The TWSolutions command computes Traveling Wave Solutions (TWS) for autonomous partial differential equations (PDE) that are rational and nonlinear in the unknowns and their derivatives. An autonomous system is one where the independent variables do not appear explicitly, only through the unknown functions and their derivatives.

A TWS is an exact, closed-form solution, expressed as a finite power series of one of a particular set of functions (default is the hyperbolic tangent, tanh) that has a linear combination of the independent variables as arguments. Concretely, given such an autonomous PDE system with $i$ unknown functions $f_i\left(x_1,\mathrm{...},x_j\right)$ of $j$ variables, by default TWSolutions computes solutions of the form

f[i](tau) = Sum(A[i,k]*tau^k, k=0..n[i]);

$f_i\left(\mathrm{\tau }\right)=\sum _{k=0}^{n_i}{A}_{i,k}{\mathrm{\tau }}^k$

where the $n_i$ are finite, the $A_{i,k}$ are constants with respect to the $x_j$ and

tau = tanh(Sum(C[k]*x[k], k=0..j));

$\mathrm{\tau }=\tanh \left(\sum _{k=0}^j{C}_k{x}_k\right)$

where the $C_k$ are constants with respect to the $x_j$.

The first argument given to TWSolutions, PDEsys, is a set or list containing one or more autonomous PDEs. The unknowns of the system can be specified anywhere after the first argument, as a set or list of functions - or function names - or as a ranking (see pdsolve,system). If the unknowns of the problem are not specified, only the differentiated unknown functions found in the PDE system are taken as the unknowns.

The mathematical function used to construct the TWS, when not specified, is tanh. To specify a different function, or a list of them to construct different types of solutions at once, use the functions = ... optional argument, where the right-hand side can be the name of a mathematical function or a set or list of them. The possible mathematical functions are: exp, ln, the trigonometric functions sin, cos, tan, csc, sec, cot, the hyperbolic versions of them, JacobiSN, JacobiCN, JacobiDN, JacobiNS, JacobiNC, JacobiND, the corresponding InverseJacobi functions, and the WeierstrassP function.

It is also possible to choose the identity as the "mathematical function" to be used, in which case, instead of a power series of a function of a linear combination of the independent variables, the TWS is constructed as a power series of the linear combination itself, resulting in a polynomial solution to the PDE system.

In some cases, or to perform further analysis and intermediate manipulations, it is useful to compute not the actual solution to the problem posed but just the ODE system equivalent to the given PDE system and the transformation mapping each other. To obtain this type of output use the optional argument output = ODE.

As explained in the previous paragraphs, once the function with which the solution will be constructed is determined, either by default (tanh) or by choice (using the optional argument function = ...), the "TWS" solution is constructed as a power series in that function. This is done by computing the polynomial solutions to the nonlinear ODE system equivalent to the given PDE system. However, a more interesting solution for the PDE system can sometimes be obtained by computing not just polynomial solutions to the nonlinear ODE system but by computing its general solution. To request TWSolutions to attempt this type of solution use the optional argument extended.

The solving process essentially transforms the given PDE system and related unknowns (functions) and parameters (there may be many, represented by symbol variables, indicated with the optional argument parameters = ...) into an nonlinear algebraic system in these unknowns, parameters, and unknown coefficients of the finite series expansion. To solve that system, an ordering or ranking for these solving variables (unknowns, parameters and coefficients) is chosen. This ordering mixes the three groups of solving variables according to convenience regarding different aspects, resulting into a sequence of solutions organized according to the ranking used. In some cases, however, a sequence of solutions organized according to the different cases for the parameters is preferred - for that purpose use the optional argument rank_parameters_lower; see the examples.

By default, TWSolutions discards constant solutions that are redundant before producing the output. If the optional argument remove_redundant = false is used, no attempt to remove redundant solutions is done. This is of use when any particular solution suffices because it avoids the costly process of removing redundant solutions, which requires running additional differential elimination processes. Alternatively, when the optional argument remove_redundant = true is given, all redundant solutions, even those that are non-constant are removed. Depending on the PDE system, this process may consume from several seconds to minutes.

All the optional arguments accepted by PDEtools,casesplit are accepted by TWSolutions. Of special interest is parameters={p1,p2, ...}, specifying a set of constant parameters that should also be considered solving variables. The solution then splits into cases with respect to these parameters through a differential elimination process (see casesplit, DifferentialAlgebra and DEtools,Rif). All constant parameters present in the system and not specified using this option are considered arbitrary.

Another optional argument of PDEtools,casesplit relevant when using TWSolutions is singsol=false. Its use avoids computing the singular cases with respect to the coefficients $\left\{A_{i,k}\,C_j\right\}$ when running the differential elimination processes required to solve the problem. Depending on the PDE system, the computation of the singular cases may consume more time than the computation of the general solution. To compute TWS the fastest way (so regardless of receiving redundant constant solutions and ignoring the singular cases), use the combination of optional arguments remove_redundant = false, singsol = false.

with(PDEtools, TWSolutions, declare);

$\left[\mathrm{TWSolutions}\,\mathrm{declare}\right]$

with(DEtools, diff_table);

$\left[\mathrm{diff\_table}\right]$

U := diff_table(u(x,t)):

sys := { U[t] + U[]*U[x] - p*U[x,x] + q*U[x,x,x] = 0 };

$\mathrm{sys}≔\left\{-p\left(\frac{{\partial }^2}{\partial x^2}u\left(x\,t\right)\right)+q\left(\frac{{\partial }^3}{\partial x^3}u\left(x\,t\right)\right)+u\left(x\,t\right)\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right)+\frac{\partial }{\partial t}u\left(x\,t\right)=0\right\}$

TWS_sol:=TWSolutions(sys);

$\mathrm{TWS\_sol}≔\left\{u\left(x\,t\right)=\mathrm{\_C4}\right\},\left\{u\left(x\,t\right)=-\frac{3{\tanh \left(\mathrm{\_C3}t-\frac{px}{10q}+\mathrm{\_C1}\right)}^2{p}^2}{25q}+\frac{6\tanh \left(\mathrm{\_C3}t-\frac{px}{10q}+\mathrm{\_C1}\right)p^2}{25q}+\frac{250\mathrm{\_C3}{q}^2+3p^3}{25qp}\right\},\left\{u\left(x\,t\right)=-\frac{3{\tanh \left(\mathrm{\_C3}t+\frac{px}{10q}+\mathrm{\_C1}\right)}^2{p}^2}{25q}-\frac{6\tanh \left(\mathrm{\_C3}t+\frac{px}{10q}+\mathrm{\_C1}\right)p^2}{25q}+\frac{-250\mathrm{\_C3}{q}^2+3p^3}{25qp}\right\}$

map(pdetest,[TWS_sol], sys);

$\left[\left\{0\right\}\,\left\{0\right\}\,\left\{0\right\}\right]$

infolevel[TWSolutions] := 2;

${\mathrm{infolevel}}_{\mathrm{TWSolutions}}≔2$

{TWSolutions(sys, parameters={p,q})};

* Using tau = tanh(t*C[2]+x*C[1]+C[0])
* Equivalent ODE system: {-q*C[1]^3*(tau^2-1)*(tau^4-2*tau^2+1)*diff(diff(diff(u(tau),tau),tau),tau)+(-p*C[1]^2*(tau^2-1)^2-q*C[1]^3*(tau^2-1)*(6*tau^3-6*tau))*diff(diff(u(tau),tau),tau)+(-2*p*C[1]^2*(tau^2-1)*tau-q*C[1]^3*(tau^2-1)*(6*tau^2-2)-u(tau)*C[1]*(tau^2-1)-C[2]*(tau^2-1))*diff(u(tau),tau)}
* Ordering for functions: [u(tau)]
* Cases for the upper bounds: [[n[1] = 2]]
* Power series solution [1]: {u(tau) = tau^2*A[1,2]+tau*A[1,1]+A[1,0]}, splitting into cases with respect to the parameters {p, q}

$\left\{\left\{p=0\,q=q\,u\left(x\,t\right)=-12{\tanh \left(\mathrm{\_C2}x+\mathrm{\_C3}t+\mathrm{\_C1}\right)}^2{\mathrm{\_C2}}^{2}q+\frac{8{\mathrm{\_C2}}^{3}q-\mathrm{\_C3}}{\mathrm{\_C2}}\right\}\,\left\{p=p\,q=0\,u\left(x\,t\right)=-2\tanh \left(\mathrm{\_C2}x+\mathrm{\_C3}t+\mathrm{\_C1}\right)\mathrm{\_C2}p-\frac{\mathrm{\_C3}}{\mathrm{\_C2}}\right\}\,\left\{p=p\,q=q\,u\left(x\,t\right)=\mathrm{\_C4}\right\}\,\left\{p=p\,q=q\,u\left(x\,t\right)=-\frac{3{\tanh \left(\mathrm{\_C3}t-\frac{px}{10q}+\mathrm{\_C1}\right)}^2{p}^2}{25q}+\frac{6\tanh \left(\mathrm{\_C3}t-\frac{px}{10q}+\mathrm{\_C1}\right)p^2}{25q}+\frac{250\mathrm{\_C3}{q}^2+3p^3}{25qp}\right\}\,\left\{p=p\,q=q\,u\left(x\,t\right)=-\frac{3{\tanh \left(\mathrm{\_C3}t+\frac{px}{10q}+\mathrm{\_C1}\right)}^2{p}^2}{25q}-\frac{6\tanh \left(\mathrm{\_C3}t+\frac{px}{10q}+\mathrm{\_C1}\right)p^2}{25q}+\frac{-250\mathrm{\_C3}{q}^2+3p^3}{25qp}\right\}\right\}$

infolevel[TWSolutions] := 0:

{TWSolutions(sys, parameters={p,q}, singsol=false)};

$\left\{\left\{p=p\,q=q\,u\left(x\,t\right)=-\frac{3{\tanh \left(\mathrm{\_C3}t-\frac{px}{10q}+\mathrm{\_C1}\right)}^2{p}^2}{25q}+\frac{6\tanh \left(\mathrm{\_C3}t-\frac{px}{10q}+\mathrm{\_C1}\right)p^2}{25q}+\frac{250\mathrm{\_C3}{q}^2+3p^3}{25qp}\right\}\,\left\{p=p\,q=q\,u\left(x\,t\right)=-\frac{3{\tanh \left(\mathrm{\_C3}t+\frac{px}{10q}+\mathrm{\_C1}\right)}^2{p}^2}{25q}-\frac{6\tanh \left(\mathrm{\_C3}t+\frac{px}{10q}+\mathrm{\_C1}\right)p^2}{25q}+\frac{-250\mathrm{\_C3}{q}^2+3p^3}{25qp}\right\}\right\}$

TWSolutions( functions_allowed );

$\left[\mathrm{JacobiCN}\,\mathrm{JacobiDN}\,\mathrm{JacobiNC}\,\mathrm{JacobiND}\,\mathrm{JacobiNS}\,\mathrm{JacobiSN}\,\mathrm{WeierstrassP}\,\mathrm{arcsinh}\,\cos \,\cosh \,\cot \,\coth \,\csc \,\mathrm{csch}\,\exp \,\mathrm{identity}\,\ln \,\sec \,\mathrm{sech}\,\sin \,\sinh \,\tan \,\tanh \right]$

TWSolutions(sys, parameters={p,q}, singsol=false, functions = [sec, JacobiSN, WeierstrassP]);

$\left\{p=p\,q=q\,u\left(x\,t\right)=\mathrm{\_C4}\right\},\left\{p=0\,q=q\,u\left(x\,t\right)=-12{\sec \left(\mathrm{\_C2}x+\mathrm{\_C3}t+\mathrm{\_C1}\right)}^2{\mathrm{\_C2}}^{2}q+\frac{4{\mathrm{\_C2}}^{3}q-\mathrm{\_C3}}{\mathrm{\_C2}}\right\},\left\{p=0\,q=q\,u\left(x\,t\right)=-12{\mathrm{JacobiSN}\left(\mathrm{\_C3}x+\mathrm{\_C4}t+\mathrm{\_C2}\,\mathrm{\_C1}\right)}^2{\mathrm{\_C1}}^{2}q{\mathrm{\_C3}}^2+\frac{4{\mathrm{\_C1}}^2{\mathrm{\_C3}}^{3}q+4{\mathrm{\_C3}}^{3}q-\mathrm{\_C4}}{\mathrm{\_C3}}\right\}$

TWSolutions(sys, parameters={p,q}, singsol=false, functions = [WeierstrassP], rank_parameters_lower);

$\left\{p=p\,q=q\,u\left(x\,t\right)=\mathrm{c\_\_6}\right\}$

TWSolutions(sys, parameters={p,q}, singsol=false, function = sec, output=ODE);

$\left[\left\{\left(q{\mathrm{\_C1}}^3{\mathrm{\tau }}^3\left({\mathrm{\tau }}^2-1\right)\left(\frac{{\ⅆ}^3}{\ⅆ{\mathrm{\tau }}^3}u\left(\mathrm{\tau }\right)\right)+3q{\mathrm{\_C1}}^3{\mathrm{\tau }}^2\left(2{\mathrm{\tau }}^2-1\right)\left(\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\tau }}^2}u\left(\mathrm{\tau }\right)\right)+\left(6q{\mathrm{\tau }}^3{\mathrm{\_C1}}^3-q{\mathrm{\_C1}}^3\mathrm{\tau }+u\left(\mathrm{\tau }\right)\mathrm{\_C1}\mathrm{\tau }+\mathrm{\_C2}\mathrm{\tau }\right)\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}u\left(\mathrm{\tau }\right)\right)\right)\sqrt{{\mathrm{\tau }}^2-1}-p{\mathrm{\_C1}}^2{\mathrm{\tau }}^2\left({\mathrm{\tau }}^2-1\right)\left(\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\tau }}^2}u\left(\mathrm{\tau }\right)\right)-p{\mathrm{\_C1}}^2\mathrm{\tau }\left(2{\mathrm{\tau }}^2-1\right)\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}u\left(\mathrm{\tau }\right)\right)\right\}\,\left\{\mathrm{\tau }=\sec \left(\mathrm{\_C1}x+\mathrm{\_C2}t+\mathrm{\_C3}\right)\,u\left(\mathrm{\tau }\right)=u\left(x\,t\right)\right\}\right]$

TWSolutions(sys, parameters={p,q}, singsol=false, function = identity, output=ODE);

$\left[\left\{q{\mathrm{\_C1}}^3\left(\frac{{\ⅆ}^3}{\ⅆ{\mathrm{\tau }}^3}u\left(\mathrm{\tau }\right)\right)-p{\mathrm{\_C1}}^2\left(\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\tau }}^2}u\left(\mathrm{\tau }\right)\right)+\left(u\left(\mathrm{\tau }\right)\mathrm{\_C1}+\mathrm{\_C2}\right)\left(\frac{\ⅆ}{\ⅆ\mathrm{\tau }}u\left(\mathrm{\tau }\right)\right)\right\}\,\left\{\mathrm{\tau }=\mathrm{\_C1}x+\mathrm{\_C2}t+\mathrm{\_C3}\,u\left(\mathrm{\tau }\right)=u\left(x\,t\right)\right\}\right]$

V := diff_table(v(x,t)):

W := diff_table(w(x,t)):

declare((u,v,w)(x,t));

$u\left(x\,t\right)\mathrm{will\; now\; be\; displayed\; as}u$

$v\left(x\,t\right)\mathrm{will\; now\; be\; displayed\; as}v$

$w\left(x\,t\right)\mathrm{will\; now\; be\; displayed\; as}w$

pde[1] := V[t] + 1/2*V[x,x,x] + 3*U[]*V[x]=0;

${\mathrm{pde}}_1≔v_t+\frac{v_{x,x,x}}{2}+3u{v}_x=0$

pde[2] := W[t] + 1/2*W[x,x,x] + 3*U[]*W[x] = 0;

${\mathrm{pde}}_2≔w_t+\frac{w_{x,x,x}}{2}+3u{w}_x=0$

pde[3] := U[t] - 1/4*U[x,x,x] - 3*U[]*U[x] - 3*diff( W[] - V[]^2, x)=0;

${\mathrm{pde}}_3≔u_t-\frac{u_{x,x,x}}{4}-3u{u}_x+6v{v}_x-3w_x=0$

sys := {pde[1], pde[2], pde[3]}:

sol := TWSolutions(sys, singsol=false);

$\mathrm{sol}≔\left\{u=-{\mathrm{\_C2}}^2{\tanh \left(\mathrm{\_C2}x+\mathrm{\_C3}t+\mathrm{\_C1}\right)}^2+\frac{{\mathrm{\_C2}}^3-\mathrm{\_C3}}{3\mathrm{\_C2}}\,v=-\frac{\sqrt{3{\mathrm{\_C2}}^4+6\mathrm{\_C2}\mathrm{\_C3}}\tanh \left(\mathrm{\_C2}x+\mathrm{\_C3}t+\mathrm{\_C1}\right)}{3}-\frac{3\mathrm{c\_\_5}}{2\sqrt{3{\mathrm{\_C2}}^4+6\mathrm{\_C2}\mathrm{\_C3}}}\,w=\mathrm{c\_\_5}\tanh \left(\mathrm{\_C2}x+\mathrm{\_C3}t+\mathrm{\_C1}\right)+\mathrm{\_C4}\right\},\left\{u=-{\mathrm{\_C2}}^2{\tanh \left(\mathrm{\_C2}x+\mathrm{\_C3}t+\mathrm{\_C1}\right)}^2+\frac{{\mathrm{\_C2}}^3-\mathrm{\_C3}}{3\mathrm{\_C2}}\,v=\frac{\sqrt{3{\mathrm{\_C2}}^4+6\mathrm{\_C2}\mathrm{\_C3}}\tanh \left(\mathrm{\_C2}x+\mathrm{\_C3}t+\mathrm{\_C1}\right)}{3}+\frac{3\mathrm{c\_\_5}}{2\sqrt{3{\mathrm{\_C2}}^4+6\mathrm{\_C2}\mathrm{\_C3}}}\,w=\mathrm{c\_\_5}\tanh \left(\mathrm{\_C2}x+\mathrm{\_C3}t+\mathrm{\_C1}\right)+\mathrm{\_C4}\right\}$

pde := U[t]^2 = 2*U[]*U[x]^2-(1+U[]^2)*U[x,x];

$\mathrm{pde}≔{u_t}^2=2u{u_x}^2-\left(u^2+1\right)u_{x,x}$

TWSolutions(pde, output = ODE);  # this will be the ODE solved

TWSolutions(pde, extended);          # solution more general than a tanh power series solution

$\left\{u=\tan \left(\frac{{\mathrm{\_C4}}^2\ln \left(-\frac{{\mathrm{c\_\_5}}^2\left(\mathrm{c\_\_7}\ln \left(\frac{\tanh \left(\mathrm{\_C4}x+\mathrm{c\_\_5}t+\mathrm{\_C3}\right)+1}{\tanh \left(\mathrm{\_C4}x+\mathrm{c\_\_5}t+\mathrm{\_C3}\right)-1}\right)-2\mathrm{c\_\_8}\right)}{2{\mathrm{\_C4}}^2}\right)}{{\mathrm{c\_\_5}}^2}\right)\right\}$

TWSolutions(pde, function = identity, output = ODE);  # this will be the ODE solved

$\left[\left\{\left({u\left(\mathrm{\tau }\right)}^2+1\right){\mathrm{c\_\_7}}^2{u}_{\mathrm{\tau },\mathrm{\tau }}+\left(-2u\left(\mathrm{\tau }\right){\mathrm{c\_\_7}}^2+{\mathrm{c\_\_8}}^2\right){u_{\mathrm{\tau }}}^2\right\}\,\left\{\mathrm{\tau }=\mathrm{c\_\_7}x+\mathrm{c\_\_8}t+\mathrm{\_C3}\,u\left(\mathrm{\tau }\right)=u\right\}\right]$

TWSolutions(pde, function = identity, extended);      # simpler extended solution