

Equation (4) has been studied by several authors. For example, Souganidis and Strauss [6] studied the stability and instability of solitary wave solutions for a general class of evolution equations that include (4) as a special case. Andrade and Pastor [7] obtained the existence of the cnoidal Jacobi elliptic function solution of the form

(5) ϕ ( ξ ) = ( α + β cn − 2 ( γ ξ , k ) ) 2 ,

and proved the orbital stability of such solutions in the energy space.

Both the KdV equation and BBM equation model the one-dimensional waves in a cold plasma [4], with the difference that the BBM equation describes much better the behavior of very short waves. Since the regularized Schamel equation can be derived by a similar manner from the Schamel equation, consequently, one can expect the same phenomena in this case. Schamel [5] stated that when u u x is replaced by ( ∣ u ∣ 3 / 2 ) x , compared to the classical KdV equation, the Schamel equation possesses a stronger nonlinearity, which reveals that the wave has a smaller width and higher velocity. Unfortunately, there are less works focusing on these topics that how these changes affect the dynamical behaviors of traveling wave solutions (TWSs) for the regularized Schamel equation.

In view of the closed relation between these equations (see the diagram as follows), we are naturally invited to a question whether there exist some connections between them with respect to the dynamical behaviors of TWS.

The objective of this paper is to study the TWS and obtain some new exact representations for the regularized Schamel equation based on the bifurcation method of dynamical systems [8,9, 10,11,12, 13,14,15, 16,17,18, 19,20]. Under different conditions of the same Hamiltonian, we give the phase portraits of the TWS (including solitary wave solutions, smooth periodic wave solutions, compactons, cuspons, loop solitons, etc.). As applications of our main results, we establish the relationships of the TWS among the regularized Schamel equation, Schamel equation, KdV equation and BBM equation through the nonlinear terms and dispersion terms. More precisely, our analyses show that the dispersion term has less effect on the existence of the TWS, while the strong nonlinearity will cause more complex dynamical behaviors.

The rest of the paper is organized as follows. In Section 2, we obtain the first integral of the corresponding planar dynamical system. In Section 3, we discuss the bifurcations of phase portraits of system (10). In Section 4, we present our main results by distinguishing two cases h = 0 and h ≠ 0 . In Section 5, by analyzing the regularization and nonlinearity, we establish the relationships of the TWS among regularized Schamel equation, Schamel equation, KdV equation and BBM equation as well as Schamel-like equation. In Section 6, we give a brief conclusion.

2 Preliminary

First, notice that the nonlinear term in (5) contains fractional power. In order to deal with the difficulty brought by this term, we set

(6) u ( x , t ) = v 2 ( x , t ) .

Second, for our purpose, we further let

(7) v ( x , t ) = ϕ ( ξ ) , ξ = x − c t ,

where c > 1 reads as the wave speed. The solutions such as (7) are called TWSs of (4). By using the transformations (6) and (7), (4) can be rewritten as follows:

(8) − c ( ϕ 2 ) ′ + c ( ϕ 2 ) ‴ + ( ϕ 2 + ϕ 3 ) ′ = 0 .

The prime in equation (8) denotes the derivative with respect to the variable ξ . Integrating (8) once, we arrive at

(9) 2 c ( ϕ ′ 2 + ϕ ϕ ″ ) + ( 1 − c ) ϕ 2 + ϕ 3 + g 1 = 0 ,

where g 1 is the constant of integration.

It is easy to show that the corresponding equivalent planar dynamical system of (9) is given by

(10) d ϕ d ξ = y , d y d ξ = − 2 c y 2 − ϕ 3 − ( 1 − c ) ϕ 2 − g 1 2 c ϕ .

Through a direct calculation, one can obtain the first integral of (10) as follows:

(11) H ( ϕ , y ) = c ϕ 2 y 2 + 1 5 ϕ 5 + 1 − c 4 ϕ 4 + 1 2 g 1 ϕ 2 = h ,

where h is the constant of integration. In addition, it can be seen that system (10) has one singular straight line ϕ = 0 , which makes the TWSs change their shapes and properties near ϕ = 0 .

3 Bifurcations of phase portraits of system (10)

To avoid the difficulty caused by the singular straight line ϕ = 0 , we consider the equivalent regular system of (10), namely,

(12) d ϕ d ζ = 2 c ϕ y , d y d ζ = − 2 c y 2 − ϕ 3 − ( 1 − c ) ϕ 2 − g 1 ,

where we have made use of the transformation d ξ = 2 c ϕ d ζ . Systems (10) and (12) have the same dynamical behaviors except in the neighborhood of ϕ = 0 . In fact, according to the geometric singular perturbation theory [21], the variable ζ can be regarded as a fast variable, while the variable ξ can be regarded as a slow variable.

To detect the information about the equilibrium points of system (12), we let

f ( ϕ ) = − ϕ 3 − ( 1 − c ) ϕ 2 − g 1 .

Obviously, we have

f ′ ( ϕ ) = − 3 ϕ 2 − 2 ( 1 − c ) ϕ .

So, when f ( 0 ) < 0 and f ( 2 ( c − 1 ) / 3 ) > 0 , namely, when 0 < g 1 < 4 ( c − 1 ) 3 27 , the function f ( ϕ ) has three different roots ϕ 1 , ϕ 2 , ϕ 3 satisfying ϕ 1 < 0 < ϕ 2 < ϕ 3 . That means under such conditions, system (12) has three equilibrium points ( ϕ i , 0 ) , i = 1 , 2 , 3 . In addition, on the straight line ϕ = 0 , there exist another two equilibrium points ( 0 , ± − g 1 / 2 c ) when g 1 < 0 .

Next, we consider the coefficient matrix M ( ϕ j , y ) of the linearized system of (12) at an equilibrium point ( ϕ j , y ) . Clearly,

J ( ϕ j , 0 ) = det M ( ϕ j , 0 ) = − 2 c ϕ j f ′ ( ϕ j ) , j = 1 , 2 , 3 , J ( 0 , ± − g 1 / 2 c ) = det M ( 0 , ± − g 1 / 2 c ) = 4 g 1 c > 0 .

Therefore, by the theory of planar dynamical systems, we know that the equilibrium points ( φ 1 , 0 ) , ( φ 2 , 0 ) are saddle points, ( φ 3 , 0 ) is a center point and ( 0 , ± − g 1 / 2 c ) are saddle points. Therefore, we can obtain the bifurcations of the phase portraits of system (10) (Figures 1, 2. 3, 4, 5, 6, 7, 8).

$Figure 1 The graphs of the phase-portrait and the wave profile when p ( ϕ ) p\left(\phi ) has three real roots, namely, 0 (double) and 5 ( c − 1 ) 4 \frac{5\left(c-1)}{4} .$

Figure 1

The graphs of the phase-portrait and the wave profile when p ( ϕ ) has three real roots, namely, 0 (double) and 5 ( c − 1 ) 4 .

$Figure 2 The graphs of the phase-portrait and the wave profile when p ( ϕ ) p\left(\phi ) has three real roots, namely, ϕ l {\phi }_{l} , ϕ m {\phi }_{m} and ϕ r {\phi }_{r} .$

Figure 2

The graphs of the phase-portrait and the wave profile when p ( ϕ ) has three real roots, namely, ϕ l , ϕ m and ϕ r .

$Figure 3 The graphs of the phase-portrait and the wave profile when P ( ϕ ) P\left(\phi ) has a double negative root ϕ 1 {\phi }_{1} , three positive roots ϕ 13 < ϕ 12 < ϕ 11 {\phi }_{13}\lt {\phi }_{12}\lt {\phi }_{11} .$

Figure 3

The graphs of the phase-portrait and the wave profile when P ( ϕ ) has a double negative root ϕ 1 , three positive roots ϕ 13 < ϕ 12 < ϕ 11 .

$Figure 4 The graphs of the phase-portrait and the wave profile when P ( ϕ ) P\left(\phi ) has two double roots ϕ 2 {\phi }_{2} , ϕ 3 {\phi }_{3} and one single root ϕ 21 {\phi }_{21} .$

Figure 4

The graphs of the phase-portrait and the wave profile when P ( ϕ ) has two double roots ϕ 2 , ϕ 3 and one single root ϕ 21 .

$Figure 5 The graphs of the phase-portrait and the wave profile when P ( ϕ ) P\left(\phi ) has two double roots ϕ 4 {\phi }_{4} , ϕ 5 {\phi }_{5} and one single root ϕ 31 {\phi }_{31} .$

Figure 5

The graphs of the phase-portrait and the wave profile when P ( ϕ ) has two double roots ϕ 4 , ϕ 5 and one single root ϕ 31 .

$Figure 6 The graphs of the phase-portrait and the wave profile when P ( ϕ ) P\left(\phi ) has a double negative root ϕ 6 {\phi }_{6} , one positive roots ϕ 41 {\phi }_{41} and two conjugate complex roots.$

Figure 6

The graphs of the phase-portrait and the wave profile when P ( ϕ ) has a double negative root ϕ 6 , one positive roots ϕ 41 and two conjugate complex roots.

$Figure 7 The graphs of the phase-portrait and the wave profile when P ( ϕ ) P\left(\phi ) has a triple positive root ϕ 7 {\phi }_{7} and two conjugate complex roots.$

Figure 7

The graphs of the phase-portrait and the wave profile when P ( ϕ ) has a triple positive root ϕ 7 and two conjugate complex roots.

$Figure 8 The graphs of the phase-portrait and the wave profile when P ( ϕ ) P\left(\phi ) has a double positive root, one positive root and two conjugate complex roots.$

Figure 8

The graphs of the phase-portrait and the wave profile when P ( ϕ ) has a double positive root, one positive root and two conjugate complex roots.

4 Main results

4.1 The case h = 0

In this case, one can solve from (11) that

(13) y 2 = − 1 c 1 5 ϕ 3 + ( 1 − c ) 4 ϕ 2 + g 1 2 = 1 5 c p ( ϕ ) ,

where p ( ϕ ) = − ϕ 3 + 5 ( 1 − c ) 4 ϕ 2 + 5 g 1 2 . Combining (13) with the first equation of (9) and integrating the resulting equation once, we have

(14) ∫ ϕ c ϕ d ϕ − ϕ 3 + 5 ( 1 − c ) 4 ϕ 2 + 5 g 1 2 = ± ξ 5 c ,

where the lower limit of integral ϕ c is an appropriate constant.

Now, we are in a position to state our first main result.

Theorem 4.1

Assume h = 0 , then system (10) possesses solitary wave solutions and periodic wave solutions. More precisely, we have

If g 1 = 0 , then system (10) possesses a smooth solitary wave solution whose implicit expression is given by
(15) 5 ( c − 1 ) 4 − ϕ − 5 ( c − 1 ) 4 5 ( c − 1 ) 4 − ϕ + 5 ( c − 1 ) 4 = exp ∓ c − 1 c ξ 2 .
If 0 < g 1 < 25 ( c − 1 ) 3 196 , then system (10) possesses a compacton solution whose explicit expression is given by
(16) ϕ = ϕ l − ϕ m sn 2 [ ± ξ / ( g 2 5 c ) , k 2 ] 1 − sn 2 [ ± ξ / ( g 2 5 c ) , k 2 ] ,
where
k 2 2 = ϕ r − ϕ m ϕ r − ϕ l , g 2 = 2 ϕ r − ϕ l .
In addition, system (10) possesses a periodic wave solution
(17) ϕ = ϕ l + ( ϕ l − ϕ m ) ( ϕ r − ϕ l ) ( ϕ r − ϕ m ) sn 2 [ ± ξ / ( g 2 5 c ) , k ] − ( ϕ r − ϕ l ) .

Proof

Assume h = 0 , there are two cases that are needed to be distinguished.

When g 1 = 0 , there exists a homoclinic orbit. The corresponding phase portrait is shown in Figure 1. Choosing the “loop-figure” as the integral path, (14) becomes
∫ ϕ 5 ( c − 1 ) 4 d ϕ ϕ 5 ( c − 1 ) 4 − ϕ = ± ξ 5 c .
Completing the aforementioned integral gives rise to a solitary wave solution given by (15).
When 0 < g 1 < 25 ( c − 1 ) 3 196 , there exists an open curve family and a family of a periodic orbit. One group of the corresponding phase portrait is shown in Figure 2. Choosing the left open curve as the integral path, (14) becomes
∫ ϕ ϕ l d ϕ ( ϕ r − ϕ ) ( ϕ m − ϕ ) ( ϕ l − ϕ ) = ± ξ 5 c ,
where ϕ l < ϕ m < ϕ r . Completing the aforementioned integral gives rise to a compacton solution given by (16).

Choosing the periodic orbit as the integral path, (14) becomes

∫ ϕ m ϕ d ϕ ( ϕ r − ϕ ) ( ϕ − ϕ m ) ( ϕ − ϕ l ) = ± ξ 5 c .

Completing the aforementioned integral gives rise to a periodic wave solution given by (17).

When g 1 ≥ 25 ( c − 1 ) 3 196 , there exists no bounded wave solution, so we omit it. The proof is thus completed.□

4.2 The case h ≠ 0

If h ≠ 0 , then it follows from (11) that

(18) y 2 = − 1 5 ϕ 5 + 1 − c 4 ϕ 4 + 1 2 g 1 ϕ 2 − h c ϕ 2 = 1 5 c ϕ 2 P ( ϕ ) ,

where P ( ϕ ) = − ϕ 5 − 5 ( 1 − c ) 4 ϕ 4 − 5 2 g 1 ϕ 2 + 5 h . Combination of (18) and the first equation of (9) yields

(19) ∫ ϕ c ϕ ϕ − ( ϕ 5 + 5 ( 1 − c ) 4 ϕ 4 + 5 2 g 1 ϕ 2 ) + 5 h d ϕ = ± ξ 5 c ,

where the lower limit of integral ϕ c is an appropriate constant.

Now, we are ready to state the second main result of this paper.

Theorem 4.2

Assume h ≠ 0 , then system (10) possesses bounded and unbounded compacton solution, cusp solution, periodic wave solution, loop soliton and so on. More precisely, we have

(1) system (10) possesses blow-up solution, the parametric expression of blow-up solution is given by

(20) ϕ = ϕ 11 − ϕ 11 − ϕ 13 sn 2 ( χ , k 4 ) , − g β 1 2 [ ( β 1 2 − α 1 2 ) Π ( φ ( χ ) , β 1 2 , k 4 ) + α 1 2 χ ] = ± ξ 5 c ,

where

α 1 2 = ϕ 11 ϕ 11 − ϕ 13 , β 1 2 = ϕ 11 − ϕ 1 ϕ 11 − ϕ 13 , k 4 2 = ϕ 11 − ϕ 12 ϕ 11 − ϕ 13 , g 3 = 2 ϕ 11 − ϕ 13 .

The parametric expression of the loop solution is given by

(21) ϕ = ϕ 13 − ϕ 12 sn 2 ( χ , k 4 ) 1 − sn 2 ( χ , k 4 ) , ϕ 13 g 3 ( ϕ 1 − ϕ 13 ) β 2 2 [ ( β 2 2 − α 2 2 ) Π ( φ ( χ ) , β 2 2 , k 4 ) + α 2 2 χ ] = ± ξ 5 c ,

where

α 2 2 = ϕ 12 ϕ 13 , β 2 2 = ϕ 1 − ϕ 12 ϕ 1 − ϕ 13 .

In addition, system (10) possesses a periodic TWS whose parametric expression is given by

(22) ϕ = ϕ 11 − ϕ 12 sn 2 ( χ , k 5 ) 1 − sn 2 ( χ , k 5 ) , ϕ 11 g ( ϕ 11 − ϕ 1 ) β 3 2 [ ( β 3 2 − α 3 2 ) Π ( φ ( χ ) , β 3 2 , k 5 ) + α 3 2 χ ] = ± ξ 5 c ,

where

α 3 2 = ϕ 12 ϕ 11 , β 3 2 = ϕ 11 − ϕ 1 ϕ 11 − ϕ 1 , k 5 2 = ϕ 12 − ϕ 13 ϕ 11 − ϕ 13 .

(2) System (10) possesses an unbounded solution, the parametric expression of which is given by

(23) − 1 ϕ 21 − ϕ 2 ln ϕ 21 − ϕ − ϕ 21 − ϕ 2 ϕ 21 − ϕ + ϕ 21 − ϕ 2 − 2 ϕ 3 − ϕ 21 arctan ϕ 21 − ϕ ϕ 3 − ϕ 21 − π 2 = ∓ ( ϕ 3 − ϕ 2 ) ξ 5 c .

The parametric expression of loop solution is given by

(24) − 1 ϕ 21 − ϕ 2 ln ϕ 21 − ϕ − ϕ 21 − ϕ 2 ϕ 21 − ϕ + ϕ 21 − ϕ 2 − 2 ϕ 3 − ϕ 21 arctan ϕ 21 − ϕ ϕ 3 − ϕ 21 = ± ( ϕ 3 − ϕ 2 ) ξ 5 c .

(3) System (10) possesses a blow-up solution, whose parametric expression is given by

(25) − 1 ϕ 31 − ϕ 4 ln ϕ 31 − ϕ − ϕ 31 − ϕ 4 ϕ 31 − ϕ + ϕ 31 − ϕ 4 + 1 ϕ 31 − ϕ 5 ϕ 31 − ϕ − ϕ 31 − ϕ 5 ϕ 31 − ϕ + ϕ 31 − ϕ 5 = ∓ ( ϕ 5 − ϕ 4 ) ξ 5 c .

The parametric expression of weak kink solution is given by

(26) 1 ϕ 31 − ϕ 4 ln ϕ 31 − ϕ 31 − ϕ 4 ϕ 31 + ϕ 31 − ϕ 4 − ln ϕ 31 − ϕ − ϕ 31 − ϕ 4 ϕ 31 − ϕ + ϕ 31 − ϕ 4 − 1 ϕ 31 − ϕ 5 ϕ 31 − ϕ 31 − ϕ 5 ϕ 31 + ϕ 31 − ϕ 5 − ϕ 31 − ϕ − ϕ 31 − ϕ 5 ϕ 31 − ϕ + ϕ 31 − ϕ 5 = ± ( ϕ 5 − ϕ 4 ) ξ 5 c .

In addition, system (10) possesses a solitary wave solution whose parametric expression is given by

(27) − 1 ϕ 31 − ϕ 4 ln ϕ 31 − ϕ − ϕ 31 − ϕ 4 ϕ 31 − ϕ + ϕ 31 − ϕ 4 − 1 ϕ 31 − ϕ 5 ln ϕ 31 − ϕ − ϕ 31 − ϕ 5 ϕ 31 − ϕ + ϕ 31 − ϕ 21 = ± ( ϕ 4 − ϕ 5 ) ξ 5 c .

(4) System (10) possesses a loop solution, whose parametric expression is given by

(28) χ + ϕ 6 ϕ 41 − A 3 − ϕ 6 1 α 4 χ − 1 α 4 Π φ , α 4 2 α 4 2 − 1 , k 6 − α 4 f 1 ( χ ) = ± A 3 ξ 5 c , ϕ ( χ ) = − ( A 3 − ϕ 41 ) + ( A 3 + ϕ 41 ) cn ( χ , k 6 ) 1 + cn ( χ , k 6 ) ,

where

A 3 2 = ( Re a − ϕ 41 ) 2 + ( Im a ) 2 , k 6 2 = A 3 − Re a + ϕ 41 2 A 3 , α 4 = ϕ 41 + A 3 − ϕ 6 ϕ 41 − A 3 − ϕ 6 ,

and

f 1 ( χ ) = 1 − α 4 2 k 6 2 + k 1 ′ 2 α 4 2 arctan k 6 2 + k 1 ′ 2 α 4 2 1 − α 4 2 sd ( χ , k 6 ) , α 4 2 / ( α 4 2 − 1 ) < k 6 2 , sd ( χ , k 6 ) , α 4 2 / ( α 4 2 − 1 ) = k 6 2 , 1 2 α 4 2 − 1 k 6 2 + k 1 ′ 2 α 4 2 ln k 6 2 + k 1 ′ 2 α 4 2 du ( χ , k 6 ) + α 4 2 − 1 sn ( χ , k 6 ) k 6 2 + k 1 ′ 2 α 4 2 du ( χ , k 6 ) − α 4 2 − 1 sn ( χ , k 6 ) , α 4 2 / ( α 4 2 − 1 ) > k 6 2 ,

sn ( χ , k 6 ) , cn ( χ , k 6 ) , sd ( χ , k 6 ) and du ( χ , k 6 ) are Jacobian elliptic functions, Π ⋅ , α 4 2 1 − α 4 2 , k 6 is the elliptic integral of the third kind.

(5) System (10) possesses a cuspon solution, whose parametric expression is given by

(29) ϕ 3 A 4 χ − 2 E ( χ ) + 2 sn χ dn χ 1 + cn χ − χ = ± A 4 ξ 5 c , ϕ ( χ ) = ( ϕ 3 − A 4 ) − ( ϕ 3 + A 4 ) cn ( χ , k 7 ) 1 − cn ( χ , k 7 ) ,

where

A 4 2 = ( Re a − ϕ 41 ) 2 + ( Im a ) 2 , k 7 2 = A 4 − Re a + ϕ 41 2 A 4 .

Proof

Assume h ≠ 0 , we divide the discussions into five cases.

(1) P ( ϕ ) has a double negative root ϕ 1 and three positive roots ϕ 13 < ϕ 12 < ϕ 11 . The corresponding phase portrait is shown in Figure 3. Taking the left open curve as the integral path, then (14) becomes

∫ − ∞ ϕ ϕ d ϕ ( ϕ 1 − ϕ ) ( ϕ 11 − ϕ ) ( ϕ 12 − ϕ ) ( ϕ 13 − ϕ ) = ± ξ 5 c .

Completing the aforementioned integral gives rise to a blow-up solution, whose parametric expression is given by (20).

Taking the middle two open curves, which tend to the straight line ϕ = 0 from the left and right hand, respectively, as the integral path, then (14) becomes

∫ ϕ ϕ 13 ϕ d ϕ ( ϕ − ϕ 1 ) ( ϕ 11 − ϕ ) ( ϕ 12 − ϕ ) ( ϕ 13 − ϕ ) = ± ξ 5 c .

Completing the aforementioned integral gives rise to a loop soliton, whose parametric expression is given by (21).

Taking the loop curve as the integral path, then (14) becomes

∫ ϕ 11 ϕ ϕ d ϕ ( ϕ − ϕ 1 ) ( ϕ − ϕ 11 ) ( ϕ − ϕ 12 ) ( ϕ − ϕ 13 ) = ± ξ 5 c .

Completing the aforementioned integral gives rise to a periodic wave solution (22).

(2) P ( ϕ ) has a double negative root ϕ 2 , one positive root ϕ 21 and a double positive root ϕ 3 . The corresponding phase portrait is shown in Figure 4. These roots satisfy ϕ 2 < ϕ 21 < ϕ 3 . Taking the left open curve as the integral path, then (14) becomes

∫ − ∞ ϕ − ϕ d ϕ ( ϕ 2 − ϕ ) ( ϕ 3 − ϕ ) ϕ 21 − ϕ = ± ξ 5 c .

Completing the aforementioned integral gives rise to a blow-up solution, whose parametric expression is given by (23).

Taking the middle two open curves as the integral path, then (14) becomes

∫ ϕ ϕ 21 ϕ d ϕ ( ϕ − ϕ 2 ) ( ϕ 3 − ϕ ) ϕ 21 − ϕ = ± ξ 5 c .

Completing the aforementioned integral gives rise to a loop soliton, whose parametric expression is given by (24).

(3) P ( ϕ ) has a double negative root ϕ 4 , one positive roots ϕ 31 and a double positive root ϕ 5 . The corresponding phase portrait is shown in Figure 5. These roots satisfy ϕ 4 < ϕ 5 < ϕ 31 . Taking the left open curve as the integral path, then (14) becomes

∫ − ∞ ϕ − ϕ d ϕ ( ϕ 4 − ϕ ) ( ϕ 5 − ϕ ) ϕ 31 − ϕ = ± ξ 5 c .

Completing the aforementioned integral gives rise to a blow-up solution, whose parametric expression is given by (25).

Taking the middle two open curves, which tend to the straight line ϕ = 0 from the left and right hand, as the integral path, then (14) becomes

∫ 0 ϕ ϕ d ϕ ( ϕ − ϕ 4 ) ( ϕ 5 − ϕ ) ϕ 31 − ϕ = ± ξ 5 c .

Completing the aforementioned integral gives rise to a weak kink solution, whose parametric expression is given by (26).

Taking the loop curve as the integral path, then (14) becomes

∫ ϕ ϕ 31 ϕ d ϕ ( ϕ − ϕ 4 ) ( ϕ − ϕ 5 ) ϕ 31 − ϕ = ± ξ 5 c .

Completing the aforementioned integral gives rise to a solitary wave solution, whose parametric expression is given by (27).

(4) P ( ϕ ) has a double negative root ϕ 6 , one positive root ϕ 41 and two conjugate complex roots. The corresponding phase portrait is shown in Figure 6. Taking the middle two open curves as the integral path, then (14) becomes

∫ ϕ ϕ 41 ϕ d ϕ ( ϕ − ϕ 6 ) ( ϕ 41 − ϕ ) ( ϕ − a ) ( ϕ − a ¯ ) = ± ξ 5 c .

Completing the aforementioned integral gives rise to a loop soliton, whose parametric expression is given by (28).

(5) P ( ϕ ) has a triple positive root ϕ 7 and two conjugate complex roots. The corresponding phase portrait is shown in Figure 7. Taking the right open curve as the integral path, then (14) becomes

∫ − ∞ ϕ ϕ d ϕ ( ϕ 7 − ϕ ) ( ϕ 7 − ϕ ) ( ϕ − a ) ( ϕ − a ¯ ) = ± ξ 5 c .

Completing the aforementioned integral gives rise to a cuspon solution, whose parametric expression is given by (29).□

Remark 4.3

Following the same calculations, one can find similar solutions such as (29) in Figure 8.

5 Applications of Theorems 4.1 and 4.2

In this section, we discuss the applications of Theorems 4.1 and 4.2. More precisely, we intend to establish the relations of TWSs among regularized Schamel equation, Schamel equation, KdV equation and Schamel-like equations. The effects of regularization and strength of nonlinearity on the type of TWS will also be considered. To do so, we recall that the bifurcation method of the dynamical system is used to derive the explicit expressions of TWS depending on two essential factors. One is the equivalent planar system of the original system, and another is the first integral. If these two integrals are the same or similar, then one has reasons to conclude that such two equations have the same type of TWS and the only difference may arise from the coefficients.

Application 1. The Schamel equation and the regularized Schamel equation

The corresponding planar system for the Schamel equation is given by

(30) d ϕ d ξ = y , d y d ξ = − 2 y 2 − ϕ 3 − ( 1 − c ) ϕ 2 − g 1 2 ϕ .

The first integral of (30) can be written in the form of

(31) H ( ϕ , y ) = ϕ 2 y 2 + 1 5 ϕ 5 + 1 − c 4 ϕ 4 + 1 2 g ϕ 2 = h ,

where h is an integral constant. Comparing (30), (31) to (10), (11), one can conclude that the Schamel equation and the regularized Schamel equation have the same type of the traveling wave equation. Moreover, the explicit expressions of TWSs for the Schamel equation can be obtained directly from Theorems 4.1 and 4.2 with a slight modification. For example, the Schamel equation has a smooth solitary wave solution whose implicit expression is given by

5 ( c − 1 ) 4 − ϕ − 5 ( c − 1 ) 4 5 ( c − 1 ) 4 − ϕ + 5 ( c − 1 ) 4 = exp ± c − 1 2 ξ ,

which is similar to (15). From the aforementioned results, we conclude that the regularization process has no influences on the construction of the TWSs. This is consistence with the statements in [3], where the author stated “The BBM equation has the same formal justification as the KdV equation.”

Application 2. The regularized Schamel equation and the BBM equation

The corresponding planar system for the BBM equation is given by

(32) d ϕ d ξ = y , d y d ξ = − 1 c 1 2 ϕ 2 + ( 1 − c ) ϕ + g .

The first integral of (32) can be written in the form of

(33) H ( ϕ , y ) = c 2 y 2 + 1 6 ϕ 3 + 1 − c 2 ϕ 2 + g ϕ = h .

While h = 0 in (11), we have

(34) H ( ϕ , y ) = c y 2 + 1 5 ϕ 3 + 1 − c 4 ϕ 2 + g 1 2 = 0 .

Combing (11) with (33), we know that the BBM equation possesses the same TWS of the regularized Schamel equation as in Theorem 4.1. For example, for h = 0 and g = 3 ( 1 − c ) 2 / 8 , the BBM equation has a periodic TWS

ϕ = 3 ( 1 − c ) 2 tan 2 c − 1 8 c ξ .

It is noteworthy that the BBM equation losses the TWS in Theorem 4.2. This is because, compared to the BBM equation, the regularized Schamel equation possesses a stronger nonlinearity, which leads to more rich dynamical behaviors.

Application 3. The Schamel-like equation and the Schamel equation

Rahman and Haider [22] obtained a Schamel-like equation (they refer to it as the mKdV equation)

(35) ∂ ϕ ∂ t + A u ∂ u ∂ x + B ∂ 3 u ∂ x 3 = 0 ,

which describes dust-ion-acoustic solitary wave in an unmagnetized dusty electronegative plasma, where u is the electrostatic wave potential, A = ( 1 − β n ) μ n v p 3 π α 3 2 , B = v p 3 2 . β n measures the number of trapped negative ions. α represents the ratio of the electron temperature to the negative ion temperature. v p is the phase speed. μ n is a parameter relating to positive ion, electron, negative ion and dust number density at equilibrium. The aforementioned equation exhibits a weak nonlinearity, smaller width, and larger propagation velocity of the nonlinear wave.

Rahman and Haider [23] obtained the following Schamel equation for ion acoustic wave,

(36) ∂ u ∂ t + 2 v 0 ( 1 − β ) 3 π ∂ ( u ) 3 / 2 ∂ x + v 0 2 ∂ 3 u ∂ x 3 = 0 ,

where u represents the electric potential, v 0 is the phase velocity of the corresponding wave and β is the trapping parameter.

Kangalgil [24] considered another Schamel-like equation

(37) u t + u 1 / 2 u x + δ u x x x = 0 ,

where δ is a constant which represents the dispersion coefficient.

By putting the coefficients and scaling factors into the dependent variable u and the independent variables x and t , equations (35)–(37) can be obtained in the tidy form

(38) u t + ( u 3 / 2 ) x + u x x x = 0 .

It follows from the same derivation of (10) and (11) that the equivalent planar system of (38) is given by

(39) d ϕ d ξ = y , d y d ξ = − 2 y 2 − ϕ 3 + c ϕ 2 − g 1 2 ϕ .

Through a direct calculation, one can obtain the first integral of (39) as follows:

(40) H ( ϕ , y ) = ϕ 2 y 2 + 1 5 ϕ 5 − c 4 ϕ 4 + 1 2 g 1 ϕ 2 = h .

Comparing (39) and (40) to (10) and (11), one can conclude that the Schamel-like equations and the Schamel equation have the same type of TWS. Furthermore, the explicit expressions of the TWS can be derived from Theorems 4.1 and 4.2, for example, equation (38) has smooth solitary wave solutions

5 c 4 − ϕ − 5 c 4 5 c 4 − ϕ + 5 c 4 = exp ± c 2 ξ .

6 Conclusion

By making use of the bifurcation method of dynamical systems, we studied the bifurcation and TWSs for the regularized Schamel equation. By performing complete phase portrait analysis, some novel TWSs are discovered. Moreover, exact expressions of solitary waves, periodic waves, cusp waves, weak kink waves, loop solitons and compactons are yielded. Finally, the effects of regularization and strength of nonlinearity on the type of TWS are investigated by studying the regularized Schamel equation, Schamel equation, KdV equation and BBM equation as well as Schamel-like equation.

Acknowledgements

This work was partially supported by the Scientific Research Fund of Hunan Provincial Education Department grant (20B512, 19C1609), Natural Science Foundation of Hunan Province grant (2021JJ30019), NSF of China grant (11626129, 11801263 and 91026015).

Conflict of interest: The authors state no conflict of interest.
Data availability statement: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

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Received: 2021-09-01

Revised: 2021-11-17

Accepted: 2021-11-25

Published Online: 2021-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2021-0136

Keywords for this article

traveling wave solutions; phase portraits; bifurcation method; regularized Schamel equation

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