# Shallow Water Waves and Conservation Laws with Dispersion Triplet
## 2 Single Nonlinearity (KdV Equations)
### Power-Law KdV Equation  
Now, \(F(q)=aq^{n}\) for any real-valued constant \(a\). From (1), this leads to  
\[q_{t}+aq^{n}q_{x}+b_{1}q_{xxx}+b_{2}q_{xxt}+b_{3}q_{xtt}=0, \tag{33}\]  
which means that (5) transforms into  
\[(n+2)(n+1)vg^{2}-2 ag^{n+2}-(n+2)(n+1)\Big{(}b_{1}-b_{2}v+b_{3}v^{2}\Big{)} \big{(}g^{\prime}\big{)}^{2}=0. \tag{34}\]  
Finally, separating variables and integrating yields a solitary wave solution as follows:  
\[q(x,t)=A\text{sech}^{\frac{2}{n}}[B(x-vt)]. \tag{35}\]  
Here, the inverse width is \(B\) and \(A\) is the amplitude of the wave; they are indicated below as  
\[A=\bigg{[}\frac{(n+1)(n+2)v}{2a}\bigg{]}^{\frac{1}{n}}, \tag{36}\]  
and  
\[B=\frac{n}{2}\sqrt{\frac{v}{b_{1}-b_{2}v+b_{3}v^{2}}}, \tag{37}\]  
which would once again imply the parameter constraint as given by (11). Figure 3 indicates the surface plot of a solitary wave (35) of the power-law KdV equation with triple dispersion terms with \(n=1/2\). The parameter values chosen are \(a=1\), \(v=1\), \(b_{1}=1\), \(b_{2}=1\), and \(b_{3}=1\). Conservation Laws  
For the power-law KdV equation, the multipliers, densities, and fluxes are given as  
Case 1:  
\[Q=1\text{ and }T^{t}=-\frac{2}{3}b_{3}q_{xt}-\frac{1}{3}b_{2}q_{xx}-q. \tag{38}\]  
Figure 3: Profile of a solitary wave for the power–law KdV equation, without radiation. Case 2:  
\[Q=q\text{ and }T^{t}=-\frac{2}{3}qb_{3}q_{xt}-\frac{1}{3}b_{2}q_{xx}q+\frac{1}{3 }b_{3}q_{x}q_{t}+\frac{1}{6}b_{2}q_{x}{}^{2}-\frac{1}{2}q^{2}. \tag{39}\]  
Case 3:  
\[Q=q^{n}aq+b_{1}q_{xx}n+b_{2}q_{xt}n+b_{3}nq_{tt}+b_{1}q_{xx}+b_{2}q_{xt}+b_{3} q_{tt}, \tag{40}\]  
and  
\[ \begin{array}{c}T^{t}=-2q^{n}ab_{2}n^{2}q_{x}^{2}-b_{1}b_{2}n^{2}qq_{xxxx}-4 q^{n+1}ab_{3}nq_{xt}-2q^{n+1}ab_{2}nq_{xx}\\ +4b_{1}b_{3}n^{2}q_{4}q_{xxx}+4b_{1}b_{3}n^{2}q_{xx}q_{xt}-6b_{1}b_{3}nqq_{xxxx} +b_{1}b_{2}n^{2}q_{x}q_{xxx}\\ -2b_{1}b_{3}n^{2}qq_{xxxt}-2b_{1}b_{3}n^{2}q_{3}q_{xtxt}+12b_{1}b_{3}nq_{xx}q_ {xt}+12b_{1}b_{3}nq_{t}q_{xxx}\\ -3b_{2}b_{3}nq_{x}q_{xtxt}+12b_{2}b_{3}nq_{t}q_{xtxt}+6b_{2}b_{3}nq_{xx}q_{tt} +4q^{n}ab_{3}q_{x}q_{t}\\ -3b_{1}b_{2}nqq_{xxxx}+3b_{1}b_{2}nq_{x}q_{xxx}-6b_{1}b_{3}nq_{t}q_{xtxt}-3b_{ 2}b_{3}n^{2}qq_{xxt}\\ -b_{2}b_{3}n^{2}q_{x}q_{xtxt}+4b_{2}b_{3}n^{2}q_{t}q_{xtxt}+2b_{2}b_{3}n^{2}q_ {xx}q_{tt}-9b_{2}b_{3}nqq_{xxt}\\ -4q^{n}ab_{3}n^{2}q_{x}q_{tt}-6b_{3}n^{2}q_{x}q_{tt}-2b_{1}b_{2}q_{xxxx}+2b_{1} b_{2}q_{x}q_{xxx}-4b_{1}b_{3}q_{t}q_{xxxt}\\ -4b_{1}b_{3}q_{x}q_{xtxt}+12b_{3}nq_{t}q_{xtxt}+8b_{1}b_{3}q_{t}q_{xxx}+8b_{1} b_{3}q_{xt}q_{xt}+12b_{3}^{2}nq_{xt}q_{tt}\\ -6b_{2}b_{3}qq_{xxt}-2b_{2}b_{3}q_{x}q_{xtxt}+8b_{2}b_{3}q_{t}q_{xtxt}+4b_{2}b _{3}q_{xx}q_{tt}-b_{2}^{2}n^{2}qq_{xxxt}\\ +b_{2}n^{2}q_{x}q_{xtxt}+2b_{1}b_{2}n^{2}q_{xx}+18b_{1}nq_{xx}+3b_{2}n^{2}q_{xt }q_{xt}-3b_{2}^{2}nqq_{xxxt}\\ +2q^{n}ab_{2}q_{x}{}^{2}+9b_{2}nq_{3}q_{t}+2q^{n+1}ab_{2}q_{xx}+6b_{1}b_{2}nq_{ xx}{}^{2}+4q^{n+1}ab_{3}q_{xt}\\ +9b_{2}nqq_{xt}+6b_{1}n^{2}q_{xx}+3b_{2}n^{2}q_{xt}+4b_{3}n^{2}n^{2}q_{xt}q_{tt} +4b_{3}n^{2}n^{2}q_{txt}\\ +6b_{2}n^{2}nq_{xtxt}+12b_{2}b_{3}nq_{t}{}^{2}+3b_{2}n^{2}q_{xt}q_{tt}-4b_{3} ^{2}q_{xtxt}+4b_{2}b_{3}n^{2}q_{x}{}^{2}\\ -2b_{3}n^{2}q_{xtxt}-2b_{3}n^{2}q_{x}q_{ttt}+2b_{2}n^{2}n^{2}q_{xx}q_{xt}-6b_{ 3}nqq_{xtxt}\\ +12b_{3}q_{t}{}^{2}+12q^{n+2}a+6b_{2}q_{xt}+8b_{3}q_{xt}q_{tt}+6b_{2}q_{x}q_{ t}+8b_{3}^{2}q_{t}q_{xtxt}\\ +8b_{2}b_{3}q_{xt}{}^{2}-4b_{3}^{2}q_{x}q_{ttt}-2b_{2}q_{xxxt}+4b_{1}b_{2}q_{ xx}{}^{2}+4b_{2}^{2}q_{xx}q_{xt}\\ +2b_{2}q_{x}q_{xtxt}+18b_{3}nq_{t}{}^{2}+6b_{3}n^{2}q_{t}{}^{2}+12b_{1}q_{xx}. \end{array}  \tag{41}\]  
For power-law, the conserved quantities are  
\[M=\int_{-\infty}^{\infty}T^{t}dx=\int_{-\infty}^{\infty}qdx=\frac{A}{B}\frac{ \Gamma\left(\frac{1}{\bar{n}}\right)\Gamma\left(\frac{1}{\bar{2}}\right)}{ \Gamma\left(\frac{1}{\bar{n}}+\frac{1}{\bar{2}}\right)}, \tag{42}\]  
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