On Compact and Noncompact Structures for the Improved Boussinesq Water Equations

Abstract

The nonlinear variants of the generalized Boussinesq water equations with positive and negative exponents are studied in this paper. The analytic expressions of the compactons, solitons, solitary patterns, and periodic solutions for the equations are obtained by using a technique based on the reduction of order of differential equations. It is shown that the nonlinear variants, or nonlinear variants together with the wave numbers, directly lead to the qualitative change in the physical structures of the solutions.

1. Introduction

One of the classical Boussinesq water equations can be written by the form 𝑢_𝑡𝑡=−𝛼𝑢_{𝑥𝑥𝑥𝑥}+𝑢_𝑥𝑥+𝛽𝑢²_𝑥𝑥, (1) where 𝑢(𝑥,𝑡) is the elevation of the free surface of fluid, the subscripts denote partial derivatives, and the constant coefficients 𝛼 and 𝛽 depend on the depth of fluid and the characteristic speed of the long waves. It is well known that Boussinesq equation (1) was originally introduced as a model for one-dimensional weakly nonlinear dispersive waves in shallow water and was subsequently applied to explain problems in the percolation of water in porous subsurface strata. It also crops up in the analysis of many other physical processes.

Various generalizations of the classical Boussinesq water equation have been proposed and studied to probe the dynamic properties of their solutions. The generalized Boussinesq's equations have been successfully applied in coastal engineering for simulating wave propagation from the deep sea to a shallow-water region and in naval architecture for computing ship waves in shallow water (See [1, 2]). Kaya [3] obtained the exact and numerical solitary-wave solutions for a generalized modified Boussinesq equation. Elgarayhi and Elhanbaly [1] discussed and analyzed new exact traveling wave solutions for the two-dimensional KdV-Burgers and Boussinesq equations. Lai and Wu [4] and Lai et al. [5] proposed an approach for constructing asymptotic solution of the Boussinesq equations. Rosenau [6] proved that the vibration of the anharmonic mass-spring chain leads to a new Boussinesq equation admitting compactons and compact breathers. It was formally shown that the collision of two compactons results in the creation of low-amplitude compacton and anticompacton pairs. One can find that many meaningful achievements that outline the structure of compactons and the collision of two compactons were reported in [6–8]. For more details about the compactons and soliton solutions for nonlinear evolution equations, the reader is referred to [4, 5, 9–11] in which many analytical and numerical methods such as the pseudospectral methods, the Galerkin method, the finite differences method, the sine-cosine ansatz, and the tanh method have been presented.

The following (2+1)-dimensional Boussinesq water system 𝑢_𝑡𝑡−𝑢_𝑥𝑥−𝑢_𝑦𝑦−𝑢²_𝑥𝑥−𝑢_{𝑥𝑥𝑥𝑥}=0 (2) was studied in [11]. The author discussed the stability of solitary wave solution of (2). It is found that pulse-like solutions to the Boussinesq water equation are stable to linear perturbations.

Wazwaz [12] studied the variants of the following improved Boussinesq water equations.(I)Variant of the (1+1)-dimensional improved Boussinesq equation with positive exponents is given by 𝑢_𝑡𝑡−𝑢_𝑥𝑥−𝑎𝑢^2𝑛_𝑥𝑥−𝑏𝑢^𝑛𝑢^𝑛_{𝑥𝑥𝑡𝑡}=0,𝑛>1, (3) and with negative exponents is given by 𝑢_𝑡𝑡−𝑢_𝑥𝑥−𝑎𝑢^−2𝑛_𝑥𝑥−𝑏𝑢^−𝑛𝑢^−𝑛_{𝑥𝑥𝑡𝑡}=0,𝑛>1, (4) where 𝑎≠0 and 𝑏≠0 are constants.(II)Variant of the (2+1) dimensional improved Boussinesq equation with positive exponents (some typing's errors occurred for (5) and (6) in [12]. This can be found through the reading of the whole paper) is given by 𝑢_𝑡𝑡−𝑢_𝑥𝑥−𝑢_𝑦𝑦−𝑎₁𝑢^2𝑛_𝑥𝑥−𝑎₂𝑢^2𝑛_𝑦𝑦−𝑏₁𝑢^𝑛𝑢^𝑛_{𝑥𝑥𝑡𝑡}−𝑏₂𝑢^𝑛𝑢^𝑛_{𝑦𝑦𝑡𝑡}=0,𝑛>1, (5)and with negative exponents is in the form 𝑢_𝑡𝑡−𝑢_𝑥𝑥−𝑢_𝑦𝑦−𝑎₁𝑢^−2𝑛_𝑥𝑥−𝑎₂𝑢^−2𝑛_𝑦𝑦−𝑏₁𝑢^−𝑛𝑢^−𝑛_{𝑥𝑥𝑡𝑡}−𝑏₂𝑢^−𝑛𝑢^−𝑛_{𝑦𝑦𝑡𝑡}=0,𝑛>1, (6) where 𝑎₁+𝑎₂≠0 and 𝑏₁+𝑏₂≠0.

Wazwaz [12] assumed that the solutions of (3), (4), (5), and (6) take the forms in terms of sine or cosine; namely, 𝑢(𝑥,𝑡)=𝜆cos^𝛽𝜇𝜉₁,𝜉₁≤𝜋,2𝜇 (7) or the ansatz 𝑢(𝑥,𝑡)=𝜆sin^𝛽𝜇𝜉₁,𝜉₁≤𝜋𝜇, (8) where 𝜉₁=𝑥−𝑐𝑡 or 𝜉₁=𝑥+𝑦−𝑐𝑡 and 𝜇 and 𝑐 are the wave number and the wave speed, respectively. Using the balancing rule, Wazwaz [12] calculated and found out the concrete values of 𝛽,𝜇,𝜆.

In this paper, by using a mathematical technique different from those in [9, 12], we derive analytical expressions of the travelling solutions with wave variable 𝜉=𝜇(𝑥−𝑐𝑡) for (3) and (4) or 𝜉=𝜇𝑥+𝜂𝑦−𝑐𝑡 for (5) and (6) where 𝜇, 𝜂, and 𝑐 are arbitrary nonzero constants. The technique used in this paper is proved very useful in constructing the exact travelling wave solutions for (3)–(6). It is shown that the nonlinear variants, or nonlinear variants together with the wave numbers, directly determine the physical structures of the solutions such as compactons, solitons, solitary patterns, and periodic solutions. Moreover, the results presented in this paper include those carried out in Wazwaz [12].

2. Variant of the (1+1)-Dimensional Improved Boussinesq Equation with Positive Exponents

Firstly, we consider solutions of the following equation: 𝑑𝑊𝑑𝑧²=𝑎₀−𝑏₀𝑊², (9) where 𝑎₀≠0 and 𝑏₀≠0 are constants. When 𝑏₀>0, (9) admits two solutions: 𝑊₁=±𝑎₀𝑏₀sin𝑏₀(,𝑊𝑧+𝐴)₂=±𝑎₀𝑏₀cos𝑏₀(𝑧+𝐴), (10) where 𝐴 is an arbitrary constant.

When 𝑏₀<0, noticing that cosh²𝑧−sinh²𝑧=1, we derive that (9) has two solutions of the form 𝑊₃=±−𝑎₀𝑏₀sinh−𝑏₀(,𝑊𝑧+𝐴)₄=±𝑖−𝑎₀𝑏₀cosh−𝑏₀(𝑧+𝐴), (11) where 𝑖=√−1.

Now, we write out the following formulas which will be used in this work: 1sinh𝑥=𝑖1sin𝑖𝑥,sin𝑥=𝑖sinh𝑖𝑥,cosh𝑥=cos𝑖𝑥,cos𝑥=cosh𝑖𝑥. (12)

Remark 1. Here we point out that we will solve (9) in the complex value district. By the theory of solution structure for the first-order ordinary differential equation, we know that the formulas (10) and (11) include all the analytical solutions of (9). In the following discussions, we let the phase number 𝐴=0 in formulas (10), (11) and the forthcoming analytical expressions of travelling wave solutions.
In this section, we seek formal travelling wave solutions for (3); namely, 𝑢(𝑥,𝑡)=𝑢(𝜉), (13) where the wave variable 𝜉 takes the form 𝜉=𝜇(𝑥−𝑐𝑡) with constants 𝜇≠0 and 𝑐≠0.
The wave variable 𝜉=𝜇(𝑥−𝑐𝑡) turns (3) into the following ODE: 𝑐²−1𝑢_𝜉𝜉−𝑎𝑢^2𝑛_𝜉𝜉−𝑏𝑐²𝜇²𝑢^𝑛𝑢^𝑛_{𝜉𝜉𝜉𝜉}=0. (14) Integrating (14) twice and setting the constants of integration to be zero, we have 𝑐²−1𝑢−𝑎𝑢^2𝑛−𝑏𝑐²𝜇²𝑢^𝑛𝑢^𝑛_𝜉𝜉=0. (15)
Since 𝑐² appears in (15), we can assume that 𝑐>0 always holds.

Case 1. When 𝑐²=1, if 𝑎/𝑏<0, the solution of (15) takes the form 𝑢^𝑛=𝑘₁exp−𝑎𝑏(𝑥−𝑐𝑡)+𝑘₂exp−−𝑎𝑏(𝑥−𝑐𝑡), (16) where 𝑘₁ and 𝑘₂ are arbitrary constants.
If 𝑎/𝑏>0, we solve (15) by 𝑢^𝑛=𝑘₃sin𝑎𝑏(𝑥−𝑐𝑡)+𝑘₄cos𝑎𝑏(𝑥−𝑐𝑡). (17) Here 𝑘₃ and 𝑘₄ are arbitrary constants.

Case 2. When 𝑐²≠1, letting 𝑑𝑢^𝑛/𝑑𝜉=𝑍, we have 𝑑²𝑢^𝑛/𝑑𝜉²=𝑍(𝑑𝑍/𝑑𝑢^𝑛) and 𝑐²−1𝑢−𝑎𝑢^2𝑛−𝑏𝑐²𝜇²𝑢^𝑛𝑍𝑑𝑍𝑑𝑢^𝑛=0. (18) Using 𝑑𝑢^𝑛=𝑛𝑢^𝑛−1𝑑𝑢, we have 𝑐²−1−𝑎𝑢^2𝑛−1−𝑏𝑐²𝜇²𝑛𝑍𝑑𝑍𝑑𝑢=0. (19) Further calculation of (19) gives rise to 𝑑𝑢^{(2𝑛−1)/2}𝑑𝜉²=(2𝑛−1)²𝑏𝑐²𝜇²(2𝑛)²2𝑛𝑐²−1−𝑎𝑢^{(2𝑛−1)/22}. (20)
If 𝑎/𝑏>0, it follows from (10) and (20) that 𝑢^{(2𝑛−1)/2}=±2𝑛𝑐²−1𝑎()sin2𝑛−12𝑛𝑐𝜇𝑎𝑏𝜇(𝑥−𝑐𝑡)+𝐴, (21) or 𝑢^{(2𝑛−1)/2}=±2𝑛𝑐²−1𝑎()cos2𝑛−12𝑛𝑐𝜇𝑎𝑏𝜇(𝑥−𝑐𝑡)+𝐴, (22) where 𝐴 is a free constant. Choosing the phase number 𝐴=0 in formulas (21) and (22) and limiting the domain of 𝜉, we have the following compacton solutions: 𝑐𝑢=2𝑛²−1𝑎sin²(2𝑛−1)2𝑛𝑐𝑎𝑏(𝑥−𝑐𝑡)^{1/(2𝑛−1)},|(𝑥−𝑐𝑡)|<2𝑛𝑐𝜋(2𝑛−1),𝑐𝑎/𝑏𝑢=0,otherwise,𝑢=2𝑛²−1𝑎cos²(2𝑛−1)2𝑛𝑐𝑎𝑏(𝑥−𝑐𝑡)^{1/(2𝑛−1)},|(𝑥−𝑐𝑡)|<𝑛𝑐𝜋(2𝑛−1),𝑎/𝑏𝑢=0,otherwise. (23)(24) However, for 𝑎/𝑏<0, from (11) and (20), we obtain the solitary pattern solutions: −𝑐𝑢=2𝑛²−1𝑎sinh²(2𝑛−1)2𝑛𝑐−𝑎𝑏(𝑥−𝑐𝑡)^{1/(2𝑛−1)},𝑐𝑢=2𝑛²−1𝑎cosh²(2𝑛−1)2𝑛𝑐−𝑎𝑏(𝑥−𝑐𝑡)^{1/(2𝑛−1)}. (25)

Remark 2. From the formulas (21)-(22), we conclude that the compactons and solitary pattern solutions are independent of wave number 𝜇 when we choose the phase number equals zero. These results extend those obtained by Wazwaz [12] in which the wave number 𝜇 was chosen as a special constant.

3. Variant of the (1+1)-Dimensional Improved Boussinesq Equation with Negative Exponents

Now, we consider the (1+1)-dimensional improved Boussinesq equation with negative exponents; namely, 𝑢_𝑡𝑡−𝑢_𝑥𝑥−𝑎𝑢^−2𝑛_𝑥𝑥−𝑏𝑢^−𝑛𝑢^−𝑛_{𝑥𝑥𝑡𝑡}=0,𝑛>1,𝑎≠0,𝑏≠0. (26) Since (3) and (26) are symmetric about 𝑛 and (−𝑛), we can readily obtain the following periodic solutions for (26): −𝑎𝑢=2𝑛𝑐²−1csc²(2𝑛+1)2𝑛𝑐𝑎𝑏(𝑥−𝑐𝑡)^1/(2𝑛+1),𝑎𝑏−𝑎>0,𝑐≠±1,𝑢=2𝑛𝑐²−1sec²(2𝑛+1)2𝑛𝑐𝑎𝑏(𝑥−𝑐𝑡)^1/(2𝑛+1),𝑎𝑏>0,𝑐≠±1. (27) For the case where 𝑎/𝑏<0, (26) has solitons of the form 𝑎𝑢=2𝑛𝑐²−1csch²(2𝑛+1)2𝑛𝑐−𝑎𝑏(𝑥−𝑐𝑡)^1/(2𝑛+1),−𝑎𝑐≠±1,𝑢=2𝑛𝑐²−1sech²(2𝑛+1)2𝑛𝑐−𝑎𝑏(𝑥−𝑐𝑡)^1/(2𝑛+1),𝑐≠±1. (28)

4. Variant of the (2+1)-Dimensional Improved Boussinesq Equation with Positive Exponents

In this section, we consider the solution of (5). The use of wave variable 𝜉=𝜇𝑥+𝜂𝑦−𝑐𝑡 carries the equation into the following ordinary differential equation (ODE): 𝑐²−𝜇²−𝜂²𝑢_𝜉𝜉−𝑎₁𝜇²+𝑎₂𝜂²𝑢^2𝑛_𝜉𝜉−𝑏₁𝜇²+𝑏₂𝜂²𝑐²𝑢^𝑛𝑢^𝑛_{𝜉𝜉𝜉𝜉}=0. (29) Integrating (29) twice and setting the constants of integration to be zero, we get 𝑐²−𝜇²−𝜂²𝑢−𝑎₁𝜇²+𝑎₂𝜂²𝑢^2𝑛−𝑏₁𝜇²+𝑏₂𝜂²𝑐²𝑢^𝑛𝑢^𝑛_𝜉𝜉=0. (30) Using the transformation 𝑑𝑢^𝑛/𝑑𝜉=𝑍, we have 𝑑²𝑢^𝑛/𝑑𝜉²=𝑍(𝑑𝑍/𝑑𝑢^𝑛) and 𝑐²−𝜇²−𝜂²𝑢−𝑎₁𝜇²+𝑎₂𝜂²𝑢^2𝑛−𝑏₁𝜇²+𝑏₂𝜂²𝑐²𝑢^𝑛𝑍𝑑𝑍𝑑𝑢^𝑛=0. (31) Furthermore, 𝑐²−𝜇²−𝜂²−𝑎₁𝜇²+𝑎₂𝜂²𝑢^2𝑛−1−𝑏₁𝜇²+𝑏₂𝜂²𝑐²𝑍𝑑𝑍𝑛𝑑𝑢=0. (32) In the following discussions, we assume that (𝑐²−𝜇²−𝜂²)≠0,𝑎₁𝜇²+𝑎₂𝜂²≠0, and 𝑏₁𝜇²+𝑏₂𝜂²≠0.

Integrating (32) yields 𝑏₁𝜇²+𝑏₂𝜂²𝑐²𝑍2𝑛²=𝑐²−𝜇²−𝜂²𝑢−𝑎₁𝜇²+𝑎₂𝜂²1𝑢2𝑛^2𝑛. (33)

It follows from (33) that 𝑏₁𝜇²+𝑏₂𝜂²𝑐²2𝑛2𝑛𝑑𝑢^{(2𝑛−1)/2}(2𝑛−1)𝑑𝜉²=𝑐²−𝜇²−𝜂²−𝑎₁𝜇²+𝑎₂𝜂²1𝑢2𝑛^2𝑛−1. (34) Setting 𝑊=𝑢^{(2𝑛−1)/2}, we have 𝑑𝑊𝑑𝜉²=(2𝑛−1)²2𝑛𝑐²𝑏₁𝜇²+𝑏₂𝜂²×𝑐²−𝜇²−𝜂²−𝑎₁𝜇²+𝑎₂𝜂²1𝑊2𝑛². (35)(1)If (𝑎₁𝜇²+𝑎₂𝜂²)/(𝑏₁𝜇²+𝑏₂𝜂²)>0 and 𝑛>1/2, it follows from (10) and (35) that 𝑊=2𝑛𝑐²−𝜇²−𝜂²𝑎₁𝜇²+𝑎₂𝜂²sin2𝑛−12𝑛𝑐𝑎₁𝜇²+𝑎₂𝜂²𝑏₁𝜇²+𝑏₂𝜂²(𝜉+𝐴),𝑊=2𝑛𝑐²−𝜇²−𝜂²𝑎₁𝜇²+𝑎₂𝜂²cos2𝑛−12𝑛𝑐𝑎₁𝜇²+𝑎₂𝜂²𝑏₁𝜇²+𝑏₂𝜂²(𝜉+𝐴). (36)Letting 𝐴=0 in formulas (36), we have the following compacton solutions: 𝑐𝑢=2𝑛²−𝜇²−𝜂²𝑎₁𝜇²+𝑎₂𝜂²×sin²2𝑛−12𝑛𝑐𝑎₁𝜇²+𝑎₂𝜂²𝑏₁𝜇²+𝑏₂𝜂²𝜇𝑥+𝜂𝑦−𝑐𝑡^{1/(2𝑛−1)},𝜇𝑥+𝜂𝑦−𝑐𝑡<2𝑛𝑐𝜋(2𝑛−1)𝑏₁𝜇²+𝑏₂𝜂²𝑎₁𝜇²+𝑎₂𝜂²,𝑐𝑢=0,otherwise,𝑢=2𝑛²−𝜇²−𝜂²𝑎₁𝜇²+𝑎₂𝜂²×cos²2𝑛−12𝑛𝑐𝑎₁𝜇²+𝑎₂𝜂²𝑏₁𝜇²+𝑏₂𝜂²𝜇𝑥+𝜂𝑦−𝑐𝑡^{1/(2𝑛−1)},𝜇𝑥+𝜂𝑦−𝑐𝑡<𝑛𝑐𝜋(2𝑛−1)𝑏₁𝜇²+𝑏₂𝜂²𝑎₁𝜇²+𝑎₂𝜂²,𝑢=0,otherwise. (37)(38)(2)For 𝑎₁𝜇²+𝑎₂𝜂²/𝑏₁𝜇²+𝑏₂𝜂²<0 and 𝑛>1/2, we obtain the solitary pattern solutions −𝑐𝑢=2𝑛²−𝜇²−𝜂²𝑎₁𝜇²+𝑎₂𝜂²×sinh²2𝑛−12𝑛𝑐−𝑎₁𝜇²+𝑎₂𝜂²𝑏₁𝜇²+𝑏₂𝜂²×𝜇𝑥+𝜂𝑦−𝑐𝑡^{1/(2𝑛−1)},𝑐𝑢=2𝑛²−𝜇²−𝜂²𝑎₁𝜇²+𝑎₂𝜂²×cosh²2𝑛−12𝑛𝑐−𝑎₁𝜇²+𝑎₂𝜂²𝑏₁𝜇²+𝑏₂𝜂²×𝜇𝑥+𝜂𝑦−𝑐𝑡^{1/(2𝑛−1)}. (39)(40)Letting 𝜇=𝜂 and 𝑐=𝜇𝑐₁ in formulas (37) to (40), we obtain the analytical expressions of compactons and solitary pattern solutions for (5) which was established in Wazwaz’s paper [12].

5. Variant of the (2+1)-Dimensional Improved Boussinesq Equation with Negative Exponents

We firstly assume that (𝑐²−𝜇²−𝜂²)≠0,𝑎₁𝜇²+𝑎₂𝜂²≠0, and 𝑏₁𝜇²+𝑏₂𝜂²≠0. Taking the symmetric property of (−𝑛) and 𝑛 into account in (5) and (6), we find the periodic solutions in the case where (𝑎₁𝜇²+𝑎₂𝜂²)/(𝑏₁𝜇²+𝑏₂𝜂²)>0 for (6) as follows: −𝑎𝑢=¹𝜇²+𝑎₂𝜂²2𝑛𝑐²−𝜇²−𝜂²×csc²2𝑛+12𝑛𝑐𝑎₁𝜇²+𝑎₂𝜂²𝑏₁𝜇²+𝑏₂𝜂²𝜇𝑥+𝜂𝑦−𝑐𝑡^1/(2𝑛+1),−𝑎𝑢=¹𝜇²+𝑎₂𝜂²2𝑛𝑐²−𝜇²−𝜂²×cos²2𝑛+12𝑛𝑐𝑎₁𝜇²+𝑎₂𝜂²𝑏₁𝜇²+𝑏₂𝜂²𝜇𝑥+𝜂𝑦−𝑐𝑡^1/(2𝑛+1), (41) where 𝑛>−1/2.

If (𝑎₁𝜇²+𝑎₂𝜂²)/(𝑏₁𝜇²+𝑏₂𝜂²)<0, we obtain the solitons 𝑎𝑢=¹𝜇²+𝑎₂𝜂²2𝑛𝑐²−𝜇²−𝜂²×csch²2𝑛+12𝑛𝑐−𝑎₁𝜇²+𝑎₂𝜂²𝑏₁𝜇²+𝑏₂𝜂²𝜇𝑥+𝜂𝑦−𝑐𝑡^1/(2𝑛+1),−𝑎𝑢=¹𝜇²+𝑎₂𝜂²2𝑛𝑐²−𝜇²−𝜂²×sech²2𝑛+12𝑛𝑐−𝑎₁𝜇²+𝑎₂𝜂²𝑏₁𝜇²+𝑏₂𝜂²𝜇𝑥+𝜂𝑦−𝑐𝑡^1/(2𝑛+1), (42) where 𝑛>−1/2.

Also, choosing 𝜇=𝜂 and 𝑐=𝜇𝑐₁ in (42) produces the analytical expressions of periodic and soliton solutions for (6) which was obtained in Wazwaz [12].

Remark 3. From the formulae (37) to (46), we can conclude that the exponent 𝑛, the wave numbers 𝜇 and 𝜂, and the coefficients 𝑎₁, 𝑎₂, 𝑏₁, and 𝑏₂ appearing in Boussinesq equation (5) or (6) are the main factors which determine the change of physical structures of solutions. More precisely, exponent 𝑛 and number (𝑎₁𝜇²+𝑎₂𝜂²)/(𝑏₁𝜇²+𝑏₂𝜂²), positive or negative, directly result in the qualitative changes in the physical structures of solutions.

6. Conclusion

In this paper, we solve the generalized Boussinesq water equations with positive or negative exponents. The analytical expressions of the travelling wave solutions for the nonlinear equations presented by (3)–(6) are established.

For the (1+1)-dimensional Boussinesq equations, two outcomes are achieved. One is that the wave number 𝜇 in the analytically expressions of the travelling waves solutions keeps as an arbitrary nonzero constant when the corresponding phase equals zero. In other words, the travelling wave solutions are independent of the wave number 𝜇 if the phase of the solutions is zero. The other is that the exponent 𝑛 together with the ratio 𝑎/𝑏, positive or negative, directly determines the qualitative changes in the physical structures of solutions.

For the (2+1)-dimensional generalized Boussinesq equations with positive or negative exponents, we obtain analytical expressions for the compactons, solitons, solitary patterns, and periodic solutions. It is shown that the exponent 𝑛, the wave numbers 𝜇 and 𝜂, and the coefficients 𝑎₁,𝑎₂,𝑏₁, and 𝑏₂ appearing in Boussinesq equation (5) and (6) are the main factors which determine the changes of physical structures of solutions. More precisely, the exponent 𝑛 and the factor (𝑎₁𝜇²+𝑎₂𝜂²)/(𝑏₁𝜇²+𝑏₂𝜂²), positive or negative, directly lead to the qualitative changes in the physical structures of the solutions.