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The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods

  • Yanjie Wang EMAIL logo , Beibei Zhang and Bo Cao
Published/Copyright: September 1, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

The exact traveling wave solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities are studied using the dynamical system and the first integral methods. Taking different parameter conditions, we obtain periodic wave solutions, exact solitary wave solutions, kink wave solutions, and anti-kink wave solutions.

Keywords: generalized Davey-Stewartson equation; the dynamical system method; the first integral method; solitary wave solution; kink wave solution; periodic wave solution
MSC 2010: 34A05; 34A30; 34B05

1 Introduction

The study of traveling wave solutions to nonlinear wave equations plays a major role in the fields of plasma, elastic media, optical fibers, etc. Mathematicians and physicists have made significant progress toward determining the exact traveling wave solutions. Various methods have been presented, such as the Darboux transformation, the inverse scattering method, the Hirota bilinear method, the tanh method, the homogeneous balance method, and the Jacobi elliptic function expansion method (see [1,2, 3,4,5, 6,7,8, 9,10,11, 12,13,14, 15,16,17, 18,19,20, 21,22,23, 24,25,26, 27,28]). To obtain exact solutions for nonlinear partial differential equations (PDEs), Jibin Li employed the dynamical system method to study nonlinear PDEs (see [29]). In addition, based on the theory of rings of commutative algebra, Feng introduced the first integral method to solve nonlinear PDEs and obtain many exact solutions (see [33]). It is worth noting that using the bifurcation theory of dynamical systems method to calculate exact solutions is a new and effective method that can be utilized to study fractional PDEs in a conformal sense (see [34,35,36, 37,38]).

In this article, we discuss the exact solutions to the following generalized Davey-Stewartson equations with arbitrary power nonlinearities:

(1.1) i u t + u x x + u y y + γ ∣ u ∣ p u + α u v + δ ∣ u ∣ 2 p u = 0 , v x x + v y y − β ( ∣ u ∣ p ) x x = 0 ,

where α , β , γ , and δ are real parameters, p is a positive integer, v is a real function, and u is a complex function. Davey-Stewartson equations are significant for describing the motion of long and short waves in shallow water. Many researchers have used various methodologies to solve these problems. For example, Mirzazadeh applied the trial equation method and the ansatz approach to establish solitary waves solitons, dark solitons, and singular solitary-wave soliton solutions to Davey-Stewartson equations (see [41]). Song and Miswas studied Davey-Stewartson equations with power-law nonlinearity and carried out several different solutions for bifurcation analysis (see [42]). Zinati and Manafian used He’s semi-inverse variational principle method, the improved tan ϕ 2 -expansion method and the generalized G ′ G -expansion method to find more exact solutions to Davey-Stewartson equations (see [43]). The generalized Kudryashov method is introduced to obtain new soliton solutions from the Davey-Stewartson equations with power-law nonlinearity (see [44]). Aghdaei and Adibi applied the generalized tan ϕ 2 and He’s semi-inverse variational methods to find the exact solitary wave solutions of Davey-Stewartson equations with power-law nonlinearity (see [45]). As a result, a more comprehensive study is necessary for equation (1.1).

This article is structured as follows: Sections 2 and 3 introduce the dynamical system and the first integral methods. Sections 4 and 5 address equation (1.1) by implementing these two proposed methods. Briefly, Section 6 outlines some of the findings.

2 Description of the dynamical system method

In this section, we consider the following nonlinear PDEs:

(2.1) P ( t , x i , u t , u x i , u x i x i , u x i x j , u t t , … ) ,

where i , j = 1 , 2 , … , n . P is a polynomial in u ( x , t ) and its partial derivatives, in which nonlinear terms and the highest order derivatives are involved. u ( x , t ) is an unknown function.

The main steps of the dynamical system method (see [30,31,32]) in this document are as follows:

Step 1. Transform

(2.2) u ( t , x 1 , u 2 , … , u n ) = ϕ ( ξ ) , ξ = ∑ i = 1 n k i x i − c t .

Equation (2.1) can be reduced to the following nonlinear ordinary differential equations (ODEs):

(2.3) D ( ξ , ϕ ξ , ϕ ξ ξ , ϕ ξ ξ ξ , … ) = 0 ,

where k i are nonzero constants and c is the wave speed. Multiple integrations are performed for equation (2.3), if equation (2.3) can be reduced to the following the second-order nonlinear ODEs:

(2.4) E ( ξ , ϕ ξ , ϕ ξ ξ ) = 0 .

Then, assuming ϕ ξ = d ϕ d ξ = y , equation (2.4) can be reduced to the following two-dimensional dynamical system:

(2.5) d ϕ d ξ = y , d y d ξ = f ( ϕ , y ) ,

where f ( ϕ , y ) is an integral expression or a fraction. If f ( ϕ , y ) is a fraction, we make f ( ϕ , y ) = F ( ϕ , y ) g ( ϕ ) , when g ( ϕ s ) = 0 , ϕ = ϕ s , d y d ξ does not exist. Then, if we transform d ζ = g ( ϕ ) d ξ , equation (2.5) can be rewritten as

(2.6) d ϕ d ζ = g ( ϕ ) y , d y d ζ = F ( ϕ , y ) ,

where ζ is a parameter. If it is possible to reduce equation (2.1) to equation (2.5) (or equation (2.6)), then we can proceed to the next step.

Step 2. If equation (2.5) is an integrable system, then we can reduce equation (2.5) (or equation (2.6)) for subsequent differential equations as follows:

(2.7) d y d ϕ = f ( ϕ , y ) y , d y d ϕ = F ( ϕ , y ) g ( ϕ ) y = f ( ϕ , y ) y .

Then equations (2.5) and (2.6) have the same first integral (i.e., Hamiltonian) as follows:

(2.8) H ( ϕ , y ) = h ,

where h is an integral constant, and we can obtain all kinds of phase portraits in the parametric space using the first integral. Since the phase orbit of the vector field that defines equation (2.5) (or equation (2.6)) determines all the traveling wave solutions of equation (2.1), we study the bifurcation of the phase diagram of equation (2.5) (or equation (2.6)) to find the traveling wave solutions of equation (2.1). As a rule, a periodic orbit corresponds to a periodic wave solution, a homoclinic orbit corresponds to a solitary wave solution, and a heteroclinic orbit (or the so-called connected orbit) corresponds to a kink (or anti-kink) wave solutions. Once we obtain all the phase orbits, we can derive the value of h or its range.

Step 3. If h is determined, then we can obtain the following relationship from equation (2.8):

(2.9) y = y ( ϕ , y ) ,

that is, d ϕ d ξ = y ( ϕ , y ) . When equation (2.9) is an integrable expression, we can substitute it into the first term of equation (2.5) and integrate it to obtain

(2.10) ∫ ϕ 0 ϕ d ϕ y ( ϕ , h ) = ∫ 0 ξ ξ d ξ ,

where ϕ 0 and 0 are initial constants. Usually, the initial constants can be taken by a root of equation (2.9) or infection points of the traveling waves. Making proper initial constants and integrating equation (2.10), we obtain the exact traveling wave solutions of equation (2.1) through the elliptic Jacobian functions.

3 Description of the first integral method

The main steps of the first integral method (see [39,40]), summarized as follows:

Step 1. The simplification of equation (2.1) gives

(3.1) u ( x , t ) = u ( ξ ) ,

where ξ = x − c t , and c is the wave speed. Then

(3.2) ∂ ∂ t ( ⋅ ) = − c ∂ ∂ ξ ( ⋅ ) , ∂ ∂ x ( ⋅ ) = ∂ ∂ ξ ( ⋅ ) , ∂ 2 ∂ t 2 ( ⋅ ) = c 2 ∂ 2 ∂ ξ 2 ( ⋅ ) , … …

Step 2. Depending on the transformation, equation (3.1) shows the following nonlinear ODEs:

(3.3) Q ( u , u ξ , u ξ ξ , … … ) = 0 .

Step 3. Assuming X and Y as new independent variables

(3.4) X ( ξ ) = u ( ξ ) , Y ( ξ ) = u ξ ( ξ ) ,

we have the following system of ODEs:

(3.5) X ξ ( ξ ) = Y ( ξ ) , Y ξ ( ξ ) = F ( X ( ξ ) , Y ( ξ ) ) .

Step 4. The general solutions are found if we obtain the first integrals to equation (3.7) under the same conditions. However, no systematic theory tells us how to find its first integrals. Thankfully, we apply the division theorem to reduce equation (3.4) to a first-order integrable ODE. Thus, by solving these equations, we arrive at the exact solutions to equation (3.1).

4 Applications of the dynamical system method

Consider the solutions to equation (1.1) with the following form:

(4.1) u ( x , y , t ) = ϕ 1 2 p ( ξ ) e i η , v ( x , y , t ) = ψ ( ξ ) ,

where ξ = x + y − 2 ( k + λ ) t , η = k x + λ y − w t , λ , k , and w are constants.

Substituting equation (4.1) into equation (1.1), we obtain that

(4.2) ψ = β 2 ϕ

and

(4.3) ( w − λ 2 − k 2 ) + 2 p 1 p − 1 ϕ − 2 ϕ ′ ϕ ′ + 2 p ϕ − 1 ϕ ″ + γ ϕ + α ψ + δ ϕ 2 = 0

where “ ′ ” is the derivative as regards ξ .

Separating the real and imaginary parts in equation (4.3), respectively, it is possible to obtain

(4.4) ϕ ″ = a ϕ ′ 2 + ϕ 2 ( A ϕ 2 + B ϕ + C ) ϕ ,

where 1 − 1 p = a < 1 , A = − p δ 2 , B = − p 2 γ + α β 2 , and C = p 2 ( λ 2 + k 2 − w ) . Equation (4.4) refers to the planar dynamical system

(4.5) d ϕ d ξ = y , d y d ξ = a y 2 + ϕ 2 ( A ϕ 2 + B ϕ + C ) ϕ ,

where equation (4.5) is a singular nonlinear traveling wave system with the singular straight line ϕ = 0 in phase plane ( ϕ , y ) . Equation (4.5) has an associated regular system

(4.6) d ϕ d ζ = ϕ y , d y d ζ = a y 2 + ϕ 2 ( A ϕ 2 + B ϕ + C ) ,

where d ξ = ϕ d ζ . Equations (4.5) and (4.6) have the same first integrals as follows:

(4.7) H = y 2 ϕ − 2 a + ϕ − 2 a A a − 2 ϕ 4 + 2 B 2 a − 3 ϕ 3 + C a − 1 ϕ 2 = h .

4.1 Bifurcations of phase portraits of equation (4.6)

Suppose that A B ≠ 0 and a ≠ 1 , 2 , 3 2 . Write that f ( ϕ ) = A ϕ 2 + B ϕ + C , and Δ = B 2 − 4 A C . Clearly, when Δ < 0 , equation (4.6) receives only one equilibrium point E 0 ( 0 , 0 ) . When Δ > 0 , equation (4.6) receives three equilibrium points E 0 ( 0 , 0 ) , E 1 ( ϕ 1 , 0 ) , and E 2 ( ϕ 2 , 0 ) , where ϕ 1 = − B − Δ 2 A and ϕ 2 = − B + Δ 2 A . When Δ = 0 , equation (4.6) receives an equilibrium point E 0 ( 0 , 0 ) and a double equilibrium point E d ( ϕ d , 0 ) , where ϕ d = − B 2 A .

Making M ( ϕ j , 0 ) the coefficient matrix of the linearized system of equation (4.6) at an equilibrium point E j , we obtain that

M = 0 ϕ j ϕ j 2 f ′ ( ϕ j ) 0 .

Then, we obtain

J ( 0 , 0 ) = det M ˆ ( 0 , 0 ) = 0 , J ( ϕ d , 0 ) = det M ( ϕ d , 0 ) = 0 ,

J ( ϕ 1 , 2 , 0 ) = det M ( ϕ 1 , 2 , 0 ) = − ϕ 1 , 2 3 f ′ ( ϕ 1 , 2 ) .

By the theory of planar dynamical systems and an equilibrium point of a planar integrable system, we obtain that if J < 0 , then the equilibrium point is a saddle point; if J > 0 and ( trace M ) 2 − 4 J < 0 ( o r > 0 ) , then it is a center point (a node point); if J = 0 and the Poincaré index of the equilibrium point is 0, then the equilibrium point is cusped. Thus, we note that the equilibrium point E 0 ( 0 , 0 ) is a high-order singular point.

Depending on the change of parameter pair ( A , B ) for a = 1 2 and a fixed C < 0 , we obtain the bifurcations of phase portraits of equation (4.6) as shown in Figure 1.

Figure 1 Bifurcations of phase portraits of equation (4.6) when C < 0 C\lt 0 . (a) A > 0 A\gt 0 and B > 0 B\gt 0 , (b) Δ > 0 , A < 0 \Delta \gt 0,A\lt 0 , and B > 0 B\gt 0 , (c) Δ = 0 , A < 0 \Delta =0,A\lt 0 , and B > 0 B\gt 0 , (d) Δ < 0 \Delta \lt 0 and A < 0 A\lt 0 , (e) Δ = 0 \Delta =0 , A < 0 A\lt 0 , B < 0 B\lt 0 (f) Δ > 0 \Delta \gt 0 , A < 0 A\lt 0 , and B < 0 B\lt 0 , (g) A > 0 A\gt 0 , B > 0 B\gt 0 .
Figure 1

Bifurcations of phase portraits of equation (4.6) when C < 0 . (a) A > 0 and B > 0 , (b) Δ > 0 , A < 0 , and B > 0 , (c) Δ = 0 , A < 0 , and B > 0 , (d) Δ < 0 and A < 0 , (e) Δ = 0 , A < 0 , B < 0 (f) Δ > 0 , A < 0 , and B < 0 , (g) A > 0 , B > 0 .

Similarly, we receive the bifurcations of phase portraits of equation (4.6) when C > 0 or a ≠ 1 2 .

4.2 Exact solutions of equation (4.5)

When a = 1 2 , we study the exact solutions of equation (4.5) in this section to obtain some solutions to equation (1.1). Then, only the case ϕ > 0 is considered. When a = 1 2 , we can deduce from equation (4.7) that

(4.8) H ( ϕ , y ) = y 2 ϕ − 2 3 ϕ 3 C + 3 2 B ϕ + A ϕ 2 = h ,

h 1 = H ( ϕ 1 , 0 ) = 1 6 A ϕ 1 ( − 8 A C + B 2 + B Δ ) , and

h 2 = H ( ϕ 2 , 0 ) = 1 6 A ϕ 2 ( − 8 A C + B 2 − B Δ ) .

This shows that

y 2 = 2 3 A ϕ 3 h 2 A + 3 C A ϕ + 3 B 2 A ϕ 2 + ϕ 3 , for A > 0 and y 2 = 2 3 ∣ A ∣ ϕ 3 h 2 ∣ A ∣ + 3 C ∣ A ∣ ϕ + 3 B 2 ∣ A ∣ ϕ 2 − ϕ 3 , for A < 0 .

Then, using the first equation of equation (4.5), we know that for A > 0 , the following function holds

(4.9) 3 2 A ξ = ∫ ϕ 0 ϕ d ϕ ϕ 3 h 2 A + 3 C A ϕ + 3 B 2 A ϕ 2 + ϕ 3 ,

or for A < 0 , the following function holds

(4.10) 3 2 A ξ = ∫ ϕ 0 ϕ d ϕ ϕ 3 h 2 ∣ A ∣ + 3 C ∣ A ∣ ϕ + 3 B 2 ∣ A ∣ ϕ 2 − ϕ 3 .

Case 1. A > 0 and B > 0 (Figure 1). If so, we obtain ϕ 1 < 0 < ϕ 2 , h 1 < 0 < h 2 .

  1. For every h ∈ ( 0 , h 2 ) , the level curves defined by H a = 1 2 ( ϕ , y ) = h contain two open branches passing through the points ( ϕ L , 0 ) and ( ϕ l , 0 ) ( ϕ l < 0 < ϕ M < ϕ 2 < ϕ L ) , respectively. A close branch contacts to the singular straight line ϕ = 0 at the equilibrium point E 0 ( 0 , 0 ) . Based on the finite-time interval theorem, we have the family of close branches that causes a family of periodic solutions to equation (4.5). Then, y 2 = 2 3 A ( ϕ L − ϕ ) ( ϕ M − ϕ ) ϕ ( ϕ − ϕ l ) . Thus, by equation (4.9), we receive the parametric representation of periodic solutions to equation (4.5) as follows:

    (4.11) ϕ ( ξ ) = ϕ M − ϕ M ( ϕ L − ϕ M ) s n 2 ( Ω 1 ξ , k ) ϕ L − ϕ M s n 2 ( Ω 1 ξ , k ) ,

    where k 2 = ϕ M ( ϕ L − ϕ l ) ϕ L ( ϕ M − ϕ l ) , Ω 1 = 1 2 3 ϕ L ( ϕ M − ϕ l ) 2 A . Equation (4.11) shows the following exact periodic wave solutions to equation (1.1):

    (4.12) u ( x , t ) = ϕ M − ϕ M ( ϕ L − ϕ M ) s n 2 ( Ω 1 ξ , k ) ϕ L − ϕ M s n 2 ( Ω 1 ξ , k ) 1 2 p e i η , v ( x , t ) = 1 3 ϕ M − ϕ M ( ϕ L − ϕ M ) s n 2 ( Ω 1 ξ , k ) ϕ L − ϕ M s n 2 ( Ω 1 ξ , k ) .

  2. Corresponding to the level curves defined by H a = 1 2 ( ϕ , y ) = h 2 , there are two heteroclinic orbits of equation (4.5) for y 2 = 2 3 A ( ϕ 2 − ϕ ) 2 ϕ ( ϕ − ϕ l ) . Hence, we obtain the parametric representation of the kink wave and anti-kink wave solutions to equation (4.5) as follows:

    (4.13) ϕ ( ξ ) = Φ 2 − 4 A 1 P 1 P 1 2 e ± ω 1 ξ + Φ l 2 e ∓ ω 1 ξ − 2 B 1 P 1 ,

    where 0 < ϕ 0 < ϕ 2 , A 1 = ϕ 2 ( ϕ 2 − ϕ l ) , B 1 = − ( 2 ϕ 2 − ϕ l ) , ω 1 = 3 A 1 2 A , and P 1 = 2 A 1 ( A 1 + B 1 ϕ 0 + ϕ 0 2 ) + B 1 ϕ 0 + 2 A 1 ϕ 0 .

Then, we draw the exact solutions to equation (1.1) as follows:

(4.14) u ( x , t ) = ϕ 2 − 4 A 1 P 1 P 1 2 e ± ω 1 ξ + ϕ l 2 e ∓ ω 1 ξ − 2 B 1 P 1 1 2 p e i η , v ( x , t ) = 1 3 ϕ 2 − 4 A 1 P 1 P 1 2 e ± ω 1 ξ + ϕ l 2 e ∓ ω 1 ξ − 2 B 1 P 1 .

Case 2. A < 0 , B > 0 , and Δ > 0 (Figure 1(b)). If so, we obtain 0 < ϕ 2 < ϕ 1 , h 1 < 0 < h 2 .

  1. When h ∈ ( h 1 , 0 ) , the level curves defined by H a = 1 2 ( ϕ , y ) = h contain a close branch enclosing the equilibrium point E 1 ( ϕ 1 , 0 ) , for which we have

    y 2 = 2 3 ∣ A ∣ ϕ − ϕ 3 + 3 B 2 ∣ A ∣ ϕ 2 + 3 C ∣ A ∣ + 3 h 2 ∣ A ∣ = 2 3 ∣ A ∣ ϕ ( r 1 − ϕ ) ( ϕ − r 2 ) ( ϕ − r 3 ) .

    Thus, we know from equation (4.10) that the family of periodic orbits has the parametric representation

    (4.15) ϕ ( ξ ) = r 2 1 − A 1 ˜ 2 s n 2 ( Ω 2 ξ , k ) ,

    where Ω 2 = 1 2 3 r 1 ( r 2 − r 3 ) 2 ∣ A ∣ , k 2 = ( r 2 − r 1 ) r 3 ( r 2 − r 3 ) r 1 , and A 1 ˜ 2 = r 1 − r 2 r 1 .

    Equation (4.15) indicates the following exact solutions to equation (1.1):

    (4.16) u ( x , t ) = r 2 1 − A 1 ˜ 2 s n 2 ( Ω 2 ξ , k ) 1 2 p e i η , v ( x , t ) = 1 3 r 2 1 − A 1 ˜ 2 s n 2 ( Ω 2 ξ , k ) .

  2. When h ∈ ( 0 , h 2 ) , we are aware that the level curves defined by H a = 1 2 ( ϕ , y ) = h contain two close branches, for which one family encloses the equilibrium point E 1 ( ϕ 1 , 0 ) , while the other contacts to singular straight line ϕ = 0 at the equilibrium point E 0 ( 0 , 0 ) . We have y 2 = 2 3 ∣ A ∣ ϕ ( r 1 − ϕ ) ( ϕ − r 2 ) ( ϕ − r 3 ) and y 2 = 2 3 ∣ A ∣ ϕ ( r 1 − ϕ ) ( r 2 − ϕ ) ( r 3 − ϕ ) , respectively . Thus, the left family of periodic orbits has the parametric representation

    (4.17) ϕ ( ξ ) = r 1 − r 1 1 − A 3 ˜ 2 s n 2 ( Ω 3 ξ , k ) ,

    and the right family of periodic orbits has the parametric representation

    (4.18) ϕ ( ξ ) = r 3 + r 2 − r 3 1 − A 2 ˜ 2 s n 2 ( Ω 3 ξ , k ) ,

    where Ω 3 = 1 2 3 r 2 ( r 1 − r 3 ) 2 ∣ A ∣ , k 2 = ( r 1 − r 2 ) r 3 ( r 1 − r 3 ) r 2 , A 2 ˜ 2 = r 1 − r 2 r 1 − r 3 , and A 3 ˜ 2 = − r 2 r 1 − r 3 .

    From equations (4.17) and (4.18), we are able to find the following two families of exact solutions:

    (4.19) u ( x , t ) = r 3 + r 2 − r 3 1 − A 2 ˜ 2 s n 2 ( Ω 3 ξ , k ) 1 2 p e i η , v ( x , t ) = 1 3 r 3 + r 2 − r 3 1 − A 2 ˜ 2 s n 2 ( Ω 3 ξ , k ) .

    (4.20) u ( x , t ) = r 1 − r 1 1 − A 2 ˜ 2 s n 2 ( Ω 3 ξ , k ) 1 2 p e i η , v ( x , t ) = 1 3 r 1 − r 1 1 − A 2 ˜ 2 s n 2 ( Ω 3 ξ , k ) .

  3. The level curves defined by H a = 1 2 ( ϕ , y ) = h 2 contain a homoclinic orbit enclosing the equilibrium points E 1 ( ϕ 1 , 0 ) and two heteroclinic orbits connecting the equilibrium points: E 2 ( ϕ 2 , 0 ) and E 0 ( 0 , 0 ) . We receive that y 2 = 2 3 ∣ A ∣ ( ϕ M − ϕ ) ( ϕ − ϕ 2 ) 2 ϕ .

    In addition, matching to the homoclinic orbit, we sketch the next solitary wave solutions

    (4.21) ϕ ( ξ ) = ϕ 2 + 2 ϕ 2 ( ϕ M − ϕ 2 ) ϕ M cosh ( Ω 0 ξ ) − ( ϕ M − 2 ϕ 2 ) ,

    where Ω 0 = 3 ϕ 2 ( ϕ M − ϕ 2 ) 2 A .

    Writing to the two heteroclinic orbits, we gain the following kink and anti-kink wave solutions:

    (4.22) ϕ ( ξ ) = ϕ 2 − 4 A 2 P 2 P 2 2 e ± ω 2 ξ + ϕ M 2 e ∓ ω 2 ξ − 2 B 2 P 2 ,

    where 0 < ϕ 0 < ϕ 2 , ω 2 = 3 A 2 2 A , A 2 = ϕ 2 ( ϕ M − ϕ 2 ) , B 2 = 2 ϕ 2 − ϕ M , and P 2 = 2 A 2 ( A 2 + B 2 ϕ 0 + ϕ 0 2 ) + B 2 ϕ 0 + 2 A 2 ϕ 0 .

    From equations (4.21) and (4.22), we obtain the exact solutions to equation (1.1) as follows:

    (4.23) u ( x , t ) = ϕ 2 − 4 A 2 P 2 P 2 2 e ± ω 2 ξ + ϕ M 2 e ∓ ω 2 ξ − 2 B 2 P 2 1 2 p e i η , v ( x , t ) = 1 3 ϕ 2 − 4 A 2 P 2 P 2 2 e ± ω 2 ξ + ϕ M 2 e ∓ ω 2 ξ − 2 B 2 P 2 .

    (4.24) u ( x , t ) = ϕ 2 + 2 ϕ 2 ( ϕ M − ϕ 2 ) ϕ M cosh ( Ω 0 ξ ) − ( ϕ M − 2 ϕ 2 ) 1 2 p e i η , v ( x , t ) = 1 3 ϕ 2 + 2 ϕ 2 ( ϕ M − ϕ 2 ) ϕ M cosh ( Ω 0 ξ ) − ( ϕ M − 2 ϕ 2 ) .

  4. When h ∈ ( h 2 , ∞ ) , we know that the level curves defined by H a = 1 2 ( ϕ , y ) = h are a global family of close orbits of equation (4.5). At the same time, it encloses the equilibrium points E 1 ( ϕ 1 , 0 ) and E 2 ( ϕ 2 , 0 ) . Besides, it also contacts the singular straight line ϕ = 0 at E 0 ( 0 , 0 ) . Next, we can obtain y 2 = 2 3 ∣ A ∣ ( ϕ M − ϕ ) [ ( ϕ − b 1 ) 2 + a 1 2 ] ϕ . Thus, equation (4.10) exists the following periodic solutions:

    (4.25) ϕ ( ξ ) = ϕ M B 3 ( 1 − c n ( Ω 4 ξ , k ) ) ( A 3 + B 3 ) − ( A 3 + B 3 ) c n ( Ω 4 ξ , k ) ,

    where Ω 4 = 1 2 3 A 3 B 3 2 ∣ A ∣ , k 2 = ϕ M 2 − ( A 3 − B 3 ) 2 4 A 3 B 3 , A 3 2 = ( ϕ M − b 1 ) 2 + a 1 2 , and B 3 2 = a 1 2 + b 1 2 .

    Hence, equation (4.25) can gain the exact solutions to equation (1.1) as follows:

    (4.26) u ( x , t ) = ϕ M B 3 ( 1 − c n ( Ω 4 ξ , k ) ) ( A 3 + B 3 ) − ( A 3 + B 3 ) c n ( Ω 4 ξ , k ) 1 2 p e i η , v ( x , t ) = 1 3 ϕ M B 3 ( 1 − c n ( Ω 4 ξ , k ) ) ( A 3 + B 3 ) − ( A 3 + B 3 ) c n ( Ω 4 ξ , k ) .

Case 3. A < 0 , B > 0 , Δ = 0 (Figure 1(c)). If so, we obtain ϕ 1 = ϕ 2 = B 2 ∣ A ∣ and h 1 = h 2 = B 2 12 A 2 .

  1. When h ∈ ( 0 , h 2 ) and h ∈ ( h 2 , ∞ ) , the level curves defined by H a = 1 2 ( ϕ , y ) = h are two families of periodic orbits of equation (4.5). Then, we recognize that they can obtain the same representation of the parameters as in equation (4.25). Thus, equation (1.1) has the same exact solutions as equation (4.26).

  2. The level curves defined by H a = 1 2 ( ϕ , y ) = h 2 are two heteroclinic orbits. Then, we know that y 2 = 2 3 ∣ A ∣ ( ϕ 2 − ϕ ) 3 ϕ . So, we can hold the following kink and anti-kink wave solutions:

    (4.27) ϕ ( ξ ) = 3 ϕ 2 3 ξ 2 8 ∣ A ∣ + 3 ϕ 2 2 ξ 2 = 3 B 3 ξ 2 64 A 4 + 6 ∣ A ∣ B 2 ξ 2 .

Equation (4.26) receives the exact solutions to equation (1.1) as follows:

(4.28) u ( x , t ) = 3 B 3 ξ 2 64 A 4 + 6 ∣ A ∣ B 2 ξ 2 1 2 p e i η , v ( x , t ) = 1 3 3 B 3 ξ 2 64 A 4 + 6 ∣ A ∣ B 2 ξ 2 .

Case 4. A < 0 and Δ < 0 (Figure 1(d)).

When h ∈ ( 0 , ∞ ) , the level curves defined by H a = 1 2 ( ϕ , y ) = h are a family of periodic orbits of equation (4.5) contacting to the singular straight line ϕ = 0 at E 0 ( 0 , 0 ) . So, we can see that it has the same parametric representations as equation (4.26). From here, we can deduce that equation (1.1) has the same exact solutions as equation (4.27).

Then, we can make a similar discussion for the cases in Figure 1(e–g). We omit them.

5 Applications of the first integral method

Accordingly, we use the first integral method to solve equation (1.1). Assume that

(5.1) u ( x , y , t ) = f ( ξ ) e i η , v ( x , y , t ) = f 0 ( ξ ) ,

where ξ = x + y − 2 ( k + λ ) t and η = k x + λ y − w t . λ , k , and w are real parameters, f ( ξ ) is a real function.

Putting equation (5.1) into equation (1.1), then making real and imaginary part be zero, respectively,

(5.2) ( w − k 2 − λ 2 ) f + 2 f ″ + γ f p + 1 + α f f 0 + δ f 2 p + 1 = 0 , f 0 = β 2 f p ,

we gain

(5.3) f ″ = − 1 2 δ f 2 p + 1 − 1 2 γ + α β 2 f p + 1 + 1 2 ( λ 2 + k 2 − w ) f .

Having simplified, we arrive at the following equation:

(5.4) f ″ = A f 2 p + 1 + B f p + 1 + C f ,

where − 1 2 δ = A ( A ≠ 0 ) , B = − 1 2 γ + α β 2 , and C = 1 2 ( λ 2 + k 2 − w ) .

Substituting equation (3.6) into equation (5.4), we are able to obtain

(5.5) X ′ = Y , Y ′ = A X 2 p + 1 + B X p + 1 + C X .

According to the first integral method, we assume that X = X ( ξ ) and Y = Y ( ξ ) are the nontrivial solutions of equation (5.5), and

(5.6) P ( X , Y ) = ∑ i = 0 m a i ( X ) Y i

is an irreducible polynomial in the complex domain C [ X , Y ] . Then, we make

(5.7) P ( X ( ξ ) , Y ( ξ ) ) = ∑ i = 0 m a i ( X ( ξ ) ) Y ( ξ ) i = 0 ,

where a i ( X ) ( i = 0 , 1 , … , m ) are polynomials of X and a m ( X ) ≠ 0 . After that, equation (5.2) is called the first integral of equation (5.5). Note that P ( X ( ξ ) , Y ( ξ ) ) is a polynomial in X and Y, and d P d ξ implies d P d ξ ( 5.6 ) = 0 .

There exists a polynomial H ( X , Y ) = h ( X ) + g ( X ) Y in C ( X , Y ) . Thus,

(5.8) d P d ξ ( 3.22 ) = d P d X d X d ξ + d P d Y d Y d ξ ( 3.22 ) = ( h ( X ) + g ( X ) Y ) ∑ i = 0 m a i ( X ) Y i .

Case 1. Let us assume m = 1 in equation (5.2). From equation (5.3), we have

(5.9) ∑ i = 0 1 a i ′ ( X ) Y i + 1 + ∑ i = 0 1 i a i ( X ) Y i − 1 ( Y ′ ( ξ ) ) = ( h ( X ) + g ( X ) Y ) ∑ i = 0 1 a i ( X ) Y i ,

where the prime denotes differentiating with respect to the variable X . Comparing to the coefficient of Y i ( i = 2 , 1 , 0 ) on equation (5.4), we receive

(5.10) a 1 ′ ( X ) = a 1 ( X ) h ( X ) ,

(5.11) a 0 ′ ( X ) = a 1 ( X ) g ( X ) + a 0 ( X ) h ( X ) , and

(5.12) a 0 ( X ) g ( X ) = a 1 ( X ) ( A X 2 p + 1 + B X p + 1 + C X ) .

Since a i ( X ) ( i = 0 , 1 ) are polynomials, we obtain that a 1 ( X ) is a constant from equation (5.3) and make a 1 ( X ) = 1 . According to equations (5.11) and (5.12), we obtain

deg [ a 0 ( x ) ] + deg [ g ( X ) ] = 2 p + 1 ,

deg [ g ( X ) ] = p ,

deg [ a 0 ( X ) ] = p + 1 .

Making it

(5.13) g ( X ) = ∑ i = 0 p A i X i ,

where A p ≠ 0 , from equation (5.11), we obtain

(5.14) a 0 ( X ) = ∑ i = 0 p 1 i + 1 A i X i + 1 + B 0 ,

where B 0 is an arbitrary integration constant. By entering a 0 ( X ) , a 1 ( X ) , and g ( X ) into equation (5.12), the coefficients of X i ( i = 0 , 1 , … , 8 ) can be compared and obtained as follows:

X 0 : A 0 B 0 = 0 , X 1 : A 0 2 + A 1 B 0 = C , X 2 : 1 2 A 0 A 1 + A 0 A 0 + A 2 B 0 = 0 , ⋮ X p : 1 p A 0 A p − 1 + 1 p − 1 A 1 A p − 2 + ⋯ + 1 2 A p − 2 A 1 + A p − 1 A 0 + A p B 0 = 0 , X p + 1 : 1 p + 1 A 0 A p + 1 p A 1 A p − 1 + ⋯ + 1 2 A p − 1 A 1 + A p A 0 + A p B 0 = B , X p + 2 : 1 p + 1 A 1 A p + 1 p A 2 A p − 1 + ⋯ + 1 3 A p − 1 A 2 + 1 2 A p A 1 = 0 , ⋮ X p + n : 1 p + 1 A n − 1 A p + 1 p A n A p − 1 + ⋯ + 1 n A p A n − 1 = 0 , ⋮ X 2 p : 1 p + 1 A p − 1 A p + 1 p A p A p − 1 = 0 , X 2 p + 1 : 1 p + 1 A p 2 = A .

Substituting a 0 ( X ) , a 1 ( X ) , and h ( X ) into equation (5.12) and making all the coefficients of X to be zero, we receive a system of nonlinear algebraic equations and use Maple to solve them. Therefore, we obtain

(5.15) A p = ± ( p + 1 ) A , A p − 1 = A p − 2 = ⋯ = A 1 = 0 , A 0 = ± C , B 0 = 0 ,

where A , C > 0 , and ( p + 1 ) B 2 = ( p + 2 ) 2 A C .

Next, we obtain a 0 ( X ) = A 0 X + 1 p + 1 A p X p + 1 . Take it into equation (5.8) and obtain

(5.16) A 0 X + 1 p + 1 A p X p + 1 + Y = 0 .

From equation (5.16), we have

(5.17) X ′ = − A 0 X − 1 p + 1 A p X p + 1 .

  1. When A p = − ( p + 1 ) A , and A 0 = − C , we hold X ′ = C X + 1 p + 1 ( p + 1 ) A X p + 1 . Let us have

    (5.18) X = − ( p + 1 ) C e p C ( ξ + ξ 01 ) A ( e p C ( ξ + ξ 01 ) ± 1 ) 1 p .

    Moreover, one can obtain the exact solutions to equation (1.1) as follows

    (5.19) u ( x , t ) = − ( p + 1 ) C e p C ( ξ + ξ 01 ) A ( e p C ( ξ + ξ 01 ) ± 1 ) 1 p e i η , v ( x , t ) = B 2 − ( p + 1 ) C e p C ( ξ + ξ 01 ) A ( e p C ( ξ + ξ 01 ) ± 1 ) ,

    where ξ 01 is an arbitrary constant.

  2. When A p = ( p + 1 ) A and A 0 = − C , we hold X ′ = C X − 1 p + 1 ( p + 1 ) A X p + 1 . Let us have

    (5.20) X = ( p + 1 ) C e p C ( ξ + ξ 02 ) A ( e p C ( ξ + ξ 02 ) ± 1 ) 1 p .

    Moreover, one can obtain the exact solutions to equation (1.1) as follows:

    (5.21) u ( x , t ) = ( p + 1 ) C e p C ( ξ + ξ 02 ) A ( e p C ( ξ + ξ 02 ) ± 1 ) 1 p e i η , v ( x , t ) = B 2 ( p + 1 ) C e p C ( ξ + ξ 02 ) A ( e p C ( ξ + ξ 02 ) ± 1 ) ,

    where ξ 02 is an arbitrary constant.

  3. When A p = ( p + 1 ) A and A 0 = C , we hold X ′ = − C X − 1 p + 1 ( p + 1 ) A X p + 1 . Let us have

    (5.22) X = − ( p + 1 ) C A ( 1 ± e p C ( ξ + ξ 03 ) ) 1 p .

    Moreover, one can obtain the exact solutions to equation (1.1) as follows:

    (5.23) u ( x , t ) = − ( p + 1 ) C A ( 1 ± e p C ( ξ + ξ 03 ) ) 1 p e i η , v ( x , t ) = B 2 − ( p + 1 ) C A ( 1 ± e p C ( ξ + ξ 03 ) ) ,

    where ξ 03 is an arbitrary constant.

  4. When A p = − ( p + 1 ) A and A 0 = C , we hold X ′ = − C X + 1 p + 1 ( p + 1 ) A X p + 1 . Let us have

    (5.24) X = ( p + 1 ) C A ( 1 ± e p C ( ξ + ξ 04 ) ) 1 p .

    Moreover, one can obtain the exact solutions to equation (1.1) as follows:

    (5.25) u ( x , t ) = ( p + 1 ) C A ( 1 ± e p C ( ξ + ξ 04 ) ) 1 p e i η , v ( x , t ) = B 2 ( p + 1 ) C A ( 1 ± e p C ( ξ + ξ 04 ) ) ,

    where ξ 04 is an arbitrary constant.

Case 2. Suppose that m = 2 . Compared to the coefficient of Y i ( i = 3 , 2 , 1 , 0 ) on equation (5.3), we have

(5.26) a 2 ′ ( X ) = a 2 ( X ) h ( X ) ,

(5.27) a 1 ′ ( X ) = a 2 ( X ) g ( X ) + a 1 ( X ) h ( X ) ,

(5.28) a 0 ′ ( X ) + 2 a 2 ( X ) Y ′ = a 1 ( X ) g ( X ) + a 0 ( X ) h ( X ) ,

(5.29) a 1 ( X ) Y ′ = a 0 ( X ) g ( X ) = a 1 ( X ) ( A X 2 p + 1 + B X p + 1 + C X ) .

Since a i ( X ) ( i = 0 , 1 , 2 ) = 0 are polynomials, we obtain deg [ a 2 ( X ) ] = 0 , h ( X ) = 0 , deg [ a 0 ( X ) ] = 2 p + 2 . Then, we make a 2 ( X ) = 1 .

Next, let us discuss a 1 ( X ) and g ( X ) in two cases.

(I) When a 1 ( X ) = 0 and g ( X ) = 0 , we obtain

a 0 ( X ) = − 1 p + 1 A X 2 p + 2 − 2 p + 2 B X p + 2 − C X 2 .

Then bringing a 0 ( X ) , a 1 ( X ) , and a 2 ( X ) into equation (5.8), we obtain

(5.30) X ′ = ± 1 p + 1 A X 2 ( X p ) 2 + 2 − 2 p + 2 B A X p + ( p + 1 ) C A .

The simultaneous integration of equation (5.30) yields

X = D e A D p + 1 p ( ξ + ξ 05 ) − 1 1 p e A D p + 1 ( ξ + ξ 05 ) ,

(5.31) X = D e A D p + 1 p ( ξ + ξ 06 ) + 1 1 p ,

where A > 0 , C > 0 , B 2 = ( p + 2 ) 2 p + 1 A C , and D = C ( p + 1 ) A . Next, we have the exact solutions to equation (1.1) as follows:

(5.32) u ( x , t ) = D e A D p + 1 p ( ξ + ξ 05 ) − 1 1 p e A D p + 1 ( ξ + ξ 05 ) e i η , v ( x , t ) = B 2 D e A D p + 1 p ( ξ + ξ 05 ) − 1 e p A D p + 1 ( ξ + ξ 05 ) .

(5.33) u ( x , t ) = D e A D p + 1 p ( ξ + ξ 06 ) + 1 1 p e i η , v ( x , t ) = B 2 D e A D p + 1 p ( ξ + ξ 06 ) + 1 .

(II) When deg [ a 1 ( X ) ] = deg [ g ( X ) ] + 1 , we make it to be two cases as follows:

When deg [ g ( X ) ] = n ( n < p ) , deg [ a 1 ( X ) ] = n + 1 , it is contradictory.

When deg [ g ( X ) ] = p and deg [ a 1 ( X ) ] = p + 1 , we make g ( X ) = ∑ i = 0 p A i X i ( A p ≠ 0 ). Then a 1 ( X ) = ∑ i = 0 p 1 i + 1 A i X i + 1 + B 0 , where B 0 is a constant. Similar ones can be launched.

  1. When

    (5.34) A p = ± 2 ( p + 1 ) A , A p − 1 = ⋯ = A 1 = A 0 = B 0 = 0 ,

    we have a 1 ( X ) = ± 2 A p + 1 X p + 1 and a 0 ( X ) = A p + 1 X 2 p + 2 , where a 0 ( X ) is a constant, C 0 = 0 and B = C = 0 . Bringing it into equation (5.8), we obtain X ′ = ± A p + 1 X p + 1 .

    Then, it is to obtain,

    (5.35) X = ± p + 1 A 1 p ( ξ + ξ 07 ) 1 p ,

    where ξ 07 is a constant. We have the exact solutions to equation (1.1) as follows:

    (5.36) u ( x , t ) = ± p + 1 A 1 p ( ξ + ξ 07 ) 1 p e i η , v ( x , t ) = ± B 2 p + 1 A 1 p ( ξ + ξ 07 ) .

  2. When

    (5.37) A p = 2 ( p + 1 ) A , A 0 = 2 C , A p − 1 = ⋯ = A 1 = B 0 = 0 ,

    we have a 1 ( X ) = 2 X A p + 1 X p + C and a 0 ( X ) = X 2 A p + 1 X p + C 2 , where a 0 ( X ) is a constant, C 0 = 0 , ( p + 1 ) B 2 = ( p + 2 ) 2 A C , and B > 0 . Taking it to equation (5.8), we obtain X ′ = − X A p + 1 X p + C .

    Then, it is to obtain

    (5.38) X = C e p C ( ξ + ξ 08 ) − A p + 1 1 p ,

    where ξ 08 is a constant. We have the exact solutions to equation (1.1) as follows:

    (5.39) u ( x , t ) = C e p C ( ξ + ξ 08 ) − A p + 1 1 p e i η , v ( x , t ) = B 2 C e p C ( ξ + ξ 08 ) − A p + 1 .

  3. When

    (5.40) A p = 2 ( p + 1 ) A , A 0 = − 2 C , A p − 1 = ⋯ = A 1 = B 0 = 0 ,

    we have a 1 ( X ) = 2 X A p + 1 X p − C and a 0 ( X ) = X 2 A p + 1 X p − C 2 , where a 0 ( X ) is a constant, C 0 = 0 , ( p + 1 ) B 2 = ( p + 2 ) 2 A C , and B < 0 . Bringing it into equation (5.8), we obtain X ′ = − X A p + 1 X p − C .

    Then, it is to obtain

    (5.41) X = C A p + 1 e p C ( ξ + ξ 09 ) − 1 1 p e C ( ξ + ξ 09 ) ,

    where ξ 09 is a constant. We have the exact solutions to equation (1.1) as follows:

    (5.42) u ( x , t ) = C A p + 1 e p C ( ξ + ξ 09 ) − 1 1 p e C ( ξ + ξ 09 ) e i η , v ( x , t ) = B 2 C A p + 1 e p C ( ξ + ξ 09 ) − 1 e p C ( ξ + ξ 09 ) .

  4. When

    (5.43) A p = − 2 ( p + 1 ) A , A 0 = 2 C , A p − 1 = ⋯ = A 1 = B 0 = 0 ,

    we have a 1 ( X ) = − 2 X A p + 1 X p − C , and a 0 ( X ) = X 2 A p + 1 X p − C 2 , where a 0 ( X ) is a constant, C 0 = 0 , ( p + 1 ) B 2 = ( p + 2 ) 2 A C , and B < 0 .

    Taking it into equation (5.8), we obtain X ′ = X A p + 1 X p − C .

    Then, it is to obtain

    (5.44) X = C − e p C ( ξ + ξ 10 ) + A p + 1 1 p ,

    where ξ 10 is a constant. We have the exact solutions to equation (1.1) as follows:

    (5.45) u ( x , t ) = C − e p C ( ξ + ξ 10 ) + A p + 1 1 p e i η , v ( x , t ) = B 2 C − e p C ( ξ + ξ 10 ) + A p + 1 .

  5. When

    (5.46) A p = − 2 ( p + 1 ) A , A 0 = − 2 C , A p − 1 = ⋯ = A 1 = B 0 = 0 ,

    we have a 1 ( X ) = − 2 X A p + 1 X p + C and a 0 ( X ) = X 2 A p + 1 X p − C 2 , where a 0 ( X ) is a constant, C 0 = 0 , ( p + 1 ) B 2 = ( p + 2 ) 2 A C , and B > 0 . Bringing it into equation (5.8), we obtain X ′ = X A p + 1 X p + C .

    Then, it is to obtain

    (5.47) X = C 1 − A p + 1 e p C ( ξ + ξ 11 ) 1 p e C ( ξ + ξ 11 ) ,

    where ξ 11 is a constant. We have the exact solutions to equation (1.1) as follows:

    (5.48) u ( x , t ) = C 1 − A p + 1 e p C ( ξ + ξ 11 ) 1 p e C ( ξ + ξ 11 ) e i η , v ( x , t ) = B 2 C 1 − A p + 1 e p C ( ξ + ξ 11 ) e p C ( ξ + ξ 11 ) .

6 Conclusion

This article obtains many new exact solutions to generalized Davey-Stewartson equations with arbitrary power nonlinearity by the dynamical system and the first integral methods. These exact solutions include periodic wave solutions, exact solitary wave solutions, kink wave solutions, anti-kink wave solutions, etc. We conclude that our results are novel and abundant by comparing the results with [41,42,43, 44,45] using the software Maple. Hence, the two methods can also be extended to solve other nonlinear PDEs and obtain more exact solutions. Nevertheless, one can note that the integer-order derivative equations considered in our dissertation are, in fact, only time-integers. Would the traveling wave solutions be consistent with our results or more for time and spatial-fractional order derivatives regarding the generalized Davey-Stewartson equations? We leave this topic for future analysis.

Acknowledgements

The authors thank the referees for their helpful comments and are very grateful to Prof. Chen Longwei and Prof. Li Jibing for their support in writing this article.

  1. Funding information: This work was partially supported by the 2021 school-level project of Ningbo Polytechnic (NZ22014).

  2. Conflict of interest: The authors state no conflict of interest.

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Received: 2021-11-15
Revised: 2022-04-10
Accepted: 2022-06-23
Published Online: 2022-09-01

© 2022 Yanjie Wang et al., published by De Gruyter

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

Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
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  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
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  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
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  59. On the general position number of two classes of graphs
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  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
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  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
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  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
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  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
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  90. Class-preserving Coleman automorphisms of some classes of finite groups
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  109. Some new characterizations of finite p-nilpotent groups
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  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
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  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
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  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
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  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
https://doi.org/10.1515/math-2022-0469
Keywords for this article
generalized Davey-Stewartson equation; the dynamical system method; the first integral method; solitary wave solution; kink wave solution; periodic wave solution
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Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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