# Shallow Water Waves and Conservation Laws with Dispersion Triplet
## 3 Dual-Nonlinearity (Gardner's Equation)
### KdV-mKdV Equation  
Here, \(F(q)=a_{1}q+a_{2}q^{2}\) for any real-valued constants \(a_{j}\) for \(j=1,2\). From (1), this leads to [8]  
\[q_{t}+\Big{(}a_{1}q+a_{2}q^{2}\Big{)}q_{x}+b_{1}q_{xxx}+b_{2}q_{xxt}+b_{3}q_{xtt} =0, \tag{45}\]  
which means that (5) transforms into  
\[6 vg^{2}-(2a_{1}+a_{2}g)g^{3}-6\Big{(}b_{1}-b_{2}v+b_{3}v^{2}\Big{)}\big{(}g^{ \prime}\big{)}^{2}=0. \tag{46}\]  
Equation (45) is the familiar KdV-mKdV equation that is here being addressed with three sources of dispersion for the first time. This equation was studied in the past, but only with spatial dispersion, using the semi-inverse variational principle [8]. Finally, separating variables and integrating yields a solitary wave solution as follows:  
\[q(x,t)=\frac{A}{D+\cos\mathrm{h}[B(x-vt)]}. \tag{47}\]  
Here, the inverse width is \(B\) with an external parameter \(D\), while \(A\) is the amplitude of the wave. The amplitude can be expressed as  
\[A=\frac{6v}{\sqrt{a_{1}^{2}+6a_{2}v}}, \tag{48}\]  
and the inverse width \(B\) is the same as (10). The parameter \(D\) is given as  
\[D=\frac{6v}{\sqrt{a_{1}^{2}+6a_{2}v}}. \tag{49}\]  
The inverse width \(B\) and the parameter \(D\) pose the constraint  
\[a_{1}^{2}+6a_{2}v>0. \tag{50}\]  
Figure 4 depicts the surface plot of a solitary wave (47) of Gardner's equation with triple dispersion terms. The parameter values chosen are \(a_{1}=1\), \(a_{2}=1\), \(v=1\), \(b_{1}=1\), \(b_{2}=1\), and \(b_{3}=1\). Conservation Laws  
Here, the multipliers, densities, and the fluxes are  
Figure 4: Profile of a solitary wave for Gardner’s equation without radiation. Case 1:  
\[Q=1\text{ and }T^{t}=-\frac{2}{3}b_{3}q_{xt}-\frac{1}{3}b_{2}q_{xx}-q. \tag{51}\]  
Case 2:  
\[Q=q\text{ and }T^{t}=-\frac{2}{3}qb_{3}q_{xt}-\frac{1}{3}b_{2}q_{xx}q+\frac{1}{3}b_{ 3}q_{x}q_{t}+\frac{1}{6}{b_{2}q_{x}}^{2}-\frac{1}{2}q^{2}. \tag{52}\]  
Case 3:  
\[Q=2a_{2}q^{3}+3a_{1}q^{2}+6b_{1}q_{xx}+6b_{2}q_{xt}+6b_{3}q_{tt}, \tag{53}\]  
and  
\[ \begin{array}{c}T^{t}=-a_{2}b_{2}q^{3}q_{xx}-3a_{2}b_{2}q^{2}q^{2}q^{2}-2a_{ 2}b_{3}q^{3}q_{xt}-6a_{2}b_{3}q^{2}q^{2}q_{t}q_{x}+6a_{2}q^{4}\\ +12a_{1}q^{3}-6b_{1}b_{2}q_{xxxx}+6b_{1}b_{2}q_{xxxx}+12b_{1}b_{2}q_{xx}{}^{2 }-12b_{1}b_{3}qq_{xxx}\\ +24b_{1}b_{3}q_{tt}q_{xxx}-12b_{1}b_{3}q_{xt}q_{xtxt}+24b_{1}b_{3}q_{xt}q_{xx }-6b_{2}q_{xxxt}+6b_{2}^{2}q_{x}q_{xtxt}\\ +12b_{2}^{2}q_{xt}q_{xx}-18b_{2}b_{3}qq_{xxtt}+24b_{2}b_{3}q_{tt}q_{xx}+12b_{2 }b_{3}q_{tt}q_{xx}\\ -6b_{2}b_{3}q_{xt}q_{xtt}+24b_{2}b_{3}q_{xt}{}^{2}-12b_{3}^{2}q_{xtxt}+24b_{3} ^{2}q_{tt}q_{xtt}+24b_{3}^{2}q_{tt}q_{xt}\\ -12b_{3}^{2}q_{tt}q_{x}+36b_{1}qq_{xx}+18b_{2}qq_{xt}+18b_{2}q_{t}q_{x}+36b_{ 3}q_{t}{}^{2}.\end{array}  \tag{54}\]  
For [PERSON]'s equation, the three conserved quantities are  
\[M=\int_{-\infty}^{\infty}T^{t}dx=\int_{-\infty}^{\infty}qdx=\frac{2A}{B}F \bigg{(}1,1;\frac{3}{2};\frac{1-D}{2}\bigg{)}, \tag{55}\]  
\[ \begin{array}{c}P=\int_{-\infty}^{\infty}T^{t}dx=\frac{1}{6}\int_{-\infty}^ {\infty}\{q^{2}-(b_{2}-6 vb_{3})q_{x}^{2}\}dx\\ =\frac{A^{2}}{45B}\Big{\{}5F\Big{(}2,2;\frac{5}{2};\frac{1-D}{2}\Big{)}-B^{2}F \Big{(}4,2;\frac{7}{2};\frac{1-D}{2}\Big{)}\Big{\}},\end{array}  \tag{56}\]  
and  
\[ \begin{array}{c}H=\int_{-\infty}^{\infty}T^{t}dx=6\int_{-\infty}^{\infty}\{ 2a_{1}q^{3}+a_{2}q^{4}-6\big{(}b_{1}-v^{2}b_{3}\big{)}q_{x}^{2}\}dx\\ =\frac{8A^{2}}{35B}\Bigg{\{}\begin{array}{c}14a_{1}A\Big{(}3,\frac{7}{2}; \frac{1-D}{2}\Big{)}+3a_{2}A^{2}F\Big{(}4,4;\frac{9}{2};\frac{1-D}{2}\Big{)}\\ -21B^{2}\big{(}b_{1}-v^{2}b_{3}\big{)}F\Big{(}4,2;\frac{7}{2};\frac{1-D}{2} \Big{)}\end{array}\Bigg{\}},\end{array}  \tag{57}\]  
where in (55)-(57), the Gauss' hypergeometric function is indicated below as [1]  
\[F(\alpha,\beta;\gamma;z)=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}(\beta)_{n}}{( \gamma)_{n}}\frac{z^{n}}{n!}, \tag{58}\]  
with the Pochhammer symbol being  
\[\left(p\right)_{n}=\left\{ \begin{array}{cc}1&n=0\,,\\ p(p+1)\cdots(p+n-1)&n>0\,.\end{array} \right.