# Traveling wave solutions of the generalized scale-invariant analogue of the
KdV equation by tanh–coth method

O. González-Gaxiola ${}^{1\;\ast}$, J. Ruiz de Chávez 2

1 Applied Mathematics and Systems Department, Universidad Autónoma
Metropolitana-Cuajimalpa,
Vasco de Quiroga 4871, 05348 Mexico City, Mexico.
∗<EMAIL_ADDRESS>
2 Department of Mathematics, Universidad Autónoma Metropolitana-Iztapalapa.
San Rafael Atlixco 186, Col. Vicentina, Iztapalapa, 09340, Mexico City, Mexico

###### Abstract

In this work, the generalized scale-invariant analogue of the Korteweg–de
Vries (gsiaKdV) equation is studied. For the first time, the tanh–coth
methodology is used to find traveling wave solutions for this nonlinear
equation. The considered generalized equation is a connection between the
well-known KdV equation and the recently investigated SIdV equation. The
obtained results show many families of solutions for the model, indicating
that this equation also shares bell-shaped solutions with KdV and SIdV, as
previously documented by other researchers. Finally, by executing the symbolic
computation, we demonstrate that the employed technique is a valuable and
effective mathematical tool that can be used to solve problems that arise in
the cross-disciplinary nonlinear sciences and applied mathematics.

Key words: KdV equation; SIdV equation; The tanh–coth method; Travelling
waves; Symbolic computation

##  1 Introduction

Many fields of science and engineering depend heavily on the mathematical
models presented by nonlinear partial differential equations (NPDE) to explain
complex phenomena. These fields include electromagnetic waves theory, ocean
dynamics, plasma physics, fluid mechanics, field theory, nonlinear optical
fibers, nuclear physics, ion acoustic waves, biological process engineering,
chemical kinetics, climatological phenomena and several other mathematical
physics problems. In 1895, Dutch mathematicians D. J. Korteweg and G. de Vries
developed the KdV equation in a formal manner, the KdV equation is a NPDE that
models the propagation of long waves on shallow water and this simultaneously
includes weak advective nonlinearity and dispersion effects given by [1]

$u_{t}+6uu_{x}+u_{xxx}=0,$ (1)

where $u$ is the perturbation wave function that depends on the spatial
variable $x$ and on time $t$. It is well known that Eq. (1) has bell-shaped
solutions of the type:

$u(x,t)=\frac{c}{2}sech^{2}\Big{(}\frac{\sqrt{c}}{2}(x-ct)\Big{)}$ (2)

where $c$ is the velocity of the wavefront [2].
Despite its antiquity, the KdV equation is still an active area of study and
research, with several articles published on the topic in recent years [3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13].
Among the numerous research associated to the KdV equation that have been
published in the last decades, several propose to generalize and/or modify the
KdV equation. In 2012, for instance, the authors of [3] present the modified
KdV equation

$u_{t}+\Big{(}\frac{2u_{xx}}{u}\Big{)}u_{x}=u_{xxx}.$ (3)

The Eq. (3) is invariant under scaling of the dependent variable; is therefore
referred to as SIdV here. Eq. (3) was discovered by surprise using computer
approaches when researchers explored equations with bell-shaped solutions
analogous to the KdV equation. Other research on Eq. (3) have been published
and can be found in [14, 15, 16].
In this article we will consider the gsiaKdV equation [17] which is a
nonlinear partial differential equation and whose dimensionless form is given
by

$u_{t}+\Big{(}3(1-\alpha)u+(1+\alpha)\frac{u_{xx}}{u}\Big{)}u_{x}=\gamma
u_{xxx}$ (4)

where $u_{xxx}$ is a dispersion term, while the term
$\big{(}3(1-\alpha)u+(1+\alpha)\frac{u_{xx}}{u}\big{)}$ can be seen as an
advecting velocity.
Firstly, let us observe that, if $\alpha=-1$ and $\gamma=-1$, then Eq. (4)
reduces to the well-known KdV equation (1). Second, we can observe that if
$\alpha=1$ and $\gamma=1$ then Eq. (4) reduces to SIdV equation (3).
Eq. (4) was investigated in [18] and the authors demonstrated the existence of
traveling waves of the bell and valley types. Using the tanh-coth method for
the first time, the main objective of this research is to find new traveling
wave type solutions for Eq. (4) with $\alpha\neq\pm 1$ and $\gamma\neq 0$.
In virtue of the significance of the previously described gsiaKdV equation, we
will conduct a study to obtain solutions of the traveling-wave type for the
first time utilizing the tanh-coth technique. In addition, some 3D and 2D
propagation profiles for the derived solutions will be discussed by selecting
various parameters that describe the solution sets achieved by the used
strategy.

##  2  Brief description of the tanh-coth method

The tanh-coth method originally established in [19, 20] provides a very useful
methodology for finding traveling wave type solutions of NPDEs. We will
explain how to implement the method in the rest of this section.
(I) Consider the general nonlinear PDE given by:

$\displaystyle G(u,u_{t},u_{x},u_{xx},u_{xxx},\ldots)=0.$ (5)

Using traveling wave variable change $u(x,t)=u(\xi)$ with $\xi=cx-\omega t$,
Eq. (5) becomes the ordinary differential equation:

$\displaystyle F(u,-\omega
u_{\xi},cu_{\xi},c^{2}u_{\xi\xi},c^{3}u_{\xi\xi\xi},\ldots)=0.$ (6)

(II) The tanh-coth method provides the solutions for Eq (6) as the finite sum

$\displaystyle
u(\xi)=S(Y)=\sum_{i=0}^{M}a_{i}Y^{i}(\xi)+\sum_{i=1}^{M}b_{i}Y^{-i}(\xi),$ (7)

where the coefficients $a_{i}$ and $b_{i}$ are constants to be determined and
$Y$ is a new dependent variable introduced by the method and is given by

$\displaystyle Y=\tanh(\xi).$ (8)

The introduction of this new dependent variable implies that:

$u_{\xi}=(1-Y^{2})\frac{dS}{dY},$
$u_{\xi\xi}=-2Y(1-Y^{2})\frac{dS}{dY}+(1-Y^{2})^{2}\frac{d^{2}S}{dY^{2}},$ (9)
$u_{\xi\xi\xi}=2(1-Y^{2})(3Y^{2}-1)\frac{dS}{dY}-6Y(1-Y^{2})^{2}\frac{d^{2}S}{dY^{2}}+(1-Y^{2})^{3}\frac{d^{3}S}{dY^{3}},$

The subsequent derivatives can be computed in a similar way.
(III) To determine the upper limit $M$ of the sum in Eq. (7), the linear terms
of highest order in the resulting equation with the highest order nonlinear
terms are balanced.
(IV) We consider $u(\xi)$ given in (7) and the necessary derivatives
$u_{\xi}$, $u_{\xi\xi}$, $u_{\xi\xi\xi}$,…, which can be calculated as in (9),
to substitute in the ordinary differential equation (6) and thus we will
obtain the polynomial equation:

$\displaystyle P[Y]=0.$ (10)

(V) We select all the terms that have the same algebraic power of $Y$ from the
polynomial equation (10), we set them equal to zero and obtain a nonlinear
system of algebraic equations with the set of unknown parameters
$\\{a_{0},\ldots,a_{M},b_{1},\ldots,b_{M},c,\omega\\}$. Using software such as
Mathematica, we can execute symbolic calculations to solve the algebraic
equations with the natural restrictions of the mathematical model.
(VI) Finally, having obtained the coefficients
$\\{a_{0},\ldots,a_{M},b_{1},\ldots,b_{M},c,\omega\\}$ and considering the
equality (7) one can obtain the exact solutions to Eq. (5).

##  3 Utilization of the tanh-coth methodology

Using the change of variable $\xi=cx-\omega t$, Eq. (4) is converted into the
ordinary differential equation:

$\displaystyle-\omega
cuu_{\xi}+3c(1-\alpha)u^{2}u_{\xi}+c^{3}(1+\alpha)u_{\xi}u_{\xi\xi}-\gamma
c^{3}uu_{\xi\xi\xi}=0.$ (11)

Integrating once with respect to $\xi$ and considering the constants of
integration as null, we obtain

$\displaystyle-\omega
cu^{2}+2c(1-\alpha)u^{3}+c^{3}(1+\alpha+\gamma)u_{\xi}^{2}-2\gamma
c^{3}uu_{\xi\xi}=0.$ (12)

Then using the characteristic variable change of the method, i.e.,
$Y=\tanh(\xi)$ and considering Eq. (7), the last differential equation is
rewritten as

$-\omega
cS^{2}+2c(1-\alpha)S^{3}+c^{3}(1+\alpha+\gamma)(1-Y^{2})^{2}\Big{(}\frac{dS}{dY}\Big{)}^{2}-2\gamma
c^{3}S\Big{[}(1-Y^{2})^{2}\frac{d^{2}S}{dY^{2}}-2Y(1-Y^{2})\frac{dS}{dY}\Big{]}=0.$
(13)

Balancing $S^{3}$ with $S\cdot\frac{d^{2}S}{dY^{2}}$ gives $M=2$.
Consequently, the tanh-coth technique enables the use of the finite sum

$\displaystyle u(\xi)=S(Y)=a_{0}+a_{1}Y+a_{2}Y^{2}+b_{1}Y^{-1}+b_{2}Y^{-2}.$
(14)

Substituting (14) with their respective derivatives into (13) and collecting
all terms with equal power of $Y$, after some algebraic simplification, we
obtain the following nonlinear system of algebraic equations:

$\displaystyle 12a_{1}b_{1}\gamma c^{3}+4a_{0}b_{2}\gamma
c^{3}+48a_{2}b_{2}\gamma
c^{3}+4a_{1}b_{1}c^{3}\alpha+16a_{2}b_{2}c^{3}\alpha+4a_{1}b_{1}c^{3}+16a_{2}b_{2}c^{3}-6a_{2}b_{1}^{2}c\alpha$
$\displaystyle-12a_{0}a_{1}b_{1}c\alpha-6a_{1}^{2}b_{2}c\alpha-12a_{0}a_{2}b_{2}c\alpha-2a_{1}b_{1}c\omega-2a_{2}b_{2}c\omega+6a_{2}b_{1}^{2}c+12a_{0}a_{1}b_{1}c+6a_{1}^{2}b_{2}c+12a_{0}a_{2}b_{2}c$
$\displaystyle+a_{1}^{2}\gamma c^{3}-4a_{0}a_{2}\gamma
c^{3}+a_{1}^{2}c^{3}\alpha+a_{1}^{2}c^{3}-2a_{0}^{3}c\alpha-
a_{0}^{2}c\omega+2a_{0}^{3}c+5b_{1}^{2}\gamma c^{3}-8b_{2}^{2}\gamma
c^{3}+b_{1}^{2}c^{3}\alpha+b_{1}^{2}c^{3}=0,$ $\displaystyle-7a_{1}^{2}\gamma
c^{3}+8a_{2}^{2}\gamma c^{3}-20a_{0}a_{2}\gamma
c^{3}+a_{1}^{2}c^{3}\alpha-8a_{2}^{2}c^{3}\alpha+a_{1}^{2}c^{3}-8a_{2}^{2}c^{3}-6a_{0}a_{2}^{2}c\alpha-6a_{1}^{2}a_{2}c\alpha-
a_{2}^{2}c\omega$ $\displaystyle+6a_{0}a_{2}^{2}c+6a_{1}^{2}a_{2}c=0,$
$\displaystyle 4a_{1}^{2}\gamma c^{3}-16a_{2}^{2}\gamma
c^{3}+8a_{0}a_{2}\gamma
c^{3}+4a_{2}^{2}c^{3}\alpha+4a_{2}^{2}c^{3}-2a_{2}^{3}c\alpha+2a_{2}^{3}\gamma=0,$
$\displaystyle 4a_{2}b_{1}\gamma\alpha+4a_{0}a_{1}\gamma
c^{3}-24a_{1}a_{2}\gamma
c^{3}+4a_{1}a_{2}c^{3}\alpha+4a_{1}a_{2}c^{3}-6a_{1}a_{2}^{2}c\alpha+6a_{1}a_{2}^{2}\omega=0,$
$\displaystyle-4a_{0}b_{1}\gamma c^{3}-20a_{2}b_{1}\gamma
c^{3}-4a_{1}b_{2}\gamma
c^{3}-4a_{2}b_{1}c^{3}\alpha-4a_{2}b_{1}c^{3}-6a_{2}^{2}b_{1}c\alpha+6a_{2}^{2}b_{1}c-8a_{0}a_{1}\gamma
c^{3}+12a_{1}a_{2}\gamma c^{3}$
$\displaystyle-8a_{1}a_{2}c^{3}\alpha-8a_{1}a_{2}c^{3}-2a_{1}^{3}c\alpha-12a_{0}a_{1}a_{2}c\alpha-2a_{1}a_{2}c\omega+2a_{1}^{3}c+12a_{0}a_{1}a_{2}c=0,$
$\displaystyle-6a_{1}b_{1}\gamma c^{3}-8a_{0}b_{2}\gamma
c^{3}-24a_{2}b_{2}\gamma
c^{3}-2a_{1}b_{1}c^{3}\alpha-8a_{2}b_{2}c^{3}\alpha-2a_{1}b_{1}c^{3}-8a_{2}b_{2}c^{3}-12a_{1}a_{2}b_{1}c\alpha-6a_{2}^{2}b_{2}c\alpha$
$\displaystyle+12a_{1}a_{2}b_{1}c+6a_{2}^{2}b_{2}c+2a_{1}^{2}\gamma
c^{3}+16a_{0}a_{2}\gamma
c^{3}-2a_{1}^{2}c^{3}\alpha+4a_{2}^{2}c^{3}\alpha-2a_{1}^{2}c^{3}+4a_{2}^{2}c^{3}-6a_{0}a_{1}^{2}c\alpha-6a_{0}^{2}a_{2}c\alpha$
$\displaystyle-
a_{1}^{2}c\omega-2a_{0}a_{2}c\omega+6a_{0}a_{1}^{2}c+6a_{0}^{2}a_{2}c-4b_{1}^{2}\gamma
c^{3}=0,$ $\displaystyle 4a_{0}b_{1}\gamma c^{3}+28a_{2}b_{1}\gamma
c^{3}-8a_{1}b_{2}\gamma
c^{3}+8a_{2}b_{1}c^{3}\alpha-4a_{1}b_{2}c^{3}\alpha+8a_{2}b_{1}c^{3}-4a_{1}b_{2}c^{3}-6a_{1}^{2}b_{1}c\alpha-12a_{0}a_{2}b_{1}c\alpha$
$\displaystyle-12a_{1}a_{2}b_{2}c\alpha-2a_{2}b_{1}c\omega+6a_{1}^{2}b_{1}c+12a_{0}a_{2}b_{1}c+12a_{1}a_{2}b_{2}c+4a_{0}a_{1}\gamma
c^{3}+4a_{1}a_{2}c^{3}\alpha+4a_{1}a_{2}c^{3}-6a_{0}^{2}a_{1}c\alpha$
$\displaystyle-2a_{0}a_{1}c\omega+6a_{0}^{2}a_{1}c-12b_{1}b_{2}\gamma
c^{3}=0,$ $\displaystyle 4a_{0}b_{1}\gamma c^{3}-12a_{2}b_{1}\gamma
c^{3}+28a_{1}b_{2}\gamma
c^{3}-4a_{2}b_{1}c^{3}\alpha+8a_{1}b_{2}c^{3}\alpha-4a_{2}b_{1}c^{3}+8a_{1}b_{2}c^{3}-6a_{1}b_{1}^{2}c\alpha-6a_{0}^{2}b_{1}c\alpha$
$\displaystyle-12a_{0}a_{1}b_{2}c\alpha-12a_{2}b_{1}b_{2}c\alpha-2a_{0}b_{1}c\omega-2a_{1}b_{2}c\omega+6a_{1}b_{1}^{2}c+6a_{0}^{2}b_{1}c+12a_{0}a_{1}b_{2}c+12a_{2}b_{1}b_{2}c$
$\displaystyle+12b_{1}b_{2}\gamma
c^{3}+4b_{1}b_{2}c^{3}\alpha+4b_{1}b_{2}c^{3}=0,$
$\displaystyle-12a_{0}b_{2}\gamma
c^{3}-6a_{0}b_{2}^{2}c\alpha+6a_{0}b_{2}^{2}c-3b_{1}^{2}\gamma
c^{3}+8b_{2}^{2}\gamma
c^{3}+b_{1}^{2}c^{3}\alpha-8b_{2}^{2}c^{3}\alpha+b_{1}^{2}c^{3}-8b_{2}^{2}c^{3}$
$\displaystyle-6b_{1}^{2}b_{2}c\alpha-b_{2}^{2}c\omega+6b_{1}^{2}b_{2}c=0,$
$\displaystyle-4a_{0}b_{1}\gamma c^{3}-16a_{1}b_{2}\gamma
c^{3}-4a_{1}b_{2}c^{3}\alpha-4a_{1}b_{2}c^{3}-6a_{1}b_{2}^{2}c\alpha-12a_{0}b_{1}b_{2}c\alpha+6a_{1}b_{2}^{2}c+12a_{0}b_{1}b_{2}c$
$\displaystyle+12b_{1}b_{2}\gamma
c^{3}-8b_{1}b_{2}c^{3}\alpha-8b_{1}b_{2}c^{3}-2b_{1}^{3}c\alpha-2b_{1}b_{2}c\omega+2b_{1}^{3}c=0,$
$\displaystyle-6a_{1}b_{1}\gamma c^{3}+16a_{0}b_{2}\gamma
c^{3}-24a_{2}b_{2}\gamma
c^{3}-2a_{1}b_{1}c^{3}\alpha-8a_{2}b_{2}c^{3}\alpha-2a_{1}b_{1}c^{3}-8a_{2}b_{2}c^{3}-6a_{0}b_{1}^{2}c\alpha-6a_{2}b_{2}^{2}c\alpha$
$\displaystyle-6a_{0}^{2}b_{2}c\alpha-12a_{1}b_{1}b_{2}c\alpha-2a_{0}b_{2}c\omega+6a_{0}b_{1}^{2}c+6a_{2}b_{2}^{2}c+6a_{0}^{2}b_{2}c+12a_{1}b_{1}b_{2}c+2b_{1}^{2}\gamma
c^{3}+8b_{2}^{2}\gamma c^{3}-2b_{1}^{2}c^{3}\alpha$
$\displaystyle+4b_{2}^{2}c^{3}\alpha-2b_{1}^{2}c^{3}+4b_{2}^{2}c^{3}-b_{1}^{2}c\omega=0,$
$\displaystyle-12b_{1}b_{2}\gamma
c^{3}+4a_{1}b_{1}b_{2}c^{3}\alpha+4b_{1}b_{2}c^{3}-6\omega
b_{1}b_{2}^{2}\alpha+6a_{0}b_{1}b_{2}^{2}c=0,$ $\displaystyle-8b_{2}^{2}\gamma
c^{3}+4a_{0}a_{1}\gamma\omega+4b_{2}^{2}c^{3}\alpha+4a_{2}b_{2}^{2}c^{3}-2b_{2}^{3}c\alpha+2b_{2}^{3}c=0.$

Using the well-known Mathematica software to solve the above system, we find
the following families of solutions:
Family 1: For $\alpha\neq 1$ and $c\neq 0$:

$a_{0}=a_{0},\;\,a_{1}=0,\,\;a_{2}=a_{2},\,\;b_{1}=0,\;\;b_{2}=-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha\
-1},\;\omega\neq 0.$

Substituting the obtained parameters into the general solution (14), we obtain
the following family of solutions

$u_{1}(x,t)=a_{0}+a_{2}\tanh^{2}(cx-\omega
t)-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha\ -1}\coth^{2}(cx-\omega t).$ (15)

Family 2: For $\alpha\neq\pm 1$ and $c\neq 0$:

$a_{0}=0,\,a_{1}=a_{1},\,a_{2}=-\frac{2c^{2}(4\gamma^{2}-\alpha^{2}-2\alpha-1)}{5(\alpha^{2}-1)},\,b_{1}=0,\;b_{2}=-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1},\;\omega\neq
0.$

Therefore, proceeding as in the previous case, the set of solutions for this
family is provided by

$u_{2}(x,t)=a_{1}\tanh(cx-\omega
t)-\frac{2c^{2}(4\gamma^{2}-\alpha^{2}-2\alpha-1)}{5(\alpha^{2}-1)}\tanh^{2}(cx-\omega
t)-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha\ -1}\coth^{2}(cx-\omega t).$ (16)

Family 3: For $\alpha\neq 1$ and $c\neq 0$:

$a_{0}=a_{0}\neq
0,\,a_{1}=a_{1},\,a_{2}=\frac{3}{2}a_{1},\,b_{1}=0,\;b_{2}=-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1},$
$\omega=\frac{8a_{0}\gamma c^{2}\alpha-8a_{0}\gamma
c^{2}+6a_{0}^{2}\alpha-16\gamma^{2}c^{4}+4c^{4}\alpha^{2}+4c^{4}}{a_{0}(\alpha-1)}.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$u_{3}(x,t)=a_{0}+a_{1}\tanh(cx-\omega t)+\frac{3}{2}a_{1}\tanh^{2}(cx-\omega
t)-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha\ -1}\coth^{2}(cx-\omega t).$ (17)

Family 4: For $\alpha\neq\pm 1$ and $c\neq 0$ :

$a_{0}=a_{0},\,a_{1}=a_{1},\,a_{2}=\frac{3}{4}a_{1},\,b_{1}=0,\;b_{2}=-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1},\;\omega=\frac{3\gamma
c^{2}-3(\alpha+1)-7\gamma}{\alpha^{2}-1}.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$u_{4}(x,t)=a_{0}+a_{1}\tanh(cx-\omega t)+\frac{3}{2}a_{1}\tanh^{2}(cx-\omega
t)-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha\ -1}\coth^{2}(cx-\omega t).$ (18)

Family 5: For $\alpha\neq\pm 1$ and $c\neq 0$:

$a_{0}=a_{0},\,a_{1}=a_{1},$ $a_{2}=\frac{16\gamma
c^{2}\alpha^{2}-16a_{0}\gamma
c^{2}-8a_{0}c^{2}\alpha^{3}-8a_{0}c^{2}\alpha^{2}+8a_{0}c^{2}\alpha+8a_{0}c^{2}-3a_{0}^{2}\alpha^{3}}{10c^{2}\left(\alpha^{2}-1\right)},$
$b_{1}=0,\;\;b_{2}=-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1},\;\;\omega=\frac{\gamma^{2}(a_{0}^{3}-4a_{1}\gamma
c^{2}-2c^{2})}{1-\alpha^{2}}.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$\begin{split}u_{5}(x,t)&=\Big{(}\frac{16\gamma c^{2}\alpha^{2}-16a_{0}\gamma
c^{2}-8a_{0}c^{2}\alpha^{3}-8a_{0}c^{2}\alpha^{2}+8a_{0}c^{2}\alpha+8a_{0}c^{2}-3a_{0}^{2}\alpha^{3}}{10c^{2}\left(\alpha^{2}-1\right)(2\gamma-\alpha-1)}\Big{)}\tanh^{2}(cx-\omega
t)\\\ &+a_{0}+a_{1}\tanh(cx-\omega t)-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha\
-1}\coth^{2}(cx-\omega t).\end{split}$ (19)

Family 6: For $\alpha\neq\pm 1$, $c\neq 0$ and $2\gamma-\alpha-1\neq 0$:

$a_{0}=a_{0},\,\;a_{1}=\frac{20a_{0}\gamma
c^{2}(\alpha-1)(2\gamma-\alpha-1)+b_{1}^{2}(\alpha-1)^{2}}{{16c^{4}(2\gamma-\alpha-1)^{2}}},\;a_{2}=a_{2},\;\;b_{1}=b_{1},$
$b_{2}=-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1},\;\omega=\frac{b_{1}^{2}(a_{0}\gamma-3(\alpha+1))}{4c^{2}(\alpha^{2}-1)^{2}}+8c^{2}(\gamma-\alpha-1).$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$\begin{split}u_{6}(x,t)&=a_{0}+\Big{(}\frac{20a_{0}\gamma
c^{2}(\alpha-1)(2\gamma-\alpha-1)+b_{1}^{2}(\alpha-1)^{2}}{{16c^{4}(2\gamma-\alpha-1)^{2}}}\Big{)}\tanh(cx-\omega
t)+a_{2}\tanh^{2}(cx-\omega t)\\\ &+b_{1}\coth(cx-\omega
t)-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1}\coth^{2}(cx-\omega
t).\end{split}$ (20)

Family 7: For $\alpha\neq 1$ and $c\neq 0$:

$a_{0}=a_{0},\;\,a_{1}=a_{1},\;\;a_{2}=\frac{3}{2}a_{1},\;\,b_{1}=0,\;\;b_{2}=-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1},\;\,\omega=\left(a_{0}a_{2}+3\gamma-2\alpha^{2}\right).$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$u_{7}(x,t)=a_{0}+a_{1}\tanh(cx-\omega t)+\frac{3}{2}a_{1}\tanh^{2}(cx-\omega
t)-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha\ -1}\coth^{2}(cx-\omega t).$ (21)

Family 8: For $\alpha\neq-1$, $c\neq 0$ and $3\gamma+1\neq 0$:

$a_{0}=a_{0},\;\,a_{1}=a_{1},\;\;a_{2}=-\frac{3(c\gamma^{2}-\alpha+3)}{2c^{2}(\alpha+1)(3\gamma+1)},\;\;b_{1}=b_{1},\;b_{2}=0,\;\;\omega=\omega.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$u_{8}(x,t)=a_{0}+a_{1}\tanh(cx-\omega
t)-\frac{3(c\gamma^{2}-\alpha+3)}{2c^{2}(\alpha+1)(3\gamma+1)}\tanh^{2}(cx-\omega
t)+b_{1}\coth(cx-\omega t).$ (22)

Family 9: For $\alpha\neq 1$, $c\neq 0$ and $\gamma\neq 0$:

$a_{0}=\frac{8a_{2}^{2}c^{2}\gamma-2a_{1}^{2}c^{2}\gamma-2a_{2}^{2}c^{2}\alpha-2a_{2}^{2}c^{2}+a_{2}^{3}\alpha-
a_{2}^{3}}{4a_{2}c^{2}\gamma},\;\;a_{1}=a_{1},\;\;a_{2}=a_{2}\neq 0,$
$b_{1}=b_{1},\;\;b_{2}=-\frac{2c^{2}(3\gamma-\alpha-1)}{3(\alpha-1)},\;\;\omega=\omega.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$\begin{split}u_{9}(x,t)&=\frac{8a_{2}^{2}c^{2}\gamma-2a_{1}^{2}c^{2}\gamma-2a_{2}^{2}c^{2}\alpha-2a_{2}^{2}c^{2}+a_{2}^{3}\alpha-
a_{2}^{3}}{4a_{2}c^{2}\gamma}+a_{1}\tanh(cx-\omega t)+a_{2}\tanh^{2}(cx-\omega
t)\\\ &+b_{1}\coth(cx-\omega
t)-\frac{2c^{2}(3\gamma-\alpha-1)}{3(\alpha-1)}\coth^{2}(cx-\omega
t).\end{split}$ (23)

Family 10: For $\alpha\neq 1$, $c\neq 0$ and $12\gamma-5\alpha-5\neq 0$:

$a_{0}=a_{0},\;\;a_{1}=a_{1},\;\;a_{2}=-\frac{2c^{2}(4\gamma^{2}-\alpha^{2}-2\alpha-1)}{(\alpha-1)(12\gamma-5\alpha-5)},\;b_{1}=0,\;b_{2}=-\frac{2c^{2}a_{1}(2\gamma-\alpha-1)}{\alpha-1},\;\;\omega=\omega.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$\begin{split}u_{10}(x,t)&=a_{0}+a_{1}\tanh(cx-\omega
t)-\frac{2c^{2}(4\gamma^{2}-\alpha^{2}-2\alpha-1)}{(\alpha-1)(12\gamma-5\alpha-5)}\tanh^{2}(cx-\omega
t)\\\ &-\frac{2c^{2}a_{1}(2\gamma-\alpha-1)}{\alpha-1}\coth^{2}(cx-\omega
t).\end{split}$ (24)

Family 11: For $\alpha\neq 1$, $c\neq 0$ and $\gamma\neq 0$:

$a_{0}=\frac{a_{2}\left(a_{2}\alpha-a_{2}+8\gamma
c^{2}-2c^{2}\alpha-2c^{2}\right)}{4c^{2}\gamma},\;\;a_{1}=0,\;\;a_{2}=a_{2},$
$b_{1}=0,\;b_{2}=-\frac{2c^{2}(3\gamma-\alpha-1)}{3(\alpha-1)},\;\omega=\omega.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$u_{11}(x,t)=\frac{a_{2}\left(a_{2}\alpha-a_{2}+8\gamma
c^{2}-2c^{2}\alpha-2c^{2}\right)}{4c^{2}\gamma}+a_{2}\tanh^{2}(cx-\omega
t)-\frac{2c^{2}(3\gamma-\alpha-1)}{3(\alpha-1)}\coth^{2}(cx-\omega t).$ (25)

Family 12: For $\alpha\neq 1$, $c\neq 0$ and $\gamma\neq 0$:

$a_{0}=\frac{b_{1}^{2}-b_{1}^{2}\alpha}{2c^{2}\gamma},\;\;a_{1}=-\frac{\sqrt{2a_{2}c^{2}(4\gamma-\alpha-1)+a_{2}^{2}(\alpha-1)}}{\sqrt{2}c\sqrt{\gamma}},$
$a_{2}=a_{2},\;\;b_{1}=b_{1},\;\;b_{2}=0,\;\;\omega=\omega.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$\begin{split}u_{12}(x,t)&=\frac{b_{1}^{2}-b_{1}^{2}\alpha}{2c^{2}\gamma}-\frac{\sqrt{2a_{2}c^{2}(4\gamma-\alpha-1)+a_{2}^{2}(\alpha-1)}}{\sqrt{2}c\sqrt{\gamma}}\tanh(cx-\omega
t)\\\ &+a_{2}\tanh^{2}(cx-\omega t)+b_{1}\coth(cx-\omega t).\end{split}$ (26)

Family 13: For $\gamma\neq 0$ and $c\neq 0$:

$a_{0}=\frac{8a_{2}^{2}c^{2}\gamma-2a_{1}^{2}c^{2}\gamma-2a_{2}^{2}c^{2}\alpha-2a_{2}^{2}c^{2}+a_{2}^{3}\alpha-
a_{2}^{3}}{4a_{2}c^{2}\gamma},\;\;a_{1}=a_{1},\;\;a_{2}=a_{2}\neq 0,$
$b_{1}=\frac{a_{1}(16a_{2}^{2}c^{2}\gamma+2a_{1}^{2}c^{2}\gamma-2a_{2}^{2}c^{2}\alpha-2a_{2}^{2}c^{2}+5a_{2}^{3}\alpha-5a_{2}^{3})}{4a_{2}^{2}c^{2}\gamma},\;\;b_{2}=0,\;\omega=\omega.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$\begin{split}u_{13}(x,t)&=\frac{8a_{2}^{2}c^{2}\gamma-2a_{1}^{2}c^{2}\gamma-2a_{2}^{2}c^{2}\alpha-2a_{2}^{2}c^{2}+a_{2}^{3}\alpha-
a_{2}^{3}}{4a_{2}c^{2}\gamma}+a_{1}\tanh(cx-\omega t)+a_{2}\tanh^{2}(cx-\omega
t)\\\
&+\frac{a_{1}(16a_{2}^{2}c^{2}\gamma+2a_{1}^{2}c^{2}\gamma-2a_{2}^{2}c^{2}\alpha-2a_{2}^{2}c^{2}+5a_{2}^{3}\alpha-5a_{2}^{3})}{4a_{2}^{2}c^{2}\gamma}\coth(cx-\omega
t).\end{split}$ (27)

Family 14: For $\alpha\neq 1$, $c\neq 0$ and $\gamma\neq 0$:

$a_{0}=-\frac{3b_{1}\alpha+c^{4}}{c^{2}\gamma(\alpha-1)},\;\;a_{1}=0,\;\;a_{2}=0,\;\;b_{1}=b_{1},\;\;b_{2}=-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1},\;\omega=\omega.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$u_{14}(x,t)=-\frac{3b_{1}\alpha+c^{4}}{c^{2}\gamma(\alpha-1)}+b_{1}\coth(cx-\omega
t)-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1}\coth^{2}(cx-\omega t).$ (28)

Family 15: For $\alpha\neq 1$, $c\neq 0$ and $9\gamma-7\alpha-7\neq 0$:

$a_{0}=0,\;\;a_{1}=0,\;\;a_{2}=\frac{2c^{2}(6\gamma^{2}-\gamma\alpha-\gamma-\alpha^{2}-2\alpha-1)}{(\alpha-1)(9\gamma-7\alpha-7)},$
$b_{1}=b_{1},\;\;b_{2}=-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1},\;\omega=\omega.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$\begin{split}u_{15}(x,t)&=\frac{2c^{2}(6\gamma^{2}-\gamma\alpha-\gamma-\alpha^{2}-2\alpha-1)}{(\alpha-1)(9\gamma-7\alpha-7)}\tanh^{2}(cx-\omega
t)+b_{1}\coth(cx-\omega t)\\\
&-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1}\coth^{2}(cx-\omega t).\end{split}$
(29)

Family 16: For $\alpha\neq\pm 1$, $c\neq 0$ and $2\gamma-\alpha-1\neq 0$:

$a_{0}=a_{0},\;\;a_{1}=a_{1},\;\;a_{2}=0,\;\;b_{1}=0,\;\;b_{2}=b_{2},\;\omega=\frac{a_{0}c\alpha+3\gamma
b_{2}^{2}}{(1-\alpha^{2})(2\gamma-\alpha-1)}.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$u_{16}(x,t)=a_{0}+a_{1}\tanh(cx+\omega t)+b_{2}\coth^{2}(cx-\omega t).$ (30)

Family 17: For $\alpha\neq 1$, $c\neq 0$ and $\gamma-\alpha-1\neq 0$:

$a_{0}=0,\;\;a_{1}=0,\;\;a_{2}=\frac{2c^{2}(9\gamma^{2}-\alpha^{2}-2\alpha-1)}{9(\alpha-1)(\gamma-\alpha-1)},\;b_{1}=b_{1},\;\;b_{2}=-\frac{2c^{2}(3\gamma-\alpha-1)}{3(\alpha-1)},\;\omega=\omega.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$u_{17}(x,t)=\frac{2c^{2}(9\gamma^{2}-\alpha^{2}-2\alpha-1)}{9(\alpha-1)(\gamma-\alpha-1)}\tanh^{2}(cx+\omega
t)+b_{1}\coth(cx-\omega
t)-\frac{2c^{2}(3\gamma-\alpha-1)}{3(\alpha-1)}\coth^{2}(cx-\omega t).$ (31)

Family 18: For $\alpha\neq 1$, $c\neq 0$ and $\gamma\neq 0$:

$a_{0}=-\frac{a_{2}(3a_{2}\alpha+10\gamma
c^{2}+2c^{2})}{2c^{2}\gamma},\;\;a_{1}=0,\;\;a_{2}=a_{2},\;b_{1}=b_{1},\;\;b_{2}=-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1},\;\;\omega=\omega.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$u_{18}(x,t)=-\frac{a_{2}(3a_{2}\alpha+10\gamma
c^{2}+2c^{2})}{2c^{2}\gamma}+a_{2}\tanh^{2}(cx+\omega t)+b_{1}\coth(cx-\omega
t)-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1}\coth^{2}(cx-\omega t).$ (32)

Family 19: For $\alpha\neq 1$, $\gamma\neq 0$, $c\neq 0$ and $2a_{1}\neq
b_{1}$:

$a_{0}=-\frac{a_{1}(a_{1}^{2}\alpha^{2}-2a_{1}^{2}\alpha+a_{1}^{2}-8\gamma^{2}c^{4}+4\gamma
c^{4}\alpha+4\gamma
c^{4})}{2c^{2}\gamma(\alpha-1)\left(2a_{1}-b_{1}\right)},\;\;a_{1}=a_{1},\;\;a_{2}=0,$
$b_{1}=b_{1},\;\;b_{2}=-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1},\;\;\omega=\omega.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$\begin{split}u_{19}(x,t)&=-\frac{a_{1}(a_{1}^{2}\alpha^{2}-2a_{1}^{2}\alpha+a_{1}^{2}-8\gamma^{2}c^{4}+4\gamma
c^{4}\alpha+4\gamma
c^{4})}{2c^{2}\gamma(\alpha-1)\left(2a_{1}-b_{1}\right)}+a_{1}\tanh(cx-\omega
t)+b_{1}\coth(cx-\omega t)\\\
&-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1}\coth^{2}(cx-\omega t).\end{split}$
(33)

Family 20: For $\alpha\neq\pm 1$, $c\neq 0$ and $\gamma\neq 0$:

$a_{0}=0,\;\;a_{1}=a_{1},\;\;a_{2}=0,\;b_{1}=\frac{2c\sqrt{a_{1}(3\gamma-\alpha-1)}}{\sqrt{3}\sqrt{\alpha-1}},\;\;b_{2}=-\frac{c^{4}(3\gamma-5\alpha-1)}{\gamma^{2}(\alpha^{2}-1)},\;\;\omega=\omega.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$u_{20}(x,t)=a_{1}\tanh(cx+\omega
t)+\frac{2c\sqrt{a_{1}(3\gamma-\alpha-1)}}{\sqrt{3}\sqrt{\alpha-1}}\coth(cx-\omega
t)-\frac{c^{4}(3\gamma-5\alpha-1)}{\gamma^{2}(\alpha^{2}-1)}\coth^{2}(cx-\omega
t).$ (34)

Family 21: For $\gamma\neq 0$ and $c\neq 0$:

$a_{0}=0,\;\;a_{1}=0,\;\;a_{2}=a_{2},\;\;b_{1}=\pm\frac{a_{2}\sqrt{\alpha+1}}{\sqrt{\gamma}},\;\;b_{2}=0,\;\;\omega=\omega.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$u_{21}(x,t)=a_{2}\tanh^{2}(cx+\omega
t)\pm\frac{a_{2}\sqrt{\alpha+1}}{\sqrt{\gamma}}\coth(cx-\omega t).$ (35)

Family 22: For $\alpha\neq\pm 1$, $c\neq 0$ and $\gamma\neq 0$:

$a_{0}=\frac{3\alpha^{3}-2\gamma
c^{2}}{c\gamma(\alpha^{2}-1)},\;\;a_{1}=0,\;\;a_{2}=-a_{0},\;\;b_{1}=0,\;b_{2}=0,\;\;\omega=\frac{3a_{0}\alpha}{4\gamma
c^{2}(\alpha-1)}.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$u_{22}(x,t)=\frac{3\alpha^{3}-2\gamma
c^{2}}{c\gamma(\alpha^{2}-1)}-\frac{3\alpha^{3}-2\gamma
c^{2}}{c\gamma(\alpha^{2}-1)}\tanh^{2}(cx+\omega t).$ (36)

Family 23: For $\alpha\neq 1$, $c\neq 0$ and $\gamma\neq 0$:

$a_{0}=\frac{3a_{1}^{4}\alpha^{2}-2\gamma
c^{4}}{4c^{2}a_{1}\gamma(\alpha-1)},\;\;a_{1}=a_{1}\neq
0,\;\;a_{2}=0,\;\;b_{1}=0,\;b_{2}=-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1},\;\;\omega=\omega.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$u_{23}(x,t)=\frac{3a_{1}^{4}\alpha^{2}-2\gamma
c^{4}}{4c^{2}a_{1}\gamma(\alpha-1)}+a_{1}\tanh(cx+\omega
t)-\frac{2c^{2}(2\gamma-\alpha-1)}{\alpha-1}\coth^{2}(cx-\omega t).$ (37)

Family 24: For $\alpha\neq\pm 1$, $\gamma\neq 0$, $c\neq 0$ and
$2\gamma-\alpha-1\neq 0$:

$a_{0}=a_{0},\;\;a_{1}=a_{1}\;\;a_{2}=-\frac{10c^{2}(2\gamma-\alpha-1)}{3(\alpha-1)},\;\;b_{1}=\frac{3a_{1}\sqrt{3a_{0}\gamma-\alpha-1}}{\sqrt{2c\gamma}},$
$b_{2}=-\frac{2a_{1}c^{2}(2\gamma-\alpha-1)}{\alpha^{2}-1},\;\;\omega=\frac{a_{0}c^{3}\alpha}{3(2\gamma-\alpha-1)^{2}}.$

Therefore, proceeding as in the previous cases, the set of solutions for this
family is provided by

$\begin{split}u_{24}(x,t)&=a_{0}+a_{1}\tanh(cx-\omega
t)-\frac{10c^{2}(2\gamma-\alpha-1)}{3(\alpha-1)}\tanh^{2}(cx-\omega t)\\\
&+\frac{3a_{1}\sqrt{3a_{0}\gamma-\alpha-1}}{\sqrt{2c\gamma}}\coth(cx-\omega
t)-\frac{2a_{1}c^{2}(2\gamma-\alpha-1)}{\alpha^{2}-1}\coth^{2}(cx-\omega
t).\end{split}$ (38)

As we can see in [21, 22, 23, 24, 25] and its references, the technique
proposed here has been effectively applied by various authors to solve
problems involving shallow water waves.

##  4 Graphical presentation of solutions and discussion

In this part, we will exhibit in 3D and 2D some of the solutions revealed in
the previous section in order to study the evolution of traveling waves and
their dependence on $\alpha$, $\gamma$, and other relevant parameters.

Example 1. To illustrate, let us consider the Family 8 with the parameters
$\alpha=1$, $\gamma=1$, and $c=0.5$. Which allows us to determine that
$a_{2}=-1.87$ and $a_{0}=1.875$. In addition to the free coefficients we
choose them as $a_{1}=0$, $b_{1}=0$, $b_{2}=0$ and $\omega=0.25$. Figure 1
shows the 3D and 2D bell-shaped solution $u_{8}(x,t)$ for these parameters.
Example 2. To illustrate, let us consider the Family 10 with the parameters
$\alpha=-1$, $\gamma=-1$, and $c=3.5$. Which allows us to determine that
$a_{2}=-4.08$ and $a_{0}=4.08$. In addition to the free coefficients we choose
them as $a_{1}=0$, $b_{1}=0$, $b_{2}=0$ and $\omega=0.33$. Figure 2 shows the
3D and 2D bell-shaped solution $u_{10}(x,t)$ for these parameters.
Example 3. To illustrate, let us consider the Family 22 with the parameters
$\alpha=4$, $\gamma=2$, and $c=1.2$. Which allows us to determine that
$a_{0}=1.61$, $a_{2}=-1.61$ and $\omega=0.56$. Figure 3 shows the 3D and 2D
bell-shaped solution $u_{22}(x,t)$ for these parameters.
Example 4. To illustrate, let us consider the Family 4 with the parameters
$\alpha=2$, $\gamma=1.5$, and $c=2.4$. Which allows us to determine that
$a_{2}=3.52$, $b_{2}=0$ and $\omega=2.14$. In addition to the free
coefficients we choose them as $a_{0}=4.7$ and $a_{1}=4.7$. Figure 4 shows the
3D and 2D kink-shaped solution $u_{4}(x,t)$ for these parameters.
Example 5. To illustrate, let us consider the Family 13 with the parameters
$\alpha=3.01$, $\gamma=2.2$, and $c=2.6$. Which allows us to determine that
$a_{0}=8.22$ and $b_{1}=6.51$. In addition to the free coefficients we choose
them as $a_{1}=5.7$, $a_{2}=2.7$ and $\omega=6.0$. Figure 5 shows the 3D and
2D singular traveling wave solution $u_{13}(x,t)$ for these parameters.
Example 6. To illustrate, let us consider the Family 5 with the parameters
$\alpha=3.4$, $\gamma=2.2$, and $c=-2.4$. Which allows us to determine that
$a_{2}=-4.73$ and $b_{2}=0$. In addition to the free coefficients we choose
them as $a_{0}=5.2$, $a_{1}=2.8$ and $\omega=-5.8$. Figure 6 shows the 3D and
2D anti-kink-shaped solution $u_{5}(x,t)$ for these parameters.
Example 7. To illustrate, let us consider the Family 16 with the parameters
$\alpha=5.5$, $\gamma=2.0$, and $c=6.0$. Which allows us to determine that
$\omega=4.15$. In addition to the free coefficients we choose them as
$a_{0}=23$, $a_{1}=-18.6$, $a_{2}=0$, $b_{1}=0$ and $b_{2}=1.05$. Figure 7
shows the 3D and 2D singular-shaped solution $u_{16}(x,t)$ for these
parameters.
Since $\alpha=1$ and $\gamma=1$, Family 8 studied in Example 1 represents a
set of solutions for the SIdV Eq. (3), whereas Family 10 studied in Example 2
represents a set of solutions for the KdV Eq. (1). In addition, the Family 22
examined in Example 3 is a solution set for the gsiaKdV equation (4), because
$\alpha\neq\pm 1$ and $\gamma\neq 0$. These three examples illustrate that the
proposed technique confirms what has been stated by previous researchers [3,
14, 17, 18], namely that Eqs. (1), (3) and (4) have solutions of the bell-
shaped type (2). Finally, the families considered in Examples 4, 5, 6, and 7
are solution sets for the gsiaKdV equation (4), because $\alpha\neq\pm 1$ and
$\gamma\neq 0$, and it is shown that Eq. (4) admits traveling wave solutions
of the kink, anti-kink, and singular anti-kink varieties [17].

Figure 1: Solution profile $u_{8}(x,t)$ in the interval $-5.0\leq x\leq 5.0$,
for the parameters selected in Example 1 (left). Wavefront contour plot
(center), and 2D plot of traveling wave solution $u_{8}(x,t)$ for $t=0.0$ and
$t=4.0$ (right). Figure 2: Solution profile $u_{10}(x,t)$ in the interval
$-1.5\leq x\leq 1.5$, for the parameters selected in Example 2 (left).
Wavefront contour plot (center), and 2D plot of traveling wave solution
$u_{10}(x,t)$ for $t=0.0$ and $t=4.0$ (right). Figure 3: Solution profile
$u_{22}(x,t)$ in the interval $-3.0\leq x\leq 3.0$, for the parameters
selected in Example 3 (left). Wavefront contour plot (center), and 2D plot of
traveling wave solution $u_{22}(x,t)$ for $t=0.0$ and $t=3.0$ (right). Figure
4: Solution profile $u_{4}(x,t)$ in the interval $-5.0\leq x\leq 5.0$, for the
parameters selected in Example 4 (left). Wavefront contour plot (center), and
2D plot of traveling wave solution $u_{4}(x,t)$ for $t=0.0$ and $t=3.0$
(right). Figure 5: Solution profile $u_{13}(x,t)$ in the interval $-3.0\leq
x\leq 3.0$, for the parameters selected in Example 5 (left). Wavefront contour
plot (center), and 2D plot of traveling wave solution $u_{13}(x,t)$ for
$t=0.0$ and $t=1.0$ (right). Figure 6: Solution profile $u_{5}(x,t)$ in the
interval $-5.0\leq x\leq 5.0$, for the parameters selected in Example 6
(left). Wavefront contour plot (center), and 2D plot of traveling wave
solution $u_{5}(x,t)$ for $t=0.0$ and $t=1.0$ (right). Figure 7: Solution
profile $u_{16}(x,t)$ in the interval $-5.0\leq x\leq 5.0$, for the parameters
selected in Example 16 (left). Wavefront contour plot (center), and 2D plot of
traveling wave solution $u_{16}(x,t)$ for $t=0.0$ (right).

##  5  Conclusions

The gsiaKdV equation is a generalization of both the KdV and SIdV equations;
the mathematical model includes the term
$\big{(}3(1-\alpha)u+(1+\alpha)\frac{u_{xx}}{u}\big{)}$ can be seen as an
advecting velocity. In this paper, we study the gsiaKdV equation for the first
time using the tanh-coth method, and the results revealed that this
generalization shares bell-shaped solutions with both the KdV equation and the
SIdV equation, as reported previously by other authors, also has other kinds
of solutions that could be of interest in the study of shallow wave motion
over the ocean. We believe that the research of novel traveling wave solutions
will shed light on new sorts of applications of this equation in applied
mathematics and engineering, as well as depict the specific behaviour of
atmospheric phenomena caused by shallow ocean waves. In future study, other
exact solution methods may be used to the studied equation.

## Data availability statement

Our article have no associated data.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Funding Information

The authors declare that no funds, grants, or other support were received
during the preparation of this manuscript.

## References

  * [1] Korteweg DJ, de Vries G. On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary wave. Philos Mag 1895; 39: 422–443.
  * [2] Helal MA, Mehanna MS. A comparative study between two different methods for solving the general Korteweg-de Vries equation (GKdV). Chaos, Solitons and Fractals 2007; 33: 725–739.
  * [3] Sen A, Ahalpara DP, Thyagaraja A, Krishnaswami GS. A KdV-like advection–dispersion equation with some remarkable properties. Commun Nonlinear Sci Numer Simul 2012; 17: 4115-4124.
  * [4] Wazwaz AM. Integrable $(3+1)-$dimensional Ito equation: variety of lump solutions and multiple-soliton solutions. Nonlinear Dyn 2022; 109; 1929-1934.
  * [5] Wazwaz AM. New $(3+1)-$dimensional Painlevé integrable fifth-order equation with third-order temporal dispersion. Nonlinear Dyn 2021; 106; 891–897.
  * [6] Wazwaz AM. Multi-soliton solutions for integrable $(3+1)-$dimensional modified seventh-order Ito and seventh-order Ito equations. Nonlinear Dyn 2022; https://doi.org/10.1007/s11071-022-07818-4
  * [7] Wazwaz AM. A new integrable nonlocal modified KdV equation: Abundant solutions with distinct physical structures. J Ocean Eng Sci 2017; 2: 1-4.
  * [8] Wazwaz AM. New sets of solitary wave solutions to the KdV, mKdV, and the generalized KdV equations. Commun Nonlinear Sci Numer Simul 2008; 13: 331-339.
  * [9] Mirzazadeh M, Eslami M, Biswas A. 1-Soliton solution of KdV6 equation. Nonlinear Dyn 2015; 80; 387-396.
  * [10] González-Gaxiola O, León-Ramírez A, Chacón-Acosta G. Application of the Kudryashov method for finding exact solutions of the Schamel-Kawahara equation. Russian J of Nonlinear Dynamics 2022; 18: 203-215.
  * [11] Kudruashov NA. Lax pair and first integrals of the traveling wave reduction for the KdV hierarchy. Appl Math Comp 2019; 350: 323-330.
  * [12] Kudruashov NA. Painlevé analysis and exact solutions of the Korteweg-de Vries equation with a source. Appl Math Lett 2025; 41: 41-45.
  * [13] Biswas A. Solitary wave solution for the generalized KdV equation with time-dependent damping and dispersion. Commun Nonlinear Sci Numer Simul 2009; 14: 3503–3506.
  * [14] Leal da Silva P, Freire IL, Sampaio JCS. A family of wave equations with some remarkable properties. Proc R Soc A 2018; 474: 20170763.
  * [15] Zhang G, He J, Wang L, Mihalache D. Kink-type solutions of the SIdV equation and their properties. R Soc open sci 2019; 6: 191040.
  * [16] Qiao Z, Fan E. Negative-order Korteweg-de Vries equations. Phys Rev E 2012; 86: 016601.
  * [17] Alzaleq L, Manoranjan V, Alzalg B. Exact traveling waves of a generalized scale-invariant analogue of the Korteweg-de Vries equation. Mathematics 2022; 10: 414.
  * [18] Fan X, Yin J. Two types of traveling wave solutions of a KdV-like advection-dispersion equation. Mathematica Aeterna 2012; 2: 273-282.
  * [19] Malfliet W, Hereman W. The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Physica Scripta 1996; 54: 563-568.
  * [20] Malfliet W, Hereman W. The tanh method: II. Perturbation technique for conservative systems. Physica Scripta 1996; 54: 569-575.
  * [21] Gözükızıl Ö F, Akçağıl Ş. The tanh-coth method for some nonlinear pseudoparabolic equations with exact solutions. Adv Differ Equ 2013; 2013: 143.
  * [22] Wazwaz AM. The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations. Appl Math Comput 2007; 184: 1002-1014.
  * [23] Wazwaz AM. The Hirota’s direct method and the tanh-coth method for multiple soliton solutions of the Sawada-Kotera-Ito seventh-order equation. Appl Math Comput 2008; 199: 133-138.
  * [24] Gomez Sierra CA, Salas AH. The generalized tanh-coth method to special types of the fifth-order KdV equation. Appl Math Comput 2008; 203: 873-880.
  * [25] Gomez Sierra CA. On a KdV equation with higher-order nonlinearity: Traveling wave solutions. J Comput Appl Math 2011; 235: 5330-5332.


#  Self-QA: Unsupervised Knowledge Guided
Language Model Alignment

Xuanyu Zhang and Qing Yang

Du Xiaoman Financial

###### Abstract

Large-scale language models like ChatGPT and GPT-4 have gained attention for
their impressive conversational and generative capabilities. However, the
creation of supervised paired question-answering data for instruction tuning
presents formidable challenges. This endeavor necessitates substantial human
effort for data annotation and wrestles with issues concerning data quality,
diversity, accuracy, and other related factors. To overcome these obstacles,
we introduce an innovative framework named Self-QA, which replaces the
traditional practice of human-written instruction seeds with a vast amount of
unsupervised knowledge, enabling the model to generate a larger quantity of
correct and domain-specific instruction data. The effectiveness of our
proposed method is demonstrated through experiments conducted on unsupervised
corpora from various domains.

##  1 Introduction

With the emergence of GPT-based Radford et al. (2018) large-scale models like
InstructGPT Ouyang et al. (2022), ChatGPT OpenAI (2022) and GPT-4 OpenAI
(2023), their remarkable conversational and generative capabilities have
garnered widespread attention. These models not only have the capacity to
understand complex language structures and grasp subtle meanings but also
possess the remarkable capability to interact naturally and fluently with
users, generating text that is both coherent and highly creative. This has
pushed the boundaries of what was previously deemed impossible. The impact of
these large-scale models extends beyond the academic realm of natural language
processing (NLP) and has a profound influence in the domains of business and
industry. They have opened up new possibilities for human-machine
interactions, intelligent customer service, and virtual assistant
applications, revolutionizing these fields and paving the way for innovation
and advancement.

Figure 1: The pipeline of Self-QA.

Despite the impressive capabilities of ChatGPT, constructing supervised fine-
tuning (SFT) data for instruction tuning presents significant challenges. The
human effort required for annotating data, along with issues related to data
quality, diversity, accuracy, and others, hinder the development of this
technique. Although Self-Instruct Wang et al. (2022) has been proposed to
mitigate this issue, it still relies on a small set of human-written seed
instructions for guidance. Furthermore, the method is limited in its ability
to control the domain coverage of generated instruction data and ensure the
correctness of the generated answers. Consequently, there is a vast amount of
untapped potential in utilizing the abundant unsupervised data, particularly
domain-specific expertise.

Model | Prompt | Domain | Correctness
---|---|---|---
customization | guarantee
Self-Instruct Wang et al. (2022) | 176 human-written seeds | × | ×
Self-Align Sun et al. (2023) | 195 human-written seeds | × | ×
Self-Chat Xu et al. (2023) | 111,502 supervised dialogues | × | ×
Self-QA (ours) | Unsupervised knowledge | ✓ | ✓
Table 1: Comparison of different self-alignment methods.

Therefore, in this paper, we introduce Self-QA, a framework to generate SFT
data from unsupervised knowledge, inspired by the human self-questioning
learning approach. Self-QA replaces manually written seeds used in other self-
alignment models Wang et al. (2022); Sun et al. (2023); Xu et al. (2023) with
a vast amount of unsupervised knowledge, alleviating the difficulty of
language models in generating instruction data according to specific
requirements. As shown in Figure 1, the unsupervised data are used
sequentially in the stage of knowledge-guided instruction generation and
machine reading comprehension. Self-QA not only reduces the reliance on human
annotators but also allows for the generation of diverse, correct, and domain-
specific instruction data. Experiments with unsupervised corpora from various
domains demonstrate the effectiveness of our proposed method.

##  2 Related Work

#### Language Models with Instruction-tuning

Recently, numerous studies Ouyang et al. (2022); Peng et al. (2023) have
investigated the effectiveness of language models in following instructions by
leveraging annotated instructional data. This approach enables the model to
learn to identify and extract relevant information from different types of
instructions and use it to generate accurate and relevant responses. It
enhances the model’s ability to understand complex instructions and generalize
to new tasks by exposing it to a wide range of instructional scenarios.
However, the reliance on human annotation in creating such instructional
datasets presents a bottleneck for scaling up and achieving broader
applicability of instruction-guided language models. To address this
limitation, researchers have explored alternative approaches that reduce the
need for extensive human involvement in generating instruction data.

#### Bootstrapped Instruction Generation

Bootstrapped instruction generation is a recently proposed class of methods
Wang et al. (2022); Sun et al. (2023); Xu et al. (2023) that reduces the cost
of human instruction annotation. For example, Self-Instruct Wang et al. (2022)
is proposed to enhance the ability of pre-trained language models to follow
instructions by utilizing their own generated samples. This technique involves
generating a set of instruction, input, and output samples from the
instruction seeds, and then carefully pruning them before fine-tuning the
model. Self-Align Sun et al. (2023) primarily employs topic-guided red-teaming
self-instruct and principle-driven self-alignment to tackle the challenges
associated with heavy human annotations. It aims to develop AI agents capable
of generating helpful, ethical, and reliable responses to user queries,
including adversarial ones, while proactively addressing harmful inquiries in
a non-evasive manner. However, as shown in Table 1, these methods often
require a small amount of supervised seed information. The instructions
generated by them cannot specify domains and content, nor can they ensure the
accuracy and professionalism of the instruction responses. Different from
them, our approach can effectively address these issues by leveraging
unsupervised knowledge.

Figure 2: Examples of transformation of unsupervised structured data.

#### Question Generation and Answering

Question generation and question answering are two closely related tasks in
natural language processing. They can be viewed as a dual problem, where the
former involves creating questions from a given passage or set of information,
and the latter involves answering questions based on a given passage or set of
information. Especially, the technique of machine reading comprehension (MRC)
Zhang (2019); Zhang and Wang (2020) is often used for question answering. For
humans, self-questioning and self-answering learning entail stimulating
individuals to formulate their own questions and answers based on the provided
information, followed by comparing their responses to the original knowledge.
This approach has showcased encouraging outcomes in augmenting individuals’
understanding of the provided information Joseph et al. (2016). For domain-
specific instruction samples, instruction and input can often be considered as
a whole. Therefore, in this paper, we assume that instructions are equivalent
to questions, and instruction outputs are equivalent to answers.

##  3 Methodology

Our proposed Self-QA consists of three different stages: knowledge-guided
instruction generation, machine reading comprehension, and filtering and
pruning.

#### Knowledge-Guided Instruction Generation

In this stage, we employ the language model itself to generate instructions
according to unsupervised text. This approach makes the generated instructions
domain-specific and content-relevant to the unsupervised text provided.
However, in the process of training and inference, instructions are fed to
language models without background knowledge, so we need to provide some
guidelines so that these instructions cannot rely on and refer to the content
in the original text. For instance, the prompt can be:

Instruction Generation Prompt The background knowledge is: {unsupervised
knowledge data} Please generate ten instruction questions as diverse as
possible based on the content of the above article. These questions can be
questions about facts or an understanding and evaluation of relevant content.
Please assume that there is no corresponding article to refer to when asking
questions, so do not use demonstrative pronouns such as “this” or “these” in
the question. Please generate questions in the following format: 1\. Question:
… 2\. Question: …

Then we can obtain several related instructions, which can be used in the next
stage. {unsupervised knowledge data} in the prompt represents sequential text.
Unstructured knowledge, such as web pages and book data, can be used directly
after undergoing cleaning processes. Structured data such as tables and
knowledge graphs Zhang et al. (2022) need to be converted into unstructured
textual data before they can be utilized. As shown in Figure 2, this can be
achieved by filling slots using templates or by concatenating each data entry
with its corresponding attribute name.

#### Machine Reading Comprehension

In this stage, the language model needs to generate answers to the generated
instruction questions according to the corresponding unsupervised knowledge.
The process can be formulated as follows:

$P({\bf A}|{\bf K},{\bf Q})=\prod_{j}P({\bf A}_{i}|{\bf A}_{\leq i},{\bf
K},{\bf Q})$ (1)

where ${\bf k,Q,A}$ represents unsupervised knowledge, instruction question,
and answer, separately. Because the whole process is the same as that of
reading comprehension, we also call this stage by this name. As in the
previous stage, the prompt for the reading comprehension stage is as follows:

Reading Comprehension Prompt The background knowledge is: {unsupervised
knowledge data} Please answer the following question based on the content of
the article above: {the generated question} Please answer this question as
thoroughly as possible, but do not change the key information in the original
text, and do not include expressions such as “based on the above article” in
the answer. Please generate the corresponding answer in the following format:
Question: … Answer: …

#### Filtering and Pruning

Although we explicitly instruct the model to assume no prior knowledge from
external documents and prohibit the use of demonstrative pronouns like “this”
in generated questions and the phrase like “based on the above content” in
generated answers, we still observed that the language model still produces
text that violates these rules. Additionally, the generated instances of
instructions also exhibit cases where they do not adhere to the required
format and become unparseable. Therefore, it is necessary to further filter
out these problematic examples.

To mitigate these issues, we implement a post-processing step to filter out
inappropriate responses and correct any formatting errors. This involves
developing heuristics and rule-based methods to identify and remove instances
that violate the instructed constraints. By applying these filters, we ensure
that the generated text adheres to the predefined guidelines and maintains the
desired level of correctness and coherence.

##  4 Discussion

###  4.1 Performance

We collect several domains of unsupervised unstructured and structured data
for experiments. An example of unsupervised knowledge and generated
instruction questions and answers are shown in Table 2. We then instruction-
tuning BLOOM-7B Scao et al. (2022) with these generated instructions. As shown
in Table 3, our model can answer the corresponding question correctly, but
ChatGPT gives a wrong answer. It is precisely because of these domain-specific
instruction-tuning data that our model has achieved better performance.

Knowledge: |  Company: DXM Founding Date: April 28, 2018 Formerly known as: Baidu Financial Headquarters Address: Haidian District, Beijing, China.
---|---
Question1: |  When was DXM founded?
Answer1: |  DXM was founded on April 28, 2018.
Question2: |  Where is the headquarters of DXM located?
Answer2: |  The headquarters of DXM is located at Haidian District, Beijing, China.
Table 2: Examples of unsupervised background knowledge and generated question and answer pairs. Human: |  Where is DXM?
---|---
ChatGPT: |  The headquarters of DXM is located in Hangzhou, China.
Our Model: |  DXM is a financial technology company headquartered in Haidian District, Beijing, China.
Table 3: Answers of different models.

###  4.2 Different Stages of Self-QA

The stage of knowledge-guided instruction generation and machine reading
comprehension can also be integrated into a single stage so that the model
only needs to be invoked once for each round of instruction generation and
answer prediction. The advantage of this is that the number of calls to the
model is reduced, because each round of instruction question and answer
generation only needs language models once. However, there are also potential
drawbacks to this approach. For instance, the model may generate output that
exceeds the predetermined length. Additionally, by combining these two tasks,
the model may not be able to focus on a single task as effectively, which can
result in less detailed and accurate answers. Therefore, the decision to
integrate two stages into a single stage should be made with careful
consideration of the specific application and task requirements.

###  4.3 Different Forms of Knowledge

In general, knowledge can be stored in large language models in a parametric
manner or separately input into the models in an explicit symbolic form. The
main focus of this paper is on how to store unsupervised knowledge in large
models using a parameterized approach. This approach enables end-to-end
processing of user questions and optimization of model parameters without the
need for external information. It offers a high level of flexibility and
adaptability to different inputs and contexts. However, this approach also
comes with potential biases and errors that can be present in the data.
Therefore, it is crucial to provide comprehensive and accurate knowledge
during the training phase to mitigate the impact of such biases on the model.
On the other hand, explicit symbolic knowledge requires the existence of
corresponding retrieval and query systems. Additionally, the model needs to
make judgments on whether to adopt the content of external knowledge. This
makes the entire process more complex.

##  5 Conclusion

In this paper, we introduced Self-QA, a framework for generating instruction-
tuning data from unsupervised knowledge. The unsupervised data are used
sequentially in the stage of knowledge-guided instruction generation and
machine reading comprehension. Our experiments demonstrate the effectiveness
of Self-QA in generating diverse, correct, and domain-specific instruction
data. By reducing the reliance on human annotators, Self-QA offers a promising
approach for improving the efficiency and scalability of instruction tuning.

## References

  * Joseph et al. (2016) Laurice M Joseph, Sheila Alber-Morgan, Jennifer Cullen, and Christina Rouse. 2016\.  The effects of self-questioning on reading comprehension: A literature review.  _Reading & Writing Quarterly_, 32(2):152–173.
  * OpenAI (2022) OpenAI. 2022.  Chatgpt.
  * OpenAI (2023) OpenAI. 2023.  Gpt-4 technical report.
  * Ouyang et al. (2022) Long Ouyang, Jeff Wu, Xu Jiang, Diogo Almeida, Carroll L. Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex Ray, John Schulman, Jacob Hilton, Fraser Kelton, Luke Miller, Maddie Simens, Amanda Askell, Peter Welinder, Paul Christiano, Jan Leike, and Ryan Lowe. 2022.  Training language models to follow instructions with human feedback.
  * Peng et al. (2023) Baolin Peng, Chunyuan Li, Pengcheng He, Michel Galley, and Jianfeng Gao. 2023.  Instruction tuning with gpt-4.  _arXiv preprint arXiv:2304.03277_.
  * Radford et al. (2018) Alec Radford, Karthik Narasimhan, Tim Salimans, Ilya Sutskever, et al. 2018.  Improving language understanding by generative pre-training.
  * Scao et al. (2022) Teven Le Scao, Angela Fan, Christopher Akiki, Ellie Pavlick, Suzana Ilić, Daniel Hesslow, Roman Castagné, Alexandra Sasha Luccioni, François Yvon, Matthias Gallé, et al. 2022.  Bloom: A 176b-parameter open-access multilingual language model.  _arXiv preprint arXiv:2211.05100_.
  * Sun et al. (2023) Zhiqing Sun, Yikang Shen, Qinhong Zhou, Hongxin Zhang, Zhenfang Chen, David Cox, Yiming Yang, and Chuang Gan. 2023.  Principle-driven self-alignment of language models from scratch with minimal human supervision.  _arXiv preprint arXiv:2305.03047_.
  * Wang et al. (2022) Yizhong Wang, Yeganeh Kordi, Swaroop Mishra, Alisa Liu, Noah A Smith, Daniel Khashabi, and Hannaneh Hajishirzi. 2022.  Self-instruct: Aligning language model with self generated instructions.  _arXiv preprint arXiv:2212.10560_.
  * Xu et al. (2023) Canwen Xu, Daya Guo, Nan Duan, and Julian McAuley. 2023.  Baize: An open-source chat model with parameter-efficient tuning on self-chat data.  _arXiv preprint arXiv:2304.01196_.
  * Zhang (2019) Xuanyu Zhang. 2019.  MC^2: Multi-perspective convolutional cube for conversational machine reading comprehension.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 6185–6190, Florence, Italy. Association for Computational Linguistics.
  * Zhang and Wang (2020) Xuanyu Zhang and Zhichun Wang. 2020.  Rception: Wide and deep interaction networks for machine reading comprehension (student abstract).  _Proceedings of the AAAI Conference on Artificial Intelligence_ , 34(10):13987–13988.
  * Zhang et al. (2022) Xuanyu Zhang, Qing Yang, and Dongliang Xu. 2022.  TranS: Transition-based knowledge graph embedding with synthetic relation representation.  In _Findings of the Association for Computational Linguistics: EMNLP 2022_ , pages 1202–1208, Abu Dhabi, United Arab Emirates. Association for Computational Linguistics.


* [1] Particle Data Group collaboration, _Review of Particle Physics_ , _PTEP_ 2022 (2022) 083C01.
  * [2] Y.-S. Tsai, _Decay Correlations of Heavy Leptons in $e^{+}e^{-}\to\ell^{+}\ell^{-}$_, _Phys. Rev. D_ 4 (1971) 2821.
  * [3] R. Shankar, _Determination of the Quark-Gluon Coupling Constant_ , _Phys. Rev. D_ 15 (1977) 755.
  * [4] E. G. Floratos, S. Narison and E. de Rafael, _Spectral Function Sum Rules in Quantum Chromodynamics. 1. Charged Currents Sector_ , _Nucl. Phys. B_ 155 (1979) 115.
  * [5] O. Nachtmann and W. Wetzel, _Quantum Chromodynamics and the Decay of the $\tau$ Lepton_, _Phys. Lett. B_ 81 (1979) 229.
  * [6] S. Narison and E. de Rafael, _Spectral Function Sum Rules in Quantum Chromodynamics. 2. Flavor Components of the Electromagnetic Current_ , _Nucl. Phys. B_ 169 (1980) 253.
  * [7] N. V. Krasnikov, A. A. Pivovarov and N. N. Tavkhelidze, _The Use of Finite Energy Sum Rules for the Description of the Hadronic Properties of QCD_ , _Z. Phys. C_ 19 (1983) 301.
  * [8] K. Schilcher and M. D. Tran, _Duality in Semileptonic $\tau$ Decay_, _Phys. Rev. D_ 29 (1984) 570.
  * [9] R. A. Bertlmann, G. Launer and E. de Rafael, _Gaussian Sum Rules in Quantum Chromodynamics and Local Duality_ , _Nucl. Phys. B_ 250 (1985) 61.
  * [10] E. Braaten, _QCD Predictions for the Decay of the tau Lepton_ , _Phys. Rev. Lett._ 60 (1988) 1606.
  * [11] E. Braaten, _The Perturbative QCD Corrections to the Ratio R for tau Decay_ , _Phys. Rev. D_ 39 (1989) 1458.
  * [12] S. Narison and A. Pich, _QCD Formulation of the tau Decay and Determination of Lambda (MS)_ , _Phys. Lett. B_ 211 (1988) 183.
  * [13] E. Braaten, S. Narison and A. Pich, _QCD analysis of the tau hadronic width_ , _Nuclear Physics B_ 373 (1992) 581.
  * [14] E. C. Poggio, H. R. Quinn and S. Weinberg, _Smearing the Quark Model_ , _Phys. Rev. D_ 13 (1976) 1958.
  * [15] I. I. Y. Bigi, M. A. Shifman, N. Uraltsev and A. I. Vainshtein, _Heavy flavor decays, OPE and duality in two-dimensional ’t Hooft model_ , _Phys. Rev. D_ 59 (1999) 054011 [hep-ph/9805241].
  * [16] O. Catà, M. Golterman and S. Peris, _Duality violations and spectral sum rules_ , _Journal of High Energy Physics_ 08 (2005) 076 [hep-ph/0506004].
  * [17] O. Catà, M. Golterman and S. Peris, _Unraveling duality violations in hadronic tau decays_ , _Phys. Rev. D_ 77 (2008) 093006 [0803.0246].
  * [18] ALEPH collaboration, _Measurement of the spectral functions of vector current hadronic tau decays_ , _Z. Phys. C_ 76 (1997) 15.
  * [19] ALEPH collaboration, _Measurement of the spectral functions of axial-vector hadronic tau decays and determination of $\alpha_{s}(M_{\tau}^{2})$_, _Eur. Phys. J. C_ 4 (1998) 409.
  * [20] OPAL collaboration, _Measurement of the strong coupling constant alpha(s) and the vector and axial vector spectral functions in hadronic tau decays_ , _Eur. Phys. J. C_ 7 (1999) 571 [hep-ex/9808019].
  * [21] F. Le Diberder and A. Pich, _Testing QCD with tau decays_ , _Physics Letters B_ 289 (1992) 165.
  * [22] A. Pich and A. Rodríguez-Sánchez, _Determination of the QCD coupling from ALEPH $\tau$ decay data_, _Physical Review D_ 94 (2016) 034027 [1605.06830].
  * [23] C. Ayala, G. Cvetic and D. Teca, _Borel–Laplace sum rules with $\tau$ decay data, using OPE with improved anomalous dimensions_, _J. Phys. G_ 50 (2023) 045004 [2206.05631].
  * [24] D. Boito, O. Catà, M. Golterman, M. Jamin, K. Maltman, J. Osborne et al., _A new determination of $\alpha_{s}$ from hadronic $\tau$ decays_, _Physical Review D_ 84 (2011) 113006 [1110.1127].
  * [25] D. Boito, M. Golterman, M. Jamin, A. Mahdavi, K. Maltman, J. Osborne et al., _An Updated determination of $\alpha_{s}$ from $\tau$ decays_, _Phys. Rev. D_ 85 (2012) 093015 [1203.3146].
  * [26] D. Boito, M. Golterman, K. Maltman, J. Osborne and S. Peris, _Strong coupling from the revised ALEPH data for hadronic $\tau$ decays_, _Physical Review D_ 91 (2015) 034003 [1410.3528].
  * [27] D. Boito, M. Golterman, K. Maltman, S. Peris, M. V. Rodrigues and W. Schaaf, _Strong coupling from an improved $\tau$ vector isovector spectral function_, _Physical Review D_ 103 (2021) 034028 [2012.10440].
  * [28] D. Boito, I. Caprini, M. Golterman, K. Maltman and S. Peris, _Hyperasymptotics and quark-hadron duality violations in QCD_ , _Physical Review D_ 97 (2018) 054007 [1711.10316].
  * [29] D. Boito, M. Golterman, K. Maltman and S. Peris, _Strong coupling from hadronic $\tau$ decays: A critical appraisal_, _Phys. Rev. D_ 95 (2017) 034024 [1611.03457].
  * [30] A. Pich, _Tau-decay determination of the strong coupling_ , _SciPost Phys. Proc._ 1 (2019) 036 [1811.10067].
  * [31] A. Pich, _Precision physics with QCD_ , _EPJ Web Conf._ 137 (2017) 01016 [1612.05010].
  * [32] A. Pich, _Precision physics with inclusive QCD processes_ , _Prog. Part. Nucl. Phys._ 117 (2021) 103846 [2012.04716].
  * [33] A. Pich and A. Rodriguez-Sanchez, _Violations of quark-hadron duality in low-energy determinations of $\alpha_{s}$_, _JHEP_ 07 (2022) 145 [2205.07587].
  * [34] A. A. Pivovarov, _Renormalization group analysis of the tau lepton decay within QCD_ , _Zeitschrift für Physik C_ 53 (1992) 461 [hep-ph/0302003].
  * [35] F. Le Diberder and A. Pich, _The perturbative QCD prediction to R(tau) revisited_ , _Phys. Lett. B_ 286 (1992) 147.
  * [36] A. Pich, _Precision Tau Physics_ , _Prog. Part. Nucl. Phys._ 75 (2014) 41 [1310.7922].
  * [37] M. Beneke and M. Jamin, _$\alpha_{s}$ and the tau hadronic width: fixed-order, contour-improved and higher-order perturbation theory_, _Journal of High Energy Physics_ 09 (2008) 044 [0806.3156].
  * [38] M. Beneke, D. Boito and M. Jamin, _Perturbative expansion of tau hadronic spectral function moments and $\alpha_{s}$ extractions_, _Journal of High Energy Physics_ 01 (2013) 125 [1210.8038].
  * [39] A. H. Hoang and C. Regner, _Borel representation of $\tau$ hadronic spectral function moments in contour-improved perturbation theory_, _Physical Review D_ 105 (2022) 096023 [2008.00578].
  * [40] A. H. Hoang and C. Regner, _On the difference between FOPT and CIPT for hadronic tau decays_ , _European Physical Journal Special Topics_ 230 (2021) 2625 [2105.11222].
  * [41] M. A. Benitez-Rathgeb, D. Boito, A. H. Hoang and M. Jamin, _Reconciling the contour-improved and fixed-order approaches for $\tau$ hadronic spectral moments. Part I. Renormalon-free gluon condensate scheme_, _Journal of High Energy Physics_ 07 (2022) 016 [2202.10957].
  * [42] M. A. Benitez-Rathgeb, D. Boito, A. H. Hoang and M. Jamin, _Reconciling the contour-improved and fixed-order approaches for $\tau$ hadronic spectral moments. Part II. Renormalon norm and application in $\alpha$s determinations_, _Journal of High Energy Physics_ 09 (2022) 223 [2207.01116].
  * [43] N. G. Gracia, A. H. Hoang and V. Mateu, _Mathematical aspects of the asymptotic expansion in contour improved perturbation theory for hadronic tau decays_ , _Phys. Rev. D_ 108 (2023) 034013 [2305.10288].
  * [44] M. Golterman, K. Maltman and S. Peris, _Difference between fixed-order and contour-improved perturbation theory_ , _Phys. Rev. D_ 108 (2023) 014007 [2305.10386].
  * [45] M. Beneke and H. Takaura, _Gradient-flow renormalon subtraction and the hadronic tau decay series_ , 9, 2023, 2309.10853.
  * [46] J. Huston, K. Rabbertz and G. Zanderighi, _Quantum Chromodynamics_ , 2312.14015.
  * [47] M. Davier, A. Höcker, B. Malaescu, C.-Z. Yuan and Z. Zhang, _Update of the ALEPH non-strange spectral functions from hadronic $\tau$ decays_, _European Physical Journal C_ 74 (2014) 2803 [1312.1501].
  * [48] BaBar collaboration, _Measurements of $e^{+}e^{-}\to K^{+}K^{-}\eta$, $K^{+}K^{-}\pi^{0}$ and $K^{0}_{s}K^{\pm}\pi^{\mp}$ cross- sections using initial state radiation events_, _Phys. Rev. D_ 77 (2008) 092002 [0710.4451].
  * [49] BaBar collaboration, _Measurement of the ${e}^{+}{e}^{{-}}{\rightarrow}{{\pi}}^{+}{{\pi}}^{{-}}{{\pi}}^{0}{{\pi}}^{0}$ cross section using initial-state radiation at BABAR_, _Phys. Rev. D_ 96 (2017) 092009 [1709.01171].
  * [50] M. N. Achasov et al., _Updated measurement of the $e^{+}e^{-}\to\omega\pi^{0}\to\pi^{0}\pi^{0}\gamma$ cross section with the SND detector_, _Phys. Rev. D_ 94 (2016) 112001 [1610.00235].
  * [51] SND collaboration, _Measurement of the $e^{+}e^{-}\to\eta\pi^{+}\pi^{-}$ cross section in the center-of-mass energy range 1.22-2.00 GeV with the SND detector at the VEPP-2000 collider_, _Phys. Rev. D_ 91 (2015) 052013 [1412.1971].
  * [52] M. N. Achasov et al., _Measurement of the $e^{+}e^{-}\to\eta\pi^{+}\pi^{-}$ cross section with the SND detector at the VEPP-2000 collider_, _Phys. Rev. D_ 97 (2018) 012008 [1711.08862].
  * [53] BaBar collaboration, _Study of the process $e^{+}e^{-}\to\pi^{+}\pi^{-}\eta$ using initial state radiation_, _Phys. Rev. D_ 97 (2018) 052007 [1801.02960].
  * [54] BaBar collaboration, _Study of the reactions $e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0}\pi^{0}\pi^{0}\gamma$ and $\pi^{+}\pi^{-}\pi^{0}\pi^{0}\eta\gamma$ at center-of-mass energies from threshold to 4.35 GeV using initial-state radiation_, _Phys. Rev. D_ 98 (2018) 112015 [1810.11962].
  * [55] S. S. Gribanov et al., _Measurement of the $e^{+}e^{-}\rightarrow\eta\pi^{+}\pi^{-}$ cross section with the CMD-3 detector at the VEPP-2000 collider_, _JHEP_ 01 (2020) 112 [1907.08002].
  * [56] BaBar collaboration, _The $e^{+}e^{-}\to 3(\pi^{+}\pi^{-}),2(\pi^{+}\pi^{-}\pi^{0})$ and $K^{+}K^{-}2(\pi^{+}\pi^{-})$ cross sections at center-of-mass energies from production threshold to 4.5-GeV measured with initial-state radiation_, _Phys. Rev. D_ 73 (2006) 052003 [hep-ex/0602006].
  * [57] CMD-3 collaboration, _Study of the process $e^{+}e^{-}\to 3(\pi^{+}\pi^{-})$ in the c.m.energy range 1.5–2.0 gev with the cmd-3 detector_, _Phys. Lett. B_ 723 (2013) 82 [1302.0053].
  * [58] M. N. Achasov et al., _Recent results from SND detector at VEPP-2000 collider_ , _EPJ Web Conf._ 212 (2019) 04002.
  * [59] CMD-3 collaboration, _Study of the process $e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0}\eta$ in the c.m. energy range 1394-2005 MeV with the CMD-3 detector_, _Phys. Lett. B_ 773 (2017) 150 [1706.06267].
  * [60] M. N. Achasov et al., _Measurement of the $e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0}\eta$ cross section below $\sqrt{s}=2$ GeV_, _Phys. Rev. D_ 99 (2019) 112004 [1903.09307].
  * [61] BaBar collaboration, _The $e^{+}e^{-}\to 2(\pi^{+}\pi^{-})\pi^{0},$ $2(\pi^{+}\pi^{-})\eta$, $K^{+}K^{-}\pi^{+}\pi^{-}\pi^{0}$ and $K^{+}K^{-}\pi^{+}\pi^{-}\eta$ Cross Sections Measured with Initial-State Radiation_, _Phys. Rev. D_ 76 (2007) 092005 [0708.2461].
  * [62] M. N. Achasov et al., _Study of the process $e^{+}e^{-}\to\omega\eta\pi^{0}$ in the energy range $\sqrt{s}<2$ GeV with the SND detector_, _Phys. Rev. D_ 94 (2016) 032010 [1606.06481].
  * [63] BaBar collaboration, _Measurement of the spectral function for the $\tau^{-}\to K^{-}K_{S}\nu_{\tau}$ decay_, _Phys. Rev. D_ 98 (2018) 032010 [1806.10280].
  * [64] P. A. Baikov, K. G. Chetyrkin and J. H. Kuhn, _Order $\alpha^{4}_{s}$ QCD Corrections to $Z$ and $\tau$ Decays_, _Physical Review Letters_ 101 (2008) 012002 [0801.1821].
  * [65] F. Herzog, B. Ruijl, T. Ueda, J. A. M. Vermaseren and A. Vogt, _On Higgs decays to hadrons and the R-ratio at N 4LO_, _Journal of High Energy Physics_ 08 (2017) 113 [1707.01044].
  * [66] D. Boito, P. Masjuan and F. Oliani, _Higher-order QCD corrections to hadronic $\tau$ decays from Padé approximants_, _Journal of High Energy Physics_ 08 (2018) 075 [1807.01567].
  * [67] I. Caprini, _Higher-order perturbative coefficients in QCD from series acceleration by conformal mappings_ , _Physical Review D_ 100 (2019) 056019 [1908.06632].
  * [68] M. Jamin, _Higher-order behaviour of two-point current correlators_ , _European Physics Journal Special Topics_ 230 (2021) 2609 [2106.01614].
  * [69] K. G. Chetyrkin, J. H. Kühn, A. Maier, P. Maierhofer, P. Marquard, M. Steinhauser et al., _Addendum to “Charm and bottom quark masses: An update”_ , _Phys. Rev._ D96 (2017) 116007 [1710.04249].
  * [70] F. Herzog, B. Ruijl, T. Ueda, J. A. M. Vermaseren and A. Vogt, _The five-loop beta function of Yang-Mills theory with fermions_ , _Journal of High Energy Physics_ 02 (2017) 090 [1701.01404].
  * [71] D. Boito and F. Oliani, _Renormalons in integrated spectral function moments and $\alpha_{s}$ extractions_, _Physical Review D_ 101 (2020) 074003 [2002.12419].
  * [72] M. A. Shifman, _Quark hadron duality_ , in _8th International Symposium on Heavy Flavor Physics_ , vol. 3, (Singapore), pp. 1447–1494, World Scientific, 7, 2000, DOI [hep-ph/0009131].
  * [73] S. Peris, D. Boito, M. Golterman and K. Maltman, _The case for duality violations in the analysis of hadronic $\tau$ decays_, _Mod. Phys. Lett. A_ 31 (2016) 1630031 [1606.08898].
  * [74] O. Catà, M. Golterman and S. Peris, _Possible duality violations in tau decay and their impact on the determination of alpha(s)_ , _Phys. Rev. D_ 79 (2009) 053002 [0812.2285].
  * [75] D. Boito, M. Golterman, M. Jamin, K. Maltman and S. Peris, _Low-energy constants and condensates from the $\tau$ hadronic spectral functions_, _Phys. Rev. D_ 87 (2013) 094008 [1212.4471].
  * [76] D. Boito, A. Francis, M. Golterman, R. Hudspith, R. Lewis, K. Maltman et al., _Low-energy constants and condensates from ALEPH hadronic $\tau$ decay data_, _Phys. Rev. D_ 92 (2015) 114501 [1503.03450].
  * [77] M. González-Alonso, A. Pich and J. Prades, _Violation of Quark-Hadron Duality and Spectral Chiral Moments in QCD_ , _Phys. Rev. D_ 81 (2010) 074007 [1001.2269].
  * [78] M. González-Alonso, A. Pich and A. Rodrίguez-Sánchez, _Updated determination of chiral couplings and vacuum condensates from hadronic $\tau$ decay data_, _Phys. Rev. D_ 94 (2016) 014017 [1602.06112].
  * [79] P. Masjuan, E. Ruiz Arriola and W. Broniowski, _Systematics of radial and angular-momentum Regge trajectories of light non-strange q\barq-states_ , _Phys. Rev. D_ 85 (2012) 094006 [1203.4782].
  * [80] S. Peris, _Violation of quark–hadron duality: The missing oscillation in the OPE_ , _Eur. Phys. J. ST_ 230 (2021) 2691.
  * [81] K. Maltman and T. Yavin, _$\alpha_{s}(M_{Z}^{2})$ from hadronic tau decays_, _Phys. Rev. D_ 78 (2008) 094020 [0807.0650].
  * [82] R. J. Hudspith, R. Lewis, K. Maltman and J. Zanotti, _A resolution of the inclusive flavor-breaking $\tau$ $|V_{us}|$ puzzle_, _Phys. Lett. B_ 781 (2018) 206 [1702.01767].
  * [83] D. Boito, M. Golterman, K. Maltman and S. Peris, _Evidence against naive truncations of the OPE from $e^{+}e^{-}\to$ hadrons below charm_, _Phys. Rev. D_ 100 (2019) 074009 [1907.03360].
  * [84] HFLAV collaboration, _Averages of b-hadron, c-hadron, and $\tau$-lepton properties as of 2021_, _Phys. Rev. D_ 107 (2023) 052008 [2206.07501].
  * [85] J. Erler, _Electroweak radiative corrections to semileptonic tau decays_ , _Rev. Mex. Fis._ 50 (2004) 200 [hep-ph/0211345].