Theoretical investigations on a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers model for a dilated artery, blood vessel or circulatory system with experimental support

Xin-Yi Gao; Yong-Jiang Guo; Wen-Rui Shan

doi:10.1088/1572-9494/acbf24

1. Introduction

1.1. Background around us and our consideration

Nonlinear waves are physically and currently interesting [1–3]. Physical studies on the pulse waves in the human arteries have started from the ancient times until the present, while recent theoretical efforts have been focused on the probes for nonlinear pulse waves in the variable-radius arteries [4–16].

We hereby consider a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers equation:

$\begin{eqnarray}&&\begin{array}{l}{v}_{t}+{\rho }_{1}(t){{vv}}_{x}+{\rho }_{2}(t){v}_{{xx}}+{\rho }_{3}(t){v}_{{xxx}}\\ \quad +\,{\left[{\rho }_{4}(x,t)v\right]}_{x}+{\rho }_{5}(t)v+{\rho }_{6}(x,t)=0,\end{array}\end{eqnarray} \tag{ 1 }$

where v(x, t), ρ₄(x, t) and ρ₆(x, t) are all the real differentiable functions of the variables x and t, the subscripts represent the partial derivatives, while ρ₁(t) ≠ 0, ρ₂(t), ρ₃(t) ≠ 0 and ρ₅(t) are all the real differentiable functions of t. As seen below, e.g., for an artery full of blood with an aneurysm, v can be the dynamical radial displacement superimposed on the original static deformation from an arterial wall, t and x can be the stretched coordinates, respectively, related to the axial coordinate and to both the axial coordinate and time parameter, ρ_i's (i = 1, ..., 6) can be linked with the axial stretch of the injured artery, blood as an incompressible Newtonian fluid, radius variation in the axial direction or aneurysmal geometry, viscosity of the fluid, thickness of the artery, mass density of the membrane material, mass density of the fluid, strain energy density of the artery, shear modulus, stretch ratio, etc.

The following special cases guide ρ₁(t) ≠ 0 and ρ₃(t) ≠ 0, while ρ₂(t), ρ₄(x, t), ρ₅(t) and ρ₆(x, t) are not restricted.

1.2. Special cases of equation (1)

There have been some special cases of equation (1) as follows:

when ${\rho }_{1}(t)={\tilde{\sigma }}_{1}$ , ${\rho }_{2}(t)={\tilde{\sigma }}_{2}$ , ${\rho }_{3}(t)={\tilde{\sigma }}_{3}$ , ${\rho }_{4}(x,t)\,={\tilde{\sigma }}_{4}(t)$ , ${\rho }_{5}(t)={\tilde{\sigma }}_{5}$ and ${\rho }_{6}(x,t)={\tilde{\sigma }}_{6}(t)$ , a forced–perturbed Korteweg–de Vries–Burgers equation with variable coefficients for the nonlinear waves in an artery full of blood with a local dilatation (representing a certain aneurysm) [15, 16]³
$\begin{eqnarray}&&\begin{array}{l}{v}_{t}+{\tilde{\sigma }}_{1}{{vv}}_{x}+{\tilde{\sigma }}_{2}{v}_{{xx}}+{\tilde{\sigma }}_{3}{v}_{{xxx}}+{\tilde{\sigma }}_{4}(t){v}_{x}\\ \quad +\,{\tilde{\sigma }}_{5}v+{\tilde{\sigma }}_{6}(t)=0,\end{array}\end{eqnarray} \tag{ 2 }$
where v(x, t) represents the dynamical radial displacement superimposed on the original static deformation from an arterial wall, t and x are the stretched coordinates which are, respectively, related to the axial coordinate and to both the axial coordinate and time parameter, the constants ${\tilde{\sigma }}_{1}$ , ${\tilde{\sigma }}_{2}$ , ${\tilde{\sigma }}_{3}$ and ${\tilde{\sigma }}_{5}$ as well as the functions ${\tilde{\sigma }}_{4}(t)$ and ${\tilde{\sigma }}_{6}(t)$ are connected with the axial stretch of the injured artery, blood as an incompressible Newtonian fluid, radius variation along the axial direction or dilatation (aneurysmal) geometry, viscosity of the fluid, thickness of the artery, mass density of the membrane material, mass density of the fluid, strain energy density of the artery, shear modulus, stretch ratio, and so on [15, 16];
when ρ₂(t) = 0, ${\rho }_{4}(x,t)={\hat{\eta }}_{4}(t)$ and ${\rho }_{6}(x,t)={\hat{\eta }}_{6}(t)$ , a variable-coefficient generalized Korteweg–de Vries model with dissipative, perturbed and external-force terms for the pulse waves in a blood vessel or dynamics in a circulatory system [17] (and references therein)
$\begin{eqnarray}&&\begin{array}{l}{v}_{t}+{\rho }_{1}(t){{vv}}_{x}+{\rho }_{3}(t){v}_{{xxx}}+{\hat{\eta }}_{4}(t){v}_{x}\\ \quad +\,{\rho }_{5}(t)v+{\hat{\eta }}_{6}(t)=0,\end{array}\end{eqnarray} \tag{ 3 }$
where v(x, t) is the wave amplitude, t and x are the scaled 'time' and scaled 'space', ρ₁(t), ρ₃(t), ρ₅(t) as well as the real functions ${\hat{\eta }}_{4}(t)$ and ${\hat{\eta }}_{6}(t)$ represent the variable coefficients of the nonlinear, dispersive, perturbed, dissipative and external-force terms, respectively [17];⁴
when ρ₂(t) = 0, a variable-coefficient Korteweg–de Vries equation for the pulse waves in a blood vessel, a circulatory system or a fluid-filled tube [18–23]
$\begin{eqnarray}&&\begin{array}{l}{v}_{t}+{\rho }_{1}(t){{vv}}_{x}+{\rho }_{3}(t){v}_{{xxx}}+{\left[{\rho }_{4}(x,t)v\right]}_{x}\\ \quad +\,{\rho }_{5}(t)v+{\rho }_{6}(x,t)=0,\end{array}\end{eqnarray} \tag{ 4 }$
where v(x, t) is the wave amplitude, t and x are the scaled time coordinate and scaled space coordinate, ρ₁(t), ρ₃(t), ρ₅(t), ρ₄(x, t) and ρ₆(x, t) represent the variable coefficients of the nonlinear, dispersive, perturbed, dissipative and external-force terms, respectively [18, 19].⁵

By the bye, more nonlinear evolution equations might be found, for instance, in [24–28].

1.3. Our work and its difference from the existing literature

However, to our knowledge, for equation (1), there has been no Bäcklund-transformation work with solitons reported as yet. No experimental comparison, either.

Our objective: In this paper, for equation (1), linking ρ_i's, we will make use of symbolic computation [29–32] to erect a Bäcklund transformation, address some solitons and present the relevant experimental support. In addition, we will employ symbolic computation to construct a family of the similarity reductions.

2. Auto-Bäcklund transformation for equation (1)

From the view of a generalized Laurent series,⁶ we introduce a Painlevé expansion,

$\begin{eqnarray}&&v(x,t)={\phi }^{-J}(x,t)\displaystyle \sum _{j=0}^{\infty }{v}_{j}(x,t){\phi }^{j}(x,t),\end{eqnarray} \tag{ 5 }$

to equation (1), around a noncharacteristic movable singular manifold given by an analytic function ϕ = 0, where J represents a positive integer, while v_j's are all the analytic functions with v₀ ≠ 0 and ϕ_x ≠ 0. Equilibrating the powers of ϕ at the lowest orders in equation (1) leads to J = 2, and cutting expansion (5) at the constant-level terms yields

$\begin{eqnarray}&&v(x,t)=\displaystyle \frac{{v}_{0}(x,t)}{\phi {\left(x,t\right)}^{2}}+\displaystyle \frac{{v}_{1}(x,t)}{\phi (x,t)}+{v}_{2}(x,t).\end{eqnarray} \tag{ 6 }$

With symbolic computation, we next introduce expression (6) to equation (1), make the coefficients of like powers of ϕ disappear and present the Painlevé-Bäcklund equations:

$\begin{eqnarray}&&{v}_{0}=-\displaystyle \frac{12{\rho }_{3}(t){\phi }_{x}^{2}}{{\rho }_{1}(t)},\end{eqnarray} \tag{ 7 }$

$\begin{eqnarray}&&{v}_{1}=\displaystyle \frac{12{\rho }_{2}(t){\phi }_{x}+60{\rho }_{3}(t){\phi }_{{xx}}}{5{\rho }_{1}(t)},\end{eqnarray} \tag{ 8 }$

$\begin{eqnarray}&&\begin{array}{l}25{\rho }_{1}(t){\rho }_{3}(t){\phi }_{x}^{2}{v}_{2}+30{\rho }_{2}(t){\rho }_{3}(t){\phi }_{x}{\phi }_{{xx}}-{\rho }_{2}{\left(t\right)}^{2}{\phi }_{x}^{2}\,\\ \,+25{\rho }_{3}(t){\phi }_{x}^{2}{\rho }_{4}(x,t)-75{\rho }_{3}{\left(t\right)}^{2}{\phi }_{{xx}}^{2}\\ \,+100{\rho }_{3}{\left(t\right)}^{2}{\phi }_{x}{\phi }_{{xxx}}+25{\rho }_{3}(t){\phi }_{t}{\phi }_{x}=0,\end{array}\end{eqnarray} \tag{ 9 }$

$\begin{eqnarray}&&\begin{array}{l}25{\rho }_{3}(t){\dot{\rho }}_{1}(t){\phi }_{x}^{2}-30{\rho }_{1}(t){\rho }_{2}(t){\rho }_{3}(t){\phi }_{{xx}}^{2}\\ \quad -\,60{\rho }_{1}(t){\rho }_{2}(t){\rho }_{3}(t){\phi }_{x}{\phi }_{{xxx}}-5{\rho }_{1}(t){\rho }_{2}(t){\phi }_{x}^{2}{\rho }_{4}(x,t)\\ \quad -\,5{\rho }_{1}{\left(t\right)}^{2}{\rho }_{2}(t){\phi }_{x}^{2}{v}_{2}-5{\rho }_{1}(t){\rho }_{2}(t){\phi }_{t}{\phi }_{x}\\ \quad -\,\,3{\rho }_{1}(t){\rho }_{2}{\left(t\right)}^{2}{\phi }_{x}{\phi }_{{xx}}-25{\rho }_{1}(t){\dot{\rho }}_{3}(t){\phi }_{x}^{2}\\ \quad -\,25{\rho }_{1}(t){\rho }_{3}(t){\phi }_{x}^{2}{\rho }_{4,x}(x,t)-75{\rho }_{1}(t){\rho }_{3}(t){\phi }_{x}{\phi }_{{xx}}{\rho }_{4}(x,t)\\ \quad -\,25{\rho }_{1}(t){\rho }_{3}(t){\rho }_{5}(t){\phi }_{x}^{2}-25{\rho }_{1}{\left(t\right)}^{2}{\rho }_{3}(t){\phi }_{x}^{2}{v}_{2,x}\\ \quad -\,75{\rho }_{1}{\left(t\right)}^{2}{\rho }_{3}(t){\phi }_{x}{\phi }_{{xx}}{v}_{2}-50{\rho }_{1}(t){\rho }_{3}(t){\phi }_{x}{\phi }_{{xt}}\\ \quad -\,25{\rho }_{1}(t){\rho }_{3}(t){\phi }_{t}{\phi }_{{xx}}+50{\rho }_{1}(t){\rho }_{3}{\left(t\right)}^{2}{\phi }_{{xx}}{\phi }_{{xxx}}\\ \quad -\,125{\rho }_{1}(t){\rho }_{3}{\left(t\right)}^{2}{\phi }_{x}{\phi }_{{xxxx}}=0,\end{array}\end{eqnarray} \tag{ 10 }$

$\begin{eqnarray}&&\begin{array}{l}{\rho }_{2}(t){\dot{\rho }}_{1}(t){\phi }_{x}+5{\rho }_{3}(t){\dot{\rho }}_{1}(t){\phi }_{{xx}}-{\rho }_{1}(t){\dot{\rho }}_{2}(t){\phi }_{x}\\ \quad -\,6{\rho }_{1}(t){\rho }_{2}(t){\rho }_{3}(t){\phi }_{{xxxx}}-{\rho }_{1}(t){\rho }_{2}(t){\phi }_{x}{\rho }_{4,x}(x,t)\\ \quad -\,{\rho }_{1}(t){\rho }_{2}(t){\phi }_{{xx}}{\rho }_{4}(x,t)-{\rho }_{1}(t){\rho }_{2}(t){\rho }_{5}(t){\phi }_{x}\\ \quad -\,{\rho }_{1}{\left(t\right)}^{2}{\rho }_{2}(t){\phi }_{x}{v}_{2,x}-{\rho }_{1}{\left(t\right)}^{2}{\rho }_{2}(t){\phi }_{{xx}}{v}_{2}\\ \quad -\,{\rho }_{1}(t){\rho }_{2}(t){\phi }_{{xt}}-{\rho }_{1}(t){\rho }_{2}{\left(t\right)}^{2}{\phi }_{{xxx}}\\ \quad -\,5{\rho }_{1}(t){\dot{\rho }}_{3}(t){\phi }_{{xx}}-5{\rho }_{1}(t){\rho }_{3}(t){\phi }_{{xx}}{\rho }_{4,x}(x,t)\\ \quad -\,5{\rho }_{1}(t){\rho }_{3}(t){\phi }_{{xxx}}{\rho }_{4}(x,t)-5{\rho }_{1}(t){\rho }_{3}(t){\rho }_{5}(t){\phi }_{{xx}}\\ \quad -\,5{\rho }_{1}{\left(t\right)}^{2}{\rho }_{3}(t){\phi }_{{xx}}{v}_{2,x}-5{\rho }_{1}{\left(t\right)}^{2}{\rho }_{3}(t){\phi }_{{xxx}}{v}_{2}\\ \quad -\,5{\rho }_{1}(t){\rho }_{3}(t){\phi }_{{xxt}}-5{\rho }_{1}(t){\rho }_{3}{\left(t\right)}^{2}{\phi }_{{xxxxx}}=0,\end{array}\end{eqnarray} \tag{ 11 }$

$\begin{eqnarray}&&\begin{array}{l}{v}_{2,t}+{\rho }_{1}(t){v}_{2}{v}_{2,x}+{\rho }_{2}(t){v}_{2,{xx}}+{\rho }_{3}(t){v}_{2,{xxx}}\,\\ \,+\,{\left[{\rho }_{4}(x,t){v}_{2}\right]}_{x}+{\rho }_{5}(t){v}_{2}+{\rho }_{6}(x,t)=0,\end{array}\,\end{eqnarray} \tag{ 12 }$

with the ' $\dot{}$ ' sign hereby denoting the derivation of t and v₂ meaning a seed solution for equation (1) [34, 35].

For an artery full of blood with an aneurysm, e.g., v(x, t) can be the dynamical radial displacement superimposed on the original static deformation from an arterial wall, and equations (7)–(12) with expression (6) formulate an auto-Bäcklund transformation, on account of the mutual consistency within equations (6)–(12), or, the explicit solvability within equations (6)–(12) with regard to ϕ, v₀ and v₁, to be seen below. By the bye, more Bäcklund transformations could be found in [37–40].

3. Three solitonic families for equation (1), their difference and experimental support^⁷

3.1. Three explicitly-solvable solitonic families for equation (1)

We now choose that

$\begin{eqnarray}&&\begin{array}{l}\phi (x,t)={{\rm{e}}}^{\,{\beta }_{1}x+{\beta }_{2}(t)}+1\qquad \mathrm{and}\\ {v}_{2}(x,t)={\beta }_{3}(t)+{\beta }_{4}x,\end{array}\end{eqnarray} \tag{ 13 }$

where β₁ and β₄ are the real constants while β₂(t) and β₃(t) are the real differentiable functions with β₁ ≠ 0 since ϕ_x ≠ 0.

Symbolic computation on auto-Bäcklund transformation(6)–(12) along with expressions (13) results in three explicitly-solvable solitonic families for equation (1):

$\begin{eqnarray}\begin{array}{rcl}\bullet \,\,{v}^{(I)}(x,t) & = & -\displaystyle \frac{3{\beta }_{1}^{2}{\rho }_{3}(t)}{{\rho }_{1}(t)}\,{\tanh }^{2}\,\left[\displaystyle \frac{{\beta }_{1}x+{\beta }_{2}(t)}{2}\right]\\ & & +\displaystyle \frac{6{\beta }_{1}{\rho }_{2}(t)}{5{\rho }_{1}(t)}\,\tanh \,\left[\displaystyle \frac{{\beta }_{1}x+{\beta }_{2}(t)}{2}\right]\\ & & +{\beta }_{4}x+{\beta }_{3}(t)\\ & & +\displaystyle \frac{3{\beta }_{1}\,\left[5{\beta }_{1}{\rho }_{3}(t)+2{\rho }_{2}(t)\right]}{5{\rho }_{1}(t)},\end{array}\end{eqnarray} \tag{ 14 }$

under the variable-coefficient constraints

$\begin{eqnarray}\begin{array}{rcl}{\rho }_{4}(x,t) & = & -{\beta }_{4}{\rho }_{1}(t)x-{\beta }_{1}^{2}{\rho }_{3}(t)-\displaystyle \frac{{\dot{\beta }}_{2}(t)}{{\beta }_{1}}\,\\ & & -\displaystyle \frac{6}{5}{\beta }_{1}{\rho }_{2}(t)-{\beta }_{3}(t){\rho }_{1}(t)+\displaystyle \frac{{\rho }_{2}{\left(t\right)}^{2}}{25{\rho }_{3}(t)},\end{array}\end{eqnarray} \tag{ 15 }$

$\begin{eqnarray}&&{\rho }_{5}(t)=\displaystyle \frac{1}{5}{\beta }_{1}^{2}{\rho }_{2}(t)+\displaystyle \frac{{\dot{\rho }}_{1}(t)}{{\rho }_{1}(t)}-\displaystyle \frac{{\rho }_{2}{\left(t\right)}^{3}}{125{\rho }_{3}{\left(t\right)}^{2}}-\displaystyle \frac{{\dot{\rho }}_{3}(t)}{{\rho }_{3}(t)},\,\end{eqnarray} \tag{ 16 }$

$\begin{eqnarray}\begin{array}{rcl}{\rho }_{6}(x,t) & = & {\beta }_{4}^{2}{\rho }_{1}(t)x-\displaystyle \frac{{\beta }_{4}{\dot{\rho }}_{1}(t)x}{{\rho }_{1}(t)}+\displaystyle \frac{{\beta }_{4}{\rho }_{2}{\left(t\right)}^{3}x}{125{\rho }_{3}{\left(t\right)}^{2}}\\ & & +\displaystyle \frac{{\beta }_{4}{\dot{\rho }}_{3}(t)x}{{\rho }_{3}(t)}-\displaystyle \frac{1}{5}{\beta }_{1}^{2}{\beta }_{4}{\rho }_{2}(t)x\\ & & -\displaystyle \frac{{\beta }_{4}{\rho }_{2}{\left(t\right)}^{2}}{25{\rho }_{3}(t)}-\displaystyle \frac{1}{5}{\beta }_{1}^{2}{\beta }_{3}(t){\rho }_{2}(t)+{\beta }_{1}^{2}{\beta }_{4}{\rho }_{3}(t)\\ & & +\displaystyle \frac{{\beta }_{4}{\dot{\beta }}_{2}(t)}{{\beta }_{1}}+\displaystyle \frac{6}{5}{\beta }_{1}{\beta }_{4}{\rho }_{2}(t)\\ & & -{\dot{\beta }}_{3}(t)+{\beta }_{4}{\beta }_{3}(t){\rho }_{1}(t)-\displaystyle \frac{{\beta }_{3}(t){\dot{\rho }}_{1}(t)}{{\rho }_{1}(t)}\\ & & +\displaystyle \frac{{\beta }_{3}(t){\rho }_{2}{\left(t\right)}^{3}}{125{\rho }_{3}{\left(t\right)}^{2}}+\displaystyle \frac{{\beta }_{3}(t){\dot{\rho }}_{3}(t)}{{\rho }_{3}(t)},\end{array}\end{eqnarray} \tag{ 17 }$

$\begin{eqnarray}&&\begin{array}{l}25{\beta }_{1}^{2}{\rho }_{2}{\left(t\right)}^{2}{\rho }_{3}{\left(t\right)}^{2}+125{\rho }_{3}{\left(t\right)}^{2}\dot{{\rho }_{2}}(t)\,\\ \,-\,125{\rho }_{2}(t){\rho }_{3}(t)\dot{{\rho }_{3}}(t)-{\rho }_{2}{\left(t\right)}^{4}=0;\end{array}\end{eqnarray} \tag{ 18 }$

$\begin{eqnarray}\begin{array}{rcl}\bullet \,\,{v}^{({II})}(x,t) & = & -\displaystyle \frac{3{\beta }_{1}^{2}{\rho }_{3}(t)}{{\rho }_{1}(t)}{\tanh }^{2}\left(\displaystyle \frac{{\beta }_{1}x+{\beta }_{5}t+{\beta }_{6}}{2}\right)\\ & & +\displaystyle \frac{6{\beta }_{1}{\rho }_{2}(t)}{5{\rho }_{1}(t)}\tanh \left(\displaystyle \frac{{\beta }_{1}x+{\beta }_{5}t+{\beta }_{6}}{2}\right)+{\beta }_{7}\\ & & +\displaystyle \frac{3{\beta }_{1}\,\left[5{\beta }_{1}{\rho }_{3}(t)+2{\rho }_{2}(t)\right]}{5{\rho }_{1}(t)},\end{array}\end{eqnarray} \tag{ 19 }$

which can be looked on as a quasi-solitary-wave⁸ case of solutions (14), if β₂(t) = β₅ t + β₆, β₃(t) = β₇, β₄ = 0, and variable-coefficient constraints (15)–(18) are reduced to

$\begin{eqnarray}&&\begin{array}{l}{\rho }_{4}(x,t)=-{\beta }_{1}^{2}{\rho }_{3}(t)-\displaystyle \frac{{\beta }_{5}}{{\beta }_{1}}-\displaystyle \frac{6}{5}{\beta }_{1}{\rho }_{2}(t)\\ \,-{\beta }_{7}{\rho }_{1}(t)+\displaystyle \frac{{\rho }_{2}{\left(t\right)}^{2}}{25{\rho }_{3}(t)},\end{array}\,\end{eqnarray} \tag{ 20 }$

$\begin{eqnarray}&&\,\begin{array}{l}{\rho }_{5}(t)=\displaystyle \frac{1}{5}{\beta }_{1}^{2}{\rho }_{2}(t)+\displaystyle \frac{{\dot{\rho }}_{1}(t)}{{\rho }_{1}(t)}-\displaystyle \frac{{\rho }_{2}{\left(t\right)}^{3}}{125{\rho }_{3}{\left(t\right)}^{2}}\\ \,-\displaystyle \frac{{\dot{\rho }}_{3}(t)}{{\rho }_{3}(t)},\end{array}\,\end{eqnarray} \tag{ 21 }$

$\begin{eqnarray}&&\begin{array}{r}{\rho }_{6}(x,t)=-\displaystyle \frac{1}{5}{\beta }_{1}^{2}{\beta }_{7}{\rho }_{2}(t)-\displaystyle \frac{{\beta }_{7}{\dot{\rho }}_{1}(t)}{{\rho }_{1}(t)}\,\\ \,+\displaystyle \frac{{\beta }_{7}{\rho }_{2}{\left(t\right)}^{3}}{125{\rho }_{3}{\left(t\right)}^{2}}+\displaystyle \frac{{\beta }_{7}{\dot{\rho }}_{3}(t)}{{\rho }_{3}(t)},\end{array}\,\end{eqnarray} \tag{ 22 }$

$\begin{eqnarray}&&\,\begin{array}{l}25{\beta }_{1}^{2}{\rho }_{2}{\left(t\right)}^{2}{\rho }_{3}{\left(t\right)}^{2}+125{\rho }_{3}{\left(t\right)}^{2}{\dot{\rho }}_{2}(t)\\ \,-\,125{\rho }_{2}(t){\rho }_{3}(t){\dot{\rho }}_{3}(t)\\ \,-{\rho }_{2}{\left(t\right)}^{4}=0,\end{array}\,\end{eqnarray} \tag{ 23 }$

where β₅, β₆ and β₇ are the real constants;

$\begin{eqnarray}\begin{array}{rcl}\bullet \,\,{v}^{({III})}(x,t) & = & -\displaystyle \frac{3{\gamma }_{0}^{2}}{25{\alpha }_{0}{\beta }_{0}}\,{\tanh }^{2}\,\left[\displaystyle \frac{{\gamma }_{0}x}{10{\beta }_{0}}\right.\\ & & \left.-\displaystyle \frac{{\gamma }_{0}\left(25{\alpha }_{0}{\beta }_{0}{\sigma }_{4}+6{\gamma }_{0}^{2}\right)t}{250{\beta }_{0}^{2}}+\displaystyle \frac{{\sigma }_{3}}{2}\right]\\ & & +\displaystyle \frac{6{\gamma }_{0}^{2}}{25{\alpha }_{0}{\beta }_{0}}\,\tanh \,\left[\displaystyle \frac{{\gamma }_{0}x}{10{\beta }_{0}}\right.\\ & & \left.-\displaystyle \frac{{\gamma }_{0}\left(25{\alpha }_{0}{\beta }_{0}{\sigma }_{4}+6{\gamma }_{0}^{2}\right)t}{250{\beta }_{0}^{2}}+\displaystyle \frac{{\sigma }_{3}}{2}\right]\\ & & +\displaystyle \frac{9{\gamma }_{0}^{2}}{25{\alpha }_{0}{\beta }_{0}}+{\sigma }_{4},\end{array}\end{eqnarray} \tag{ 24 }$

which can be regarded as a constant-coefficient-shock-wave case of solutions (19), if ρ₁(t) = α₀, ρ₂(t) = γ₀, ρ₃(t) = β₀, ${\beta }_{1}=\tfrac{{\gamma }_{0}}{5{\beta }_{0}}$ , ${\beta }_{7}=-\tfrac{5{\beta }_{5}{\beta }_{0}}{{\alpha }_{0}{\gamma }_{0}}-\tfrac{6{\gamma }_{0}^{2}}{25{\alpha }_{0}{\beta }_{0}}$ , ${\beta }_{5}=-\tfrac{{\gamma }_{0}\left(25{\alpha }_{0}{\beta }_{0}{\sigma }_{4}+6{\gamma }_{0}^{2}\right)}{125{\beta }_{0}^{2}}$ and β₆ = σ₃ [so that we also end up with ρ₄(x, t) = ρ₅(t) = ρ₆(x, t) = 0], where σ₃, σ₄, α₀, γ₀ and β₀ are the real constants.

Those solutions indicate that, e.g., for an artery full of blood with an aneurysm, v(x, t) is the solitonic radial displacement superimposed on the original static deformation from an arterial wall.

3.2. Difference among those solitonic families

The difference among solitonic solutions (14), (19) and (24) is ascribable to the respective variable-coefficient constraints among ρ₁(t), ρ₂(t), ρ₃(t), ρ₄(x, t), ρ₅(t) and ρ₆(x, t), while ρ_i's are linked with the axial stretch of the injured artery, blood as an incompressible Newtonian fluid, radius variation in the axial direction or aneurysmal geometry, viscosity of the fluid, thickness of the artery, mass density of the membrane material, mass density of the fluid, strain energy density of the artery, shear modulus, stretch ratio, etc.

3.3. Experimental support

Especially, we call the attention that the shock-wave structures from solutions (24) have been shown to agree well with those dusty-plasma-experimentally reported, as detailed in [41, 42] and references among the rest. Graphs describing the dynamical behaviors of solutions (24), versus those experimental graphs, have been worked out and presented in [41, 42].

We need to say that such a dusty-plasma-experimental agreement directly supports the correctness/validation of auto-Bäcklund transformation (6)–(12) and solitonic solutions (24), which in fact supports the correctness/validation of our above analytic work towards the blood vessel or circulatory system.

By the way, other relevant solitonic issues might be found in [43–53].

4. Similarity reductions for equation (1)

Our assumption

$\begin{eqnarray}&&v(x,t)=\theta (x,t)+\kappa (x,t)q[r(x,t)],\end{eqnarray} \tag{ 25 }$

which is similar to those in [54–59], can lead to certain similarity reductions for equation (1), with θ(x, t), κ(x, t) ≠ 0 and r(x, t) ≠ 0 as the real differentiable functions to be determined.

Making use of symbolic computation and inserting assumption (25) into equation (1) turn into

$\begin{eqnarray}&&\begin{array}{l}{\rho }_{3}(t)\kappa {r}_{x}^{3}q\prime\prime\prime +{\rho }_{1}(t){\kappa }^{2}{r}_{x}{qq}^{\prime} +{\rho }_{1}(t)\kappa {\kappa }_{x}{q}^{2}\\ \quad +\,{r}_{x}\left[{\rho }_{2}(t)\kappa {r}_{x}+3{\rho }_{3}(t)\left({\kappa }_{x}{r}_{x}+\kappa {r}_{{xx}}\right)\right]q^{\prime\prime} \\ \quad +\,\left[\kappa {r}_{t}+{\rho }_{1}(t)\kappa \theta {r}_{x}+{\rho }_{2}(t)\left(2{\kappa }_{x}{r}_{x}+\kappa {r}_{{xx}}\right)\right.\\ \quad \left.+\,\,{\rho }_{3}(t)\left(3{\kappa }_{{xx}}{r}_{x}+3{\kappa }_{x}{r}_{{xx}}+\kappa {r}_{{xxx}}\right)+{\rho }_{4}(x,t)\kappa {r}_{x}\right]q^{\prime} \\ \quad +\,\left\{{\kappa }_{t}+{\rho }_{1}(t)\left({\kappa }_{x}\theta +\kappa {\theta }_{x}\right)+{\rho }_{2}(t){\kappa }_{{xx}}\right.\\ \quad \left.+\,{\rho }_{3}(t){\kappa }_{{xxx}}+{\left[{\rho }_{4}(x,t)\kappa \right]}_{x}+{\rho }_{5}(t)\kappa \right\}q\\ \quad +\left\{{\theta }_{t}+{\rho }_{1}(t)\theta {\theta }_{x}+{\rho }_{2}(t){\theta }_{{xx}}+{\rho }_{3}(t){\theta }_{{xxx}}\right.\\ \quad \left.+\,{\left[{\rho }_{4}(x,t)\theta \right]}_{x}+{\rho }_{5}(t)\theta +{\rho }_{6}(x,t)\right\}=0,\end{array}\end{eqnarray} \tag{ 26 }$

in which the ''' sign hereby stands for the derivation with respect to r.

Taking into consideration that equation (26) can be designed to reduce to a single ordinary differential equation (ODE) as for q(r), one requires those ratios of derivatives and/or powers of q(r) to mean some functions with respect to r only, so that

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{1}(r){\rho }_{3}(t)\kappa {r}_{x}^{3}={\rho }_{1}(t){\kappa }^{2}{r}_{x},\,\end{eqnarray} \tag{ 27a }$

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{2}(r){\rho }_{3}(t)\kappa {r}_{x}^{3}={\rho }_{1}(t)\kappa {\kappa }_{x},\,\end{eqnarray} \tag{ 27b }$

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{3}(r){\rho }_{3}(t)\kappa {r}_{x}^{3}={r}_{x}\left[{\rho }_{2}(t)\kappa {r}_{x}+3{\rho }_{3}(t)\left({\kappa }_{x}{r}_{x}+\kappa {r}_{{xx}}\right)\right],\end{eqnarray} \tag{ 27c }$

$\begin{eqnarray}\begin{array}{lcl}\,{{\rm{\Gamma }}}_{4}(r){\rho }_{3}(t)\kappa {r}_{x}^{3} & = & \kappa {r}_{t}+{\rho }_{1}(t)\kappa \theta {r}_{x}+{\rho }_{2}(t)\left(2{\kappa }_{x}{r}_{x}+\kappa {r}_{{xx}}\right)\ \\ & & +{\rho }_{3}(t)\left(3{\kappa }_{{xx}}{r}_{x}+3{\kappa }_{x}{r}_{{xx}}+\kappa {r}_{{xxx}}\right)\\ & & +{\rho }_{4}(x,t)\kappa {r}_{x},\end{array}\,\end{eqnarray} \tag{ 27d }$

$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{5}(r){\rho }_{3}(t)\kappa {r}_{x}^{3} & = & {\kappa }_{t}+{\rho }_{1}(t)\left({\kappa }_{x}\theta +\kappa {\theta }_{x}\right)\\ & & +{\rho }_{2}(t){\kappa }_{{xx}}+{\rho }_{3}(t){\kappa }_{{xxx}}+{\left[{\rho }_{4}(x,t)\kappa \right]}_{x}\\ & & +{\rho }_{5}(t)\kappa ,\end{array}\end{eqnarray} \tag{ 27e }$

$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{6}(r){\rho }_{3}(t)\kappa {r}_{x}^{3} & = & {\theta }_{t}+{\rho }_{1}(t)\theta {\theta }_{x}+{\rho }_{2}(t){\theta }_{{xx}}\,\\ & & +{\rho }_{3}(t){\theta }_{{xxx}}+{\left[{\rho }_{4}(x,t)\theta \right]}_{x}\\ & & +{\rho }_{5}(t)\theta +{\rho }_{6}(x,t),\end{array}\end{eqnarray} \tag{ 27f }$

with Γ_χ(r)'s (χ = 1, ..., 6) as merely the real functions of r, of course to be determined.

Grounded on the 2nd freedom of remark 3 in [59], equation (27a ) gives rise to

$\begin{eqnarray}&&\kappa (x,t)=\displaystyle \frac{{\rho }_{3}(t)}{{\rho }_{1}(t)}{r}_{x}^{2},\qquad {{\rm{\Gamma }}}_{1}(r)=1.\end{eqnarray} \tag{ 28 }$

Because of the 1st freedom of remark 3 in [59], equation (27b ) brings about

$\begin{eqnarray}&&r(x,t)={\lambda }_{1}(t)x+{\lambda }_{2}(t),\qquad {{\rm{\Gamma }}}_{2}(r)=0,\end{eqnarray} \tag{ 29 }$

and then equation (27c ) results in

$\begin{eqnarray}&&{\lambda }_{1}(t)={\xi }_{1}\displaystyle \frac{{\rho }_{2}(t)}{{\rho }_{3}(t)},\qquad {{\rm{\Gamma }}}_{3}(r)=\displaystyle \frac{1}{{\xi }_{1}},\end{eqnarray} \tag{ 30 }$

with ξ₁ indicating a real non-zero constant, while λ₁(t) and λ₂(t) implying two real non-zero differentiable functions with respect to t.

Because the 1st freedom of remark 3 in [59] helps us reduce equation (27d ) to

$\begin{eqnarray}&&\begin{array}{c}{\rho }_{2}(t)={\mu }_{1}{\rho }_{3}(t),\,{\rho }_{4}(x,t)={\rho }_{4}(t)\,{\rm{only}},\\ \theta (x,t)=-\displaystyle \frac{1}{{{\mu }_{1}}^{2}{{\xi }_{1}}^{2}{\rho }_{1}\left(t\right)}\left[[{\mu }_{1}{\xi }_{1}{\lambda }_{2}^{{\rm{{\prime} }}}\left(t\right)+{\mu }_{1}^{2}{\xi }_{1}^{2}{\rho }_{4}\left(t\right)\right],\,{{\rm{\Gamma }}}_{4}(r)=0,\end{array}\end{eqnarray} \tag{ 31 }$

equation (27e ) turns to

$\begin{eqnarray}&&{\rho }_{1}(t)={\mu }_{2}{\rho }_{3}(t),\,\,{\rho }_{5}(t)={\mu }_{3}{\rho }_{3}(t),\,\,{{\rm{\Gamma }}}_{5}(r)=\displaystyle \frac{{\mu }_{3}}{{\mu }_{1}^{3}{\xi }_{1}^{3}},\end{eqnarray} \tag{ 32 }$

and then equation (27f ) develops into

$\begin{eqnarray}&&\begin{array}{l}{\rho }_{6}(x,t)={\mu }_{4}\left[x-\displaystyle \int {\rho }_{4}(t){\rm{d}}t\right]{\rho }_{3}(t),\\ {\lambda }_{2}(t)=-{\mu }_{1}{\xi }_{1}\displaystyle \int {\rho }_{4}(t){\rm{d}}t,\\ {{\rm{\Gamma }}}_{6}(r)=\displaystyle \frac{{\mu }_{2}{\mu }_{4}}{{\mu }_{1}^{6}{\xi }_{1}^{6}}r,\end{array}\end{eqnarray} \tag{ 33 }$

with μ₁ and μ₂ as two real non-zero constants, while μ₃ and μ₄ as two real constants.

For an artery full of blood with an aneurysm, e.g., v(x, t) can be the dynamical radial displacement superimposed on the original static deformation from an arterial wall, and in general, under the variable-coefficient constraints

$\begin{eqnarray}&&\begin{array}{l}{\rho }_{1}(t)={\mu }_{2}{\rho }_{3}(t),\,\,{\rho }_{2}(t)={\mu }_{1}{\rho }_{3}(t),\\ {\rho }_{4}(x,t)={\rho }_{4}(t)\,\,\mathrm{only},\,\,\,{\rho }_{5}(t)={\mu }_{3}{\rho }_{3}(t),\\ {\rho }_{6}(x,t)={\mu }_{4}\left[x-\displaystyle \int {\rho }_{4}(t){\rm{d}}t\right]{\rho }_{3}(t),\end{array}\end{eqnarray} \tag{ 34 }$

we build up the following family of the similarity reductions for equation (1):

$\begin{eqnarray}&&v(x,t)=\displaystyle \frac{{\mu }_{1}^{2}{\xi }_{1}^{2}}{{\mu }_{2}}q[r(x,t)],\,\end{eqnarray} \tag{ 35a }$

$\begin{eqnarray}&&r(x,t)={\mu }_{1}{\xi }_{1}\left[x-\int {\rho }_{4}(t){\rm{d}}t\right],\,\end{eqnarray} \tag{ 35b }$

$\begin{eqnarray}&&q\prime\prime\prime +{qq}^{\prime} +\displaystyle \frac{1}{{\xi }_{1}}q^{\prime\prime} +\displaystyle \frac{{\mu }_{3}}{{\mu }_{1}^{3}{\xi }_{1}^{3}}q+\displaystyle \frac{{\mu }_{2}{\mu }_{4}}{{\mu }_{1}^{6}{\xi }_{1}^{6}}r=0.\,\end{eqnarray} \tag{ 35c }$

ODE (35c ) has been presented in [60] and thus can be considered as a known ODE.

5. Conclusions

With respect to the nonlinear waves in an artery full of blood with a certain aneurysm, pulses in a blood vessel, or features in a circulatory system, this paper has symbolically computed out auto-Bäcklund transformation (6)–(12) and solitonic solutions (14), (19) and (24) for equation (1), i.e., a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers equation. We have also built up similarity reductions (35), from equation (1) to a known ODE. Aiming, e.g., at the dynamical radial displacement superimposed on the original static deformation from an arterial wall, our results rely on the axial stretch of the injured artery, blood as an incompressible Newtonian fluid, radius variation along the axial direction or aneurysmal geometry, viscosity of the fluid, thickness of the artery, mass density of the membrane material, mass density of the fluid, strain energy density of the artery, shear modulus, stretch ratio, etc. Relevant variable-coefficient constraints have also been given. Finally, we have highlighted that the shock-wave structures from our solutions agree well with those dusty-plasma-experimentally reported.

Acknowledgments

We express our sincere thanks to the Editors and Reviewers for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11871116 and 11772017, and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2019XD-A11.

Theoretical investigations on a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers model for a dilated artery, blood vessel or circulatory system with experimental support

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Abstract

1. Introduction

1.1. Background around us and our consideration

1.2. Special cases of equation (1)

1.3. Our work and its difference from the existing literature

2. Auto-Bäcklund transformation for equation (1)

3. Three solitonic families for equation (1), their difference and experimental support^⁷

3.1. Three explicitly-solvable solitonic families for equation (1)

3.2. Difference among those solitonic families

3.3. Experimental support

4. Similarity reductions for equation (1)

5. Conclusions

Acknowledgments

Footnotes

Theoretical investigations on a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers model for a dilated artery, blood vessel or circulatory system with experimental support

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Submit

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Share this article

Author e-mails

Author affiliations

Author notes

Dates

Abstract

1. Introduction

1.1. Background around us and our consideration

1.2. Special cases of equation (1)

1.3. Our work and its difference from the existing literature

2. Auto-Bäcklund transformation for equation (1)

3. Three solitonic families for equation (1), their difference and experimental support 7

3.1. Three explicitly-solvable solitonic families for equation (1)

3.2. Difference among those solitonic families

3.3. Experimental support

4. Similarity reductions for equation (1)

5. Conclusions

Acknowledgments

Footnotes

3. Three solitonic families for equation (1), their difference and experimental support^⁷