加德納-KP方程（Gardner-KP equation）是一個非線性偏微分方程[1]
( u t + 6 u u x + 6 u 2 ∗ u x + u x x x ) x + u y y = 0 {\displaystyle (u_{t}+6uu_{x}+6u^{2}*u_{x}+u_{x}xx)_{x}+u_{y}y=0}
加德納-KP方程有行波解：
u ( x , y , t ) = − 1 / 2 − C 2 ∗ s e c h ( C 1 + C 2 ∗ x + C 3 ∗ y − ( 1 / 2 ) ∗ ( − 3 ∗ C 2 2 + 2 ∗ C 3 2 + 2 ∗ C 2 4 ) ∗ t / C 2 ) {\displaystyle {u(x,y,t)=-1/2-_{C}2*sech(_{C}1+_{C}2*x+_{C}3*y-(1/2)*(-3*_{C}2^{2}+2*_{C}3^{2}+2*_{C}2^{4})*t/_{C}2)}} u ( x , y , t ) = − 1 / 2 − C 3 ∗ J a c o b i D N ( C 2 + C 3 ∗ x + C 4 ∗ y + ( 1 / 2 ) ∗ ( 3 ∗ C 3 2 − 2 ∗ C 4 2 + 2 ∗ C 3 4 ∗ C 1 2 − 4 ∗ C 3 4 ) ∗ t / C 3 , C 1 ) {\displaystyle {u(x,y,t)=-1/2-_{C}3*JacobiDN(_{C}2+_{C}3*x+_{C}4*y+(1/2)*(3*_{C}3^{2}-2*_{C}4^{2}+2*_{C}3^{4}*_{C}1^{2}-4*_{C}3^{4})*t/_{C}3,_{C}1)}} u ( x , y , t ) = − 1 / 2 + C 3 ∗ J a c o b i D N ( C 2 + C 3 ∗ x + C 4 ∗ y + ( 1 / 2 ) ∗ ( 3 ∗ C 3 2 − 2 ∗ C 4 2 + 2 ∗ C 3 4 ∗ C 1 2 − 4 ∗ C 3 4 ) ∗ t / C 3 , C 1 ) {\displaystyle {u(x,y,t)=-1/2+_{C}3*JacobiDN(_{C}2+_{C}3*x+_{C}4*y+(1/2)*(3*_{C}3^{2}-2*_{C}4^{2}+2*_{C}3^{4}*_{C}1^{2}-4*_{C}3^{4})*t/_{C}3,_{C}1)}} u ( x , y , t ) = − 1 / 2 − I ∗ C 2 ∗ c o t h ( C 1 + C 2 ∗ x + C 3 ∗ y + ( 1 / 2 ) ∗ ( 3 ∗ C 2 2 − 2 ∗ C 3 2 + 4 ∗ C 2 4 ) ∗ t / C 2 ) {\displaystyle {u(x,y,t)=-1/2-I*_{C}2*coth(_{C}1+_{C}2*x+_{C}3*y+(1/2)*(3*_{C}2^{2}-2*_{C}3^{2}+4*_{C}2^{4})*t/_{C}2)}} u ( x , y , t ) = − 1 / 2 − I ∗ C 2 ∗ c s c ( C 1 + C 2 ∗ x + C 3 ∗ y + ( 1 / 2 ) ∗ ( 3 ∗ C 2 2 − 2 ∗ C 3 2 + 2 ∗ C 2 4 ) ∗ t / C 2 ) {\displaystyle {u(x,y,t)=-1/2-I*_{C}2*csc(_{C}1+_{C}2*x+_{C}3*y+(1/2)*(3*_{C}2^{2}-2*_{C}3^{2}+2*_{C}2^{4})*t/_{C}2)}} u ( x , y , t ) = − 1 / 2 − I ∗ C 2 ∗ t a n ( C 1 + C 2 ∗ x + C 3 ∗ y − ( 1 / 2 ) ∗ ( − 3 ∗ C 2 2 + 2 ∗ C 3 2 + 4 ∗ C 2 4 ) ∗ t / C 2 ) {\displaystyle {u(x,y,t)=-1/2-I*_{C}2*tan(_{C}1+_{C}2*x+_{C}3*y-(1/2)*(-3*_{C}2^{2}+2*_{C}3^{2}+4*_{C}2^{4})*t/_{C}2)}} u ( x , y , t ) = − 1 / 2 − I ∗ C 3 ∗ J a c o b i N D ( C 2 + C 3 ∗ x + C 4 ∗ y + ( 1 / 2 ) ∗ ( 3 ∗ C 3 2 − 2 ∗ C 4 2 ) ∗ t / C 3 , s q r t ( 2 ) ) {\displaystyle {u(x,y,t)=-1/2-I*_{C}3*JacobiND(_{C}2+_{C}3*x+_{C}4*y+(1/2)*(3*_{C}3^{2}-2*_{C}4^{2})*t/_{C}3,sqrt(2))}} u ( x , y , t ) = − 1 / 2 − ( 1 / 2 ∗ I ) ∗ ( 2 ) ∗ C 3 ∗ J a c o b i N C ( C 2 + C 3 ∗ x + C 4 ∗ y + ( 1 / 2 ) ∗ ( 3 ∗ C 3 2 − 2 ∗ C 4 2 ) ∗ t / C 3 , ( 1 / 2 ) ∗ ( 2 ) ) {\displaystyle {u(x,y,t)=-1/2-(1/2*I)*{\sqrt {(}}2)*_{C}3*JacobiNC(_{C}2+_{C}3*x+_{C}4*y+(1/2)*(3*_{C}3^{2}-2*_{C}4^{2})*t/_{C}3,(1/2)*{\sqrt {(}}2))}}
