双重sinh-Gordon方程

双重sinh-Gordon方程（Double sinh-Gordon equation）是一个非线性偏微分方程。.

$u_{xt}=asinh(u)+bsinh(2u)$

行波解

• $v{=}_{C}5\ast JacobiCN{(}_{C}2{+}_{C}3\ast x-(a{\ast }_C{5}^2-2\ast b{\ast }_C{5}^2-2\ast b-a)\ast t/{(}_{C}3\ast {(}_C{5}^2-1)),\sqrt{(}(-2\ast a{\ast }_C{5}^2+a{\ast }_C{5}^4+a+2\ast b-2\ast b{\ast }_C{5}^4)\ast (a{\ast }_C{5}^2-2\ast b{\ast }_C{5}^2-a)){\ast }_{C}5/(-2\ast a{\ast }_C{5}^2+a{\ast }_C{5}^4+a+2\ast b-2\ast b{\ast }_C{5}^4))$
• $v{=}_{C}5\ast JacobiDN{(}_{C}2{+}_{C}3\ast x{-}_C{5}^2\ast (a{\ast }_C{5}^2-2\ast b{\ast }_C{5}^2-a)\ast t/{(}_{C}3\ast (-2{\ast }_C{5}^2+1{+}_C{5}^4)),\sqrt{(}(-2\ast a{\ast }_C{5}^2+a{\ast }_C{5}^4+a+2\ast b-2\ast b{\ast }_C{5}^4)\ast (a{\ast }_C{5}^2-2\ast b{\ast }_C{5}^2-a))/((a{\ast }_C{5}^2-2\ast b{\ast }_C{5}^2-a){\ast }_{C}5))$
• $v{=}_{C}5\ast JacobiNC{(}_{C}2{+}_{C}3\ast x+(a{\ast }_C{5}^2-2\ast b{\ast }_C{5}^2-2\ast b-a)\ast t/{(}_{C}3\ast {(}_C{5}^2-1)),\sqrt{(}-(-2\ast a{\ast }_C{5}^2+a{\ast }_C{5}^4+a+2\ast b-2\ast b{\ast }_C{5}^4)\ast (a{\ast }_C{5}^2-2\ast b-a))/(-2\ast a{\ast }_C{5}^2+a{\ast }_C{5}^4+a+2\ast b-2\ast b{\ast }_C{5}^4))$
• $v{=}_{C}5\ast JacobiND{(}_{C}2{+}_{C}3\ast x-(a{\ast }_C{5}^2-2\ast b-a)\ast t/{(}_{C}3\ast (-2{\ast }_C{5}^2+1{+}_C{5}^4)),\sqrt{(}-(-2\ast a{\ast }_C{5}^2+a{\ast }_C{5}^4+a+2\ast b-2\ast b{\ast }_C{5}^4)\ast (a{\ast }_C{5}^2-2\ast b-a))/(a{\ast }_C{5}^2-2\ast b-a))$
• $v=\sqrt{(}a\ast (2\ast b+a))\ast csc{(}_{C}1{+}_{C}2\ast x-(2\ast b+a)\ast t{/}_{C}2)/a$
• $v=\sqrt{(}a\ast (2\ast b+a))\ast csc{(}_{C}2{+}_{C}3\ast x-(2\ast b+a)\ast t{/}_{C}3)/a$
• $v=\sqrt{(}a\ast (2\ast b+a))\ast sec{(}_{C}1{+}_{C}2\ast x-(2\ast b+a)\ast t{/}_{C}2)/a$
• $v=\sqrt{(}a\ast (2\ast b+a))\ast sech{(}_{C}1{+}_{C}2\ast x+(2\ast b+a)\ast t{/}_{C}2)/a$
• $v=\sqrt{(}-a\ast (2\ast b+a))\ast csch{(}_{C}1{+}_{C}2\ast x+(2\ast b+a)\ast t{/}_{C}2)/a$
• $v=\sqrt{(}(a-2\ast b)\ast a)\ast cosh{(}_{C}2{+}_{C}3\ast x-(a-2\ast b)\ast t{/}_{C}3)/(a-2\ast b)$
• $v=\sqrt{(}(a-2\ast b)\ast (2\ast b+a))\ast tanh{(}_{C}1{+}_{C}2\ast x+(1/8)\ast (a^2-4\ast b^2)\ast t/{(}_{C}2\ast b))/(a-2\ast b)$

其中 $v=tanh((1/2)\ast u)$

特解

• $u(x,t)=2arctanh(1.5\ast JacobiCN(1.2+1.3\ast x+3.2307692307692307692\ast t,1.0555973258234951998))$
• $u(x,t)=2arctanh(1.5\ast JacobiDN(1.2+1.3\ast x+3.6000000000000000000\ast t,.94733093343134184593))$
• $u(x,t)=2\ast arctanh(1.5\ast JacobiNC(-1.2-1.3\ast x+3.2307692307692307692\ast t,.33806170189140663100\ast I))$
• $u(x,t)=2\ast arctanh(1.5\ast JacobiND(1.2+1.3\ast x+.36923076923076923077\ast t,2.9580398915498080213\ast I))$
• $u(x,t)=-2\ast arctanh(\sqrt{(}3)\ast csc(15.1-1.2\ast x+2.5000000000000000000\ast t))$
• $u(x,t)=-2\ast arctanh(\sqrt{(}3)\ast csc(-1.2-1.3\ast x+2.3076923076923076923\ast t))$
• $u(x,t)=2\ast arctanh(\sqrt{(}3)\ast sec(15.1-1.2\ast x+2.5000000000000000000\ast t))$
• $u(x,t)=2\ast arctanh(\sqrt{(}3)\ast sech(-15.1+1.2\ast x+2.5000000000000000000\ast t))$
• $u(x,t)=2\ast arctanh(\sqrt{(}3)\ast sech(1.2+1.3\ast x+2.3076923076923076923\ast t))$
• $u(x,t)=2\ast arctanh(\sqrt{(}-3)\ast csch(-15.1+1.2\ast x+2.5000000000000000000\ast t))$
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行波图