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:
$\begin{eqnarray}\begin{array}{l}{v}_{t}+{\rho }_{1}(t){{vv}}_{x}+{\rho }_{2}(t){v}_{{xx}}+{\rho }_{3}(t){v}_{{xxx}}\\ \quad +\,{\left[{\rho }_{4}(x,t)v\right]}_{x}+{\rho }_{5}(t)v+{\rho }_{6}(x,t)=0,\end{array}\end{eqnarray}$
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, ρi'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(3 Reference [16] has discussed the situation with ${\tilde{\sigma }}_{2}=0$.) $\begin{eqnarray}\begin{array}{l}{v}_{t}+{\tilde{\sigma }}_{1}{{vv}}_{x}+{\tilde{\sigma }}_{2}{v}_{{xx}}+{\tilde{\sigma }}_{3}{v}_{{xxx}}+{\tilde{\sigma }}_{4}(t){v}_{x}\\ \quad +\,{\tilde{\sigma }}_{5}v+{\tilde{\sigma }}_{6}(t)=0,\end{array}\end{eqnarray}$ 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) $\begin{eqnarray}\begin{array}{l}{v}_{t}+{\rho }_{1}(t){{vv}}_{x}+{\rho }_{3}(t){v}_{{xxx}}+{\hat{\eta }}_{4}(t){v}_{x}\\ \quad +\,{\rho }_{5}(t)v+{\hat{\eta }}_{6}(t)=0,\end{array}\end{eqnarray}$ 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(4Reference [17] has also, with references therein, listed out other applications of equation ( |
| • | 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] $\begin{eqnarray}\begin{array}{l}{v}_{t}+{\rho }_{1}(t){{vv}}_{x}+{\rho }_{3}(t){v}_{{xxx}}+{\left[{\rho }_{4}(x,t)v\right]}_{x}\\ \quad +\,{\rho }_{5}(t)v+{\rho }_{6}(x,t)=0,\end{array}\end{eqnarray}$ 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(5References [18, 19] have also, with references therein, listed out other applications of equation ( |
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 ρi'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(6Similar to those in [33–36].) we introduce a Painlevé expansion,1 ), around a noncharacteristic movable singular manifold given by an analytic function φ = 0, where J represents a positive integer, while vj's are all the analytic functions with v0 ≠ 0 and φx ≠ 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
$\begin{eqnarray}v(x,t)={\phi }^{-J}(x,t)\displaystyle \sum _{j=0}^{\infty }{v}_{j}(x,t){\phi }^{j}(x,t),\end{eqnarray}$
to equation ( $\begin{eqnarray}v(x,t)=\displaystyle \frac{{v}_{0}(x,t)}{\phi {\left(x,t\right)}^{2}}+\displaystyle \frac{{v}_{1}(x,t)}{\phi (x,t)}+{v}_{2}(x,t).\end{eqnarray}$
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:1 ) [34, 35].
$\begin{eqnarray}{v}_{0}=-\displaystyle \frac{12{\rho }_{3}(t){\phi }_{x}^{2}}{{\rho }_{1}(t)},\end{eqnarray}$
$\begin{eqnarray}{v}_{1}=\displaystyle \frac{12{\rho }_{2}(t){\phi }_{x}+60{\rho }_{3}(t){\phi }_{{xx}}}{5{\rho }_{1}(t)},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}25{\rho }_{1}(t){\rho }_{3}(t){\phi }_{x}^{2}{v}_{2}+30{\rho }_{2}(t){\rho }_{3}(t){\phi }_{x}{\phi }_{{xx}}-{\rho }_{2}{\left(t\right)}^{2}{\phi }_{x}^{2}\,\\ \,+25{\rho }_{3}(t){\phi }_{x}^{2}{\rho }_{4}(x,t)-75{\rho }_{3}{\left(t\right)}^{2}{\phi }_{{xx}}^{2}\\ \,+100{\rho }_{3}{\left(t\right)}^{2}{\phi }_{x}{\phi }_{{xxx}}+25{\rho }_{3}(t){\phi }_{t}{\phi }_{x}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}25{\rho }_{3}(t){\dot{\rho }}_{1}(t){\phi }_{x}^{2}-30{\rho }_{1}(t){\rho }_{2}(t){\rho }_{3}(t){\phi }_{{xx}}^{2}\\ \quad -\,60{\rho }_{1}(t){\rho }_{2}(t){\rho }_{3}(t){\phi }_{x}{\phi }_{{xxx}}-5{\rho }_{1}(t){\rho }_{2}(t){\phi }_{x}^{2}{\rho }_{4}(x,t)\\ \quad -\,5{\rho }_{1}{\left(t\right)}^{2}{\rho }_{2}(t){\phi }_{x}^{2}{v}_{2}-5{\rho }_{1}(t){\rho }_{2}(t){\phi }_{t}{\phi }_{x}\\ \quad -\,\,3{\rho }_{1}(t){\rho }_{2}{\left(t\right)}^{2}{\phi }_{x}{\phi }_{{xx}}-25{\rho }_{1}(t){\dot{\rho }}_{3}(t){\phi }_{x}^{2}\\ \quad -\,25{\rho }_{1}(t){\rho }_{3}(t){\phi }_{x}^{2}{\rho }_{4,x}(x,t)-75{\rho }_{1}(t){\rho }_{3}(t){\phi }_{x}{\phi }_{{xx}}{\rho }_{4}(x,t)\\ \quad -\,25{\rho }_{1}(t){\rho }_{3}(t){\rho }_{5}(t){\phi }_{x}^{2}-25{\rho }_{1}{\left(t\right)}^{2}{\rho }_{3}(t){\phi }_{x}^{2}{v}_{2,x}\\ \quad -\,75{\rho }_{1}{\left(t\right)}^{2}{\rho }_{3}(t){\phi }_{x}{\phi }_{{xx}}{v}_{2}-50{\rho }_{1}(t){\rho }_{3}(t){\phi }_{x}{\phi }_{{xt}}\\ \quad -\,25{\rho }_{1}(t){\rho }_{3}(t){\phi }_{t}{\phi }_{{xx}}+50{\rho }_{1}(t){\rho }_{3}{\left(t\right)}^{2}{\phi }_{{xx}}{\phi }_{{xxx}}\\ \quad -\,125{\rho }_{1}(t){\rho }_{3}{\left(t\right)}^{2}{\phi }_{x}{\phi }_{{xxxx}}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\rho }_{2}(t){\dot{\rho }}_{1}(t){\phi }_{x}+5{\rho }_{3}(t){\dot{\rho }}_{1}(t){\phi }_{{xx}}-{\rho }_{1}(t){\dot{\rho }}_{2}(t){\phi }_{x}\\ \quad -\,6{\rho }_{1}(t){\rho }_{2}(t){\rho }_{3}(t){\phi }_{{xxxx}}-{\rho }_{1}(t){\rho }_{2}(t){\phi }_{x}{\rho }_{4,x}(x,t)\\ \quad -\,{\rho }_{1}(t){\rho }_{2}(t){\phi }_{{xx}}{\rho }_{4}(x,t)-{\rho }_{1}(t){\rho }_{2}(t){\rho }_{5}(t){\phi }_{x}\\ \quad -\,{\rho }_{1}{\left(t\right)}^{2}{\rho }_{2}(t){\phi }_{x}{v}_{2,x}-{\rho }_{1}{\left(t\right)}^{2}{\rho }_{2}(t){\phi }_{{xx}}{v}_{2}\\ \quad -\,{\rho }_{1}(t){\rho }_{2}(t){\phi }_{{xt}}-{\rho }_{1}(t){\rho }_{2}{\left(t\right)}^{2}{\phi }_{{xxx}}\\ \quad -\,5{\rho }_{1}(t){\dot{\rho }}_{3}(t){\phi }_{{xx}}-5{\rho }_{1}(t){\rho }_{3}(t){\phi }_{{xx}}{\rho }_{4,x}(x,t)\\ \quad -\,5{\rho }_{1}(t){\rho }_{3}(t){\phi }_{{xxx}}{\rho }_{4}(x,t)-5{\rho }_{1}(t){\rho }_{3}(t){\rho }_{5}(t){\phi }_{{xx}}\\ \quad -\,5{\rho }_{1}{\left(t\right)}^{2}{\rho }_{3}(t){\phi }_{{xx}}{v}_{2,x}-5{\rho }_{1}{\left(t\right)}^{2}{\rho }_{3}(t){\phi }_{{xxx}}{v}_{2}\\ \quad -\,5{\rho }_{1}(t){\rho }_{3}(t){\phi }_{{xxt}}-5{\rho }_{1}(t){\rho }_{3}{\left(t\right)}^{2}{\phi }_{{xxxxx}}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{v}_{2,t}+{\rho }_{1}(t){v}_{2}{v}_{2,x}+{\rho }_{2}(t){v}_{2,{xx}}+{\rho }_{3}(t){v}_{2,{xxx}}\,\\ \,+\,{\left[{\rho }_{4}(x,t){v}_{2}\right]}_{x}+{\rho }_{5}(t){v}_{2}+{\rho }_{6}(x,t)=0,\end{array}\,\end{eqnarray}$
with the ‘$\dot{}$' sign hereby denoting the derivation of t and v2 meaning a seed solution for equation (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 support7(7For simplicity, derivations are elided.)
3.1. Three explicitly-solvable solitonic families for equation (1 )
We now choose that
$\begin{eqnarray}\begin{array}{l}\phi (x,t)={{\rm{e}}}^{\,{\beta }_{1}x+{\beta }_{2}(t)}+1\qquad \mathrm{and}\\ {v}_{2}(x,t)={\beta }_{3}(t)+{\beta }_{4}x,\end{array}\end{eqnarray}$
where β1 and β4 are the real constants while β2(t) and β3(t) are the real differentiable functions with β1 ≠ 0 since φx ≠ 0.Symbolic computation on auto-Bäcklund transformation(6 )–(12 ) along with expressions (13 ) results in three explicitly-solvable solitonic families for equation (1 ):14 ), if β2(t) = β5t + β6, β3(t) = β7, β4 = 0, and variable-coefficient constraints (15 )–(18 ) are reduced to19 ), 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.
$\begin{eqnarray}\begin{array}{rcl}\bullet \,\,{v}^{(I)}(x,t) & = & -\displaystyle \frac{3{\beta }_{1}^{2}{\rho }_{3}(t)}{{\rho }_{1}(t)}\,{\tanh }^{2}\,\left[\displaystyle \frac{{\beta }_{1}x+{\beta }_{2}(t)}{2}\right]\\ & & +\displaystyle \frac{6{\beta }_{1}{\rho }_{2}(t)}{5{\rho }_{1}(t)}\,\tanh \,\left[\displaystyle \frac{{\beta }_{1}x+{\beta }_{2}(t)}{2}\right]\\ & & +{\beta }_{4}x+{\beta }_{3}(t)\\ & & +\displaystyle \frac{3{\beta }_{1}\,\left[5{\beta }_{1}{\rho }_{3}(t)+2{\rho }_{2}(t)\right]}{5{\rho }_{1}(t)},\end{array}\end{eqnarray}$
under the variable-coefficient constraints $\begin{eqnarray}\begin{array}{rcl}{\rho }_{4}(x,t) & = & -{\beta }_{4}{\rho }_{1}(t)x-{\beta }_{1}^{2}{\rho }_{3}(t)-\displaystyle \frac{{\dot{\beta }}_{2}(t)}{{\beta }_{1}}\,\\ & & -\displaystyle \frac{6}{5}{\beta }_{1}{\rho }_{2}(t)-{\beta }_{3}(t){\rho }_{1}(t)+\displaystyle \frac{{\rho }_{2}{\left(t\right)}^{2}}{25{\rho }_{3}(t)},\end{array}\end{eqnarray}$
$\begin{eqnarray}{\rho }_{5}(t)=\displaystyle \frac{1}{5}{\beta }_{1}^{2}{\rho }_{2}(t)+\displaystyle \frac{{\dot{\rho }}_{1}(t)}{{\rho }_{1}(t)}-\displaystyle \frac{{\rho }_{2}{\left(t\right)}^{3}}{125{\rho }_{3}{\left(t\right)}^{2}}-\displaystyle \frac{{\dot{\rho }}_{3}(t)}{{\rho }_{3}(t)},\,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{6}(x,t) & = & {\beta }_{4}^{2}{\rho }_{1}(t)x-\displaystyle \frac{{\beta }_{4}{\dot{\rho }}_{1}(t)x}{{\rho }_{1}(t)}+\displaystyle \frac{{\beta }_{4}{\rho }_{2}{\left(t\right)}^{3}x}{125{\rho }_{3}{\left(t\right)}^{2}}\\ & & +\displaystyle \frac{{\beta }_{4}{\dot{\rho }}_{3}(t)x}{{\rho }_{3}(t)}-\displaystyle \frac{1}{5}{\beta }_{1}^{2}{\beta }_{4}{\rho }_{2}(t)x\\ & & -\displaystyle \frac{{\beta }_{4}{\rho }_{2}{\left(t\right)}^{2}}{25{\rho }_{3}(t)}-\displaystyle \frac{1}{5}{\beta }_{1}^{2}{\beta }_{3}(t){\rho }_{2}(t)+{\beta }_{1}^{2}{\beta }_{4}{\rho }_{3}(t)\\ & & +\displaystyle \frac{{\beta }_{4}{\dot{\beta }}_{2}(t)}{{\beta }_{1}}+\displaystyle \frac{6}{5}{\beta }_{1}{\beta }_{4}{\rho }_{2}(t)\\ & & -{\dot{\beta }}_{3}(t)+{\beta }_{4}{\beta }_{3}(t){\rho }_{1}(t)-\displaystyle \frac{{\beta }_{3}(t){\dot{\rho }}_{1}(t)}{{\rho }_{1}(t)}\\ & & +\displaystyle \frac{{\beta }_{3}(t){\rho }_{2}{\left(t\right)}^{3}}{125{\rho }_{3}{\left(t\right)}^{2}}+\displaystyle \frac{{\beta }_{3}(t){\dot{\rho }}_{3}(t)}{{\rho }_{3}(t)},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}25{\beta }_{1}^{2}{\rho }_{2}{\left(t\right)}^{2}{\rho }_{3}{\left(t\right)}^{2}+125{\rho }_{3}{\left(t\right)}^{2}\dot{{\rho }_{2}}(t)\,\\ \,-\,125{\rho }_{2}(t){\rho }_{3}(t)\dot{{\rho }_{3}}(t)-{\rho }_{2}{\left(t\right)}^{4}=0;\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\bullet \,\,{v}^{({II})}(x,t) & = & -\displaystyle \frac{3{\beta }_{1}^{2}{\rho }_{3}(t)}{{\rho }_{1}(t)}{\tanh }^{2}\left(\displaystyle \frac{{\beta }_{1}x+{\beta }_{5}t+{\beta }_{6}}{2}\right)\\ & & +\displaystyle \frac{6{\beta }_{1}{\rho }_{2}(t)}{5{\rho }_{1}(t)}\tanh \left(\displaystyle \frac{{\beta }_{1}x+{\beta }_{5}t+{\beta }_{6}}{2}\right)+{\beta }_{7}\\ & & +\displaystyle \frac{3{\beta }_{1}\,\left[5{\beta }_{1}{\rho }_{3}(t)+2{\rho }_{2}(t)\right]}{5{\rho }_{1}(t)},\end{array}\end{eqnarray}$
which can be looked on as a quasi-solitary-wave8(8The word ‘quasi' implies that there still exist ρ1(t), ρ2(t) and ρ3(t), beyond the travelling-wave format.) case of solutions ( $\begin{eqnarray}\begin{array}{l}{\rho }_{4}(x,t)=-{\beta }_{1}^{2}{\rho }_{3}(t)-\displaystyle \frac{{\beta }_{5}}{{\beta }_{1}}-\displaystyle \frac{6}{5}{\beta }_{1}{\rho }_{2}(t)\\ \,-{\beta }_{7}{\rho }_{1}(t)+\displaystyle \frac{{\rho }_{2}{\left(t\right)}^{2}}{25{\rho }_{3}(t)},\end{array}\,\end{eqnarray}$
$\begin{eqnarray}\,\begin{array}{l}{\rho }_{5}(t)=\displaystyle \frac{1}{5}{\beta }_{1}^{2}{\rho }_{2}(t)+\displaystyle \frac{{\dot{\rho }}_{1}(t)}{{\rho }_{1}(t)}-\displaystyle \frac{{\rho }_{2}{\left(t\right)}^{3}}{125{\rho }_{3}{\left(t\right)}^{2}}\\ \,-\displaystyle \frac{{\dot{\rho }}_{3}(t)}{{\rho }_{3}(t)},\end{array}\,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{\rho }_{6}(x,t)=-\displaystyle \frac{1}{5}{\beta }_{1}^{2}{\beta }_{7}{\rho }_{2}(t)-\displaystyle \frac{{\beta }_{7}{\dot{\rho }}_{1}(t)}{{\rho }_{1}(t)}\,\\ \,+\displaystyle \frac{{\beta }_{7}{\rho }_{2}{\left(t\right)}^{3}}{125{\rho }_{3}{\left(t\right)}^{2}}+\displaystyle \frac{{\beta }_{7}{\dot{\rho }}_{3}(t)}{{\rho }_{3}(t)},\end{array}\,\end{eqnarray}$
$\begin{eqnarray}\,\begin{array}{l}25{\beta }_{1}^{2}{\rho }_{2}{\left(t\right)}^{2}{\rho }_{3}{\left(t\right)}^{2}+125{\rho }_{3}{\left(t\right)}^{2}{\dot{\rho }}_{2}(t)\\ \,-\,125{\rho }_{2}(t){\rho }_{3}(t){\dot{\rho }}_{3}(t)\\ \,-{\rho }_{2}{\left(t\right)}^{4}=0,\end{array}\,\end{eqnarray}$
where β5, β6 and β7 are the real constants; $\begin{eqnarray}\begin{array}{rcl}\bullet \,\,{v}^{({III})}(x,t) & = & -\displaystyle \frac{3{\gamma }_{0}^{2}}{25{\alpha }_{0}{\beta }_{0}}\,{\tanh }^{2}\,\left[\displaystyle \frac{{\gamma }_{0}x}{10{\beta }_{0}}\right.\\ & & \left.-\displaystyle \frac{{\gamma }_{0}\left(25{\alpha }_{0}{\beta }_{0}{\sigma }_{4}+6{\gamma }_{0}^{2}\right)t}{250{\beta }_{0}^{2}}+\displaystyle \frac{{\sigma }_{3}}{2}\right]\\ & & +\displaystyle \frac{6{\gamma }_{0}^{2}}{25{\alpha }_{0}{\beta }_{0}}\,\tanh \,\left[\displaystyle \frac{{\gamma }_{0}x}{10{\beta }_{0}}\right.\\ & & \left.-\displaystyle \frac{{\gamma }_{0}\left(25{\alpha }_{0}{\beta }_{0}{\sigma }_{4}+6{\gamma }_{0}^{2}\right)t}{250{\beta }_{0}^{2}}+\displaystyle \frac{{\sigma }_{3}}{2}\right]\\ & & +\displaystyle \frac{9{\gamma }_{0}^{2}}{25{\alpha }_{0}{\beta }_{0}}+{\sigma }_{4},\end{array}\end{eqnarray}$
which can be regarded as a constant-coefficient-shock-wave case of solutions (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 ρi'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.
4. Similarity reductions for equation (1 )
Making use of symbolic computation and inserting assumption (25 ) into equation (1 ) turn into
$\begin{eqnarray}\begin{array}{l}{\rho }_{3}(t)\kappa {r}_{x}^{3}q\prime\prime\prime +{\rho }_{1}(t){\kappa }^{2}{r}_{x}{qq}^{\prime} +{\rho }_{1}(t)\kappa {\kappa }_{x}{q}^{2}\\ \quad +\,{r}_{x}\left[{\rho }_{2}(t)\kappa {r}_{x}+3{\rho }_{3}(t)\left({\kappa }_{x}{r}_{x}+\kappa {r}_{{xx}}\right)\right]q^{\prime\prime} \\ \quad +\,\left[\kappa {r}_{t}+{\rho }_{1}(t)\kappa \theta {r}_{x}+{\rho }_{2}(t)\left(2{\kappa }_{x}{r}_{x}+\kappa {r}_{{xx}}\right)\right.\\ \quad \left.+\,\,{\rho }_{3}(t)\left(3{\kappa }_{{xx}}{r}_{x}+3{\kappa }_{x}{r}_{{xx}}+\kappa {r}_{{xxx}}\right)+{\rho }_{4}(x,t)\kappa {r}_{x}\right]q^{\prime} \\ \quad +\,\left\{{\kappa }_{t}+{\rho }_{1}(t)\left({\kappa }_{x}\theta +\kappa {\theta }_{x}\right)+{\rho }_{2}(t){\kappa }_{{xx}}\right.\\ \quad \left.+\,{\rho }_{3}(t){\kappa }_{{xxx}}+{\left[{\rho }_{4}(x,t)\kappa \right]}_{x}+{\rho }_{5}(t)\kappa \right\}q\\ \quad +\left\{{\theta }_{t}+{\rho }_{1}(t)\theta {\theta }_{x}+{\rho }_{2}(t){\theta }_{{xx}}+{\rho }_{3}(t){\theta }_{{xxx}}\right.\\ \quad \left.+\,{\left[{\rho }_{4}(x,t)\theta \right]}_{x}+{\rho }_{5}(t)\theta +{\rho }_{6}(x,t)\right\}=0,\end{array}\end{eqnarray}$
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
$\begin{eqnarray}{{\rm{\Gamma }}}_{1}(r){\rho }_{3}(t)\kappa {r}_{x}^{3}={\rho }_{1}(t){\kappa }^{2}{r}_{x},\,\end{eqnarray}$
$\begin{eqnarray}{{\rm{\Gamma }}}_{2}(r){\rho }_{3}(t)\kappa {r}_{x}^{3}={\rho }_{1}(t)\kappa {\kappa }_{x},\,\end{eqnarray}$
$\begin{eqnarray}{{\rm{\Gamma }}}_{3}(r){\rho }_{3}(t)\kappa {r}_{x}^{3}={r}_{x}\left[{\rho }_{2}(t)\kappa {r}_{x}+3{\rho }_{3}(t)\left({\kappa }_{x}{r}_{x}+\kappa {r}_{{xx}}\right)\right],\end{eqnarray}$
$\begin{eqnarray}\begin{array}{lcl}\,{{\rm{\Gamma }}}_{4}(r){\rho }_{3}(t)\kappa {r}_{x}^{3} & = & \kappa {r}_{t}+{\rho }_{1}(t)\kappa \theta {r}_{x}+{\rho }_{2}(t)\left(2{\kappa }_{x}{r}_{x}+\kappa {r}_{{xx}}\right)\ \\ & & +{\rho }_{3}(t)\left(3{\kappa }_{{xx}}{r}_{x}+3{\kappa }_{x}{r}_{{xx}}+\kappa {r}_{{xxx}}\right)\\ & & +{\rho }_{4}(x,t)\kappa {r}_{x},\end{array}\,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{5}(r){\rho }_{3}(t)\kappa {r}_{x}^{3} & = & {\kappa }_{t}+{\rho }_{1}(t)\left({\kappa }_{x}\theta +\kappa {\theta }_{x}\right)\\ & & +{\rho }_{2}(t){\kappa }_{{xx}}+{\rho }_{3}(t){\kappa }_{{xxx}}+{\left[{\rho }_{4}(x,t)\kappa \right]}_{x}\\ & & +{\rho }_{5}(t)\kappa ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{6}(r){\rho }_{3}(t)\kappa {r}_{x}^{3} & = & {\theta }_{t}+{\rho }_{1}(t)\theta {\theta }_{x}+{\rho }_{2}(t){\theta }_{{xx}}\,\\ & & +{\rho }_{3}(t){\theta }_{{xxx}}+{\left[{\rho }_{4}(x,t)\theta \right]}_{x}\\ & & +{\rho }_{5}(t)\theta +{\rho }_{6}(x,t),\end{array}\end{eqnarray}$
with Γχ(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 to27b ) brings about27c ) results in
$\begin{eqnarray}\kappa (x,t)=\displaystyle \frac{{\rho }_{3}(t)}{{\rho }_{1}(t)}{r}_{x}^{2},\qquad {{\rm{\Gamma }}}_{1}(r)=1.\end{eqnarray}$
Because of the 1st freedom of remark 3 in [59], equation ( $\begin{eqnarray}r(x,t)={\lambda }_{1}(t)x+{\lambda }_{2}(t),\qquad {{\rm{\Gamma }}}_{2}(r)=0,\end{eqnarray}$
and then equation ( $\begin{eqnarray}{\lambda }_{1}(t)={\xi }_{1}\displaystyle \frac{{\rho }_{2}(t)}{{\rho }_{3}(t)},\qquad {{\rm{\Gamma }}}_{3}(r)=\displaystyle \frac{1}{{\xi }_{1}},\end{eqnarray}$
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 ) to27e ) turns to27f ) develops into
$\begin{eqnarray}\begin{array}{c}{\rho }_{2}(t)={\mu }_{1}{\rho }_{3}(t),\,{\rho }_{4}(x,t)={\rho }_{4}(t)\,{\rm{only}},\\ \theta (x,t)=-\displaystyle \frac{1}{{{\mu }_{1}}^{2}{{\xi }_{1}}^{2}{\rho }_{1}\left(t\right)}\left[[{\mu }_{1}{\xi }_{1}{\lambda }_{2}^{{\rm{{\prime} }}}\left(t\right)+{\mu }_{1}^{2}{\xi }_{1}^{2}{\rho }_{4}\left(t\right)\right],\,{{\rm{\Gamma }}}_{4}(r)=0,\end{array}\end{eqnarray}$
equation ( $\begin{eqnarray}{\rho }_{1}(t)={\mu }_{2}{\rho }_{3}(t),\,\,{\rho }_{5}(t)={\mu }_{3}{\rho }_{3}(t),\,\,{{\rm{\Gamma }}}_{5}(r)=\displaystyle \frac{{\mu }_{3}}{{\mu }_{1}^{3}{\xi }_{1}^{3}},\end{eqnarray}$
and then equation ( $\begin{eqnarray}\begin{array}{l}{\rho }_{6}(x,t)={\mu }_{4}\left[x-\displaystyle \int {\rho }_{4}(t){\rm{d}}t\right]{\rho }_{3}(t),\\ {\lambda }_{2}(t)=-{\mu }_{1}{\xi }_{1}\displaystyle \int {\rho }_{4}(t){\rm{d}}t,\\ {{\rm{\Gamma }}}_{6}(r)=\displaystyle \frac{{\mu }_{2}{\mu }_{4}}{{\mu }_{1}^{6}{\xi }_{1}^{6}}r,\end{array}\end{eqnarray}$
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 constraints1 ):35c ) has been presented in [60] and thus can be considered as a known ODE.
$\begin{eqnarray}\begin{array}{l}{\rho }_{1}(t)={\mu }_{2}{\rho }_{3}(t),\,\,{\rho }_{2}(t)={\mu }_{1}{\rho }_{3}(t),\\ {\rho }_{4}(x,t)={\rho }_{4}(t)\,\,\mathrm{only},\,\,\,{\rho }_{5}(t)={\mu }_{3}{\rho }_{3}(t),\\ {\rho }_{6}(x,t)={\mu }_{4}\left[x-\displaystyle \int {\rho }_{4}(t){\rm{d}}t\right]{\rho }_{3}(t),\end{array}\end{eqnarray}$
we build up the following family of the similarity reductions for equation ( $\begin{eqnarray}v(x,t)=\displaystyle \frac{{\mu }_{1}^{2}{\xi }_{1}^{2}}{{\mu }_{2}}q[r(x,t)],\,\end{eqnarray}$
$\begin{eqnarray}r(x,t)={\mu }_{1}{\xi }_{1}\left[x-\int {\rho }_{4}(t){\rm{d}}t\right],\,\end{eqnarray}$
$\begin{eqnarray}q\prime\prime\prime +{qq}^{\prime} +\displaystyle \frac{1}{{\xi }_{1}}q^{\prime\prime} +\displaystyle \frac{{\mu }_{3}}{{\mu }_{1}^{3}{\xi }_{1}^{3}}q+\displaystyle \frac{{\mu }_{2}{\mu }_{4}}{{\mu }_{1}^{6}{\xi }_{1}^{6}}r=0.\,\end{eqnarray}$
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.