# Shallow Water Waves and Conservation Laws with Dispersion Triplet
## 2 Single Nonlinearity (KdV Equations)
### KdV Equation  
For this model, \(F(q)=aq\) for any real-valued constant \(a\). From (1), this leads to [4]  
\[q_{t}+aqq_{x}+b_{1}q_{xxx}+b_{2}q_{xxt}+b_{3}q_{xtt}=0, \tag{6}\]  
which would mean that (5) transforms into  
\[3\text{{v}}g^{2}-a\text{{g}}^{3}-3\Big{(}b_{1}-b_{2}v+b_{3}v^{2}\Big{)}\big{(} \text{{g}}^{\prime}\big{)}^{2}=0. \tag{7}\]  
Finally, separating variables and integrating yields a solitary wave solution as follows:  
\[q(x,t)=A\text{sech}\,\text{h}^{2}[B(x-vt)]. \tag{8}\]  
Here, the inverse width is \(B\) and \(A\) is the amplitude of the wave; they are presented below as  
\[A=\frac{3\text{{v}}}{a}, \tag{9}\]  
and  
\[B=\frac{1}{2}\sqrt{\frac{v}{b_{1}-b_{2}v+b_{3}v^{2}}}. \tag{10}\]  
The width of the wave therefore introduces the following parameter constraint:  
\[v\Big{(}b_{1}-b_{2}v+b_{3}v^{2}\Big{)}>0. \tag{11}\]  
Figure 1 depicts the surface plot of a solitary wave (8) of the KdV equation with triple dispersion terms. The parameter values chosen are \(a=1\), \(v=1\), \(b_{1}=1\), \(b_{2}=1\), and \(b_{3}=1\).  
Conservation Laws  
For the conserved flow that renders a closed form of the respective models, \(\left(T^{t},T^{x}\right)\) satisfies \(D_{t}T^{t}+D_{x}T^{x}=0\) along the solutions. \(T^{t}\) represents the conserved density and  
Figure 1: Profile of a solitary wave for the KdV equation, without radiation.  
is the conserved flux [2; 4; 5; 9; 14; 15]. We present the various densities that arise from the multipliers \(Q(t,x,u,u_{t},u_{x},u_{xx},u_{tt},u_{xt})\) when  
\[D_{t}T^{t}+D_{x}T^{x}=Q(q_{t}+aqq_{x}+b_{1}q_{xxx}+b_{2}q_{xxt}+b_{3}q_{xtt}). \tag{12}\]  
Case 1:  
\[Q=1\text{ and }T^{t}=-\frac{2}{3}b_{3}q_{xt}-\frac{1}{3}b_{2}q_{xx}-q. \tag{13}\]  
Case 2:  
\[Q=q\text{ and }T^{t}=-\frac{2}{3}qb_{3}q_{xt}-\frac{1}{3}b_{2}q_{xx}q+\frac{1}{ 3}b_{3}q_{x}q_{t}+\frac{1}{6}b_{2}q_{x}{}^{2}-\frac{1}{2}q^{2}. \tag{14}\]  
Case 3:  
\[Q=\frac{1}{2b_{2}}\Big{(}aq^{2}+2b_{1}q_{xx}+2b_{2}q_{xt}+2b_{3}q_{xt}\Big{)}, \tag{15}\]  
and  
\[\begin{array}{c}T^{t}=2 aq^{3}-b_{1}b_{2}qq_{xxxx}+b_{1}b_{2}q_{x}q_{xxx}+2b_ {1}b_{2}q_{xx}{}^{2}-2b_{1}b_{3}qq_{xxxt}\\ +4b_{1}b_{3}q_{t}q_{xxx}-2b_{1}b_{3}q_{x}q_{xxt}+4b_{1}b_{3}q_{xt}q_{xx}-b_{2} ^{2}qq_{xxxt}+b_{2}^{2}q_{x}q_{xxt}\\ +2b_{2}{}^{2}q_{xt}q_{xx}-3b_{2}b_{3}q_{xtxt}+4b_{2}b_{3}q_{tt}q_{xx}+2b_{2}b_ {3}q_{tt}q_{xx}-b_{2}b_{3}q_{t}q_{xtt}\\ +4b_{2}b_{3}q_{xt}{}^{2}-2b_{3}^{2}qq_{xt}+4b_{3}^{2}q_{t}q_{xtt}+4b_{3}q_{tt} ^{2}q_{tt}q_{xt}\\ -2b_{3}{}^{2}q_{tt}q_{x}+6b_{1}q_{xxx}+3b_{2}q_{xt}+3b_{2}q_{t}q_{x}+6b_{3}q_{ t}{}^{2}.\end{array} \tag{16}\]  
Therefore, the respective conserved quantities are  
\[M=\int_{-\infty}^{\infty}T^{t}dx=\int_{-\infty}^{\infty}qdx=\frac{2A}{B}, \tag{17}\]  
\[P=\int_{-\infty}^{\infty}T^{t}dx=\frac{1}{6}\int_{-\infty}^{\infty}\{3q^{2}-( 3b_{2}+2 vb_{3})q_{x}^{2}\}dx=\frac{2A^{2}}{45B}\{15-4B^{2}(3b_{2}+2 vb_{3})\}, \tag{18}\]  
and  
\[H=\int_{-\infty}^{\infty}T^{t}dx=\int_{-\infty}^{\infty}\Big{\{}2 aq^{3}-6\Big{(} b_{1}-v^{2}b_{3}\Big{)}q_{x}^{2}\Big{\}}dx=\frac{32A^{2}}{15B}\Big{\{}aA-3B^{2} \Big{(}b_{1}-v^{2}b_{3}\Big{)}\Big{\}}. \tag{19}\]  
Here, \(H\) gives the Hamiltonian, \(P\) stems from linear momentum, and \(M\) is from mass. To evaluate the integrals in (17)-(19), the solitary wave solution given by (8) is utilized.