Reality conditions on the coefficient matrix in the Wronskian and Casoratian condition equations for the KdV equation and the Toda lattice

The Wronskian technique generates solutions of the Korteweg-de Vries (KdV) equation and the Toda lattice from entry vectors that satisfy linear condition equations with a constant coefficient matrix $A$: a real matrix $A$ always gives a real tau function and hence a real solution, and the one known complex family — eigenvalues occurring in conjugate pairs arranged in block canonical form — gives the real but singular complexiton solutions. Zhang's 2006 notes pose, and explicitly leave open, the classification question: precisely which complex matrices $A$ generate real solutions in the Wronskian (KdV) and Casoratian (Toda) settings?

Background

The Korteweg-de Vries (KdV) equation $u_t+6uu_x+u_{xxx}=0$ is the prototype integrable nonlinear evolution equation. In Hirota's bilinear method one substitutes $u=2(\ln\tau)_{xx}$, which turns the equation into a quadratic (bilinear) equation for the tau function $\tau$. The Wronskian technique of Freeman and Nimmo (1983) takes $\tau$ to be a Wronskian determinant $\tau=W(\phi_1,\ldots,\phi_N)=\det[\phi_j^{(i-1)}]_{i,j=1}^N$ of $N$ functions of $x$ and $t$; the bilinear identity then reduces to Grassmann-Plücker relations, so $\tau$ solves the bilinear KdV equation as soon as the entry vector $\phi=(\phi_1,\ldots,\phi_N)^T$ satisfies a linear system, called the condition equations. For the KdV equation these can be written (Zhang 2006, generalizing Sirianunpiboon, Howard and Roy 1988 and Ma and You 2004) as $-\phi_{xx}=A\phi,\qquad \phi_t=-4\phi_{xxx},$ where $A$ is a constant $N\times N$ matrix; replacing $A$ by a similar matrix gives the same class of solutions, so $A$ may be taken in canonical (diagonal or Jordan) form. The eigenvalues of $A$ play the role of the eigenvalues of the stationary Schrödinger operator $-\partial_x^2$ at zero potential, and the spectrum organizes the classical solution types: negative real eigenvalues give solitons (negatons in the generalized setting), positive real eigenvalues give positons, zero eigenvalues give rational solutions, and complex eigenvalues occurring in conjugate pairs give complexitons, introduced by Ma in 2002. KdV complexitons are real solutions built from conjugate pairs of complex wave numbers; they are breather-like but necessarily singular, because trigonometric factors appear in the entries. Ma and You (2004) determined all solutions of the condition equations when $A$ takes a Jordan form, and Zhang's 2006 notes derived explicit general and 'effective' (minimal-parameter) solutions, their limits, and the fully parallel theory for the Toda lattice $x_{n,tt}=e^{x_{n-1}-x_n}-e^{x_n-x_{n+1}}$, a chain of unit masses with exponential nearest-neighbour interaction: there the tau function is a Casoratian determinant (the discrete analogue of a Wronskian, built from shifts in $n$ instead of $x$-derivatives), the entry vector satisfies difference condition equations with a constant coefficient matrix $A$, and the same eigenvalue classification holds, with complexitons of the Toda lattice constructed by Ma and Maruno (2004) and positon-negaton type solutions by Maruno, Ma and Oikawa (2004).

Reality enters as follows. If $A$ is real and the free parameters of the entries are real, then $\tau$ is real and $u=2(\ln\tau)_{xx}$ is a real solution. If $A$ is complex, reality becomes delicate: the complexiton canonical form (conjugate-pair eigenvalues arranged in real $2\times2$ blocks) yields a tau function that is real or pure imaginary, hence a real $u$, but Zhang showed that in this construction the conjugate blocks cannot simply be replaced by arbitrary complex blocks without losing the guarantee of a real solution. Zhang therefore posed (Remarks 3.7 and 4.5 of the notes) the classification question of which complex matrices $A$ generate real solutions in the Wronskian (KdV) and Casoratian (Toda) settings. For the modified KdV (mKdV) equation, whose Wronskian condition equations contain an explicit complex-conjugation operation, the analogous issue was later settled by introducing an auxiliary matrix $A=B\bar B$ (Zhang, Zhao, Sun and Zhou 2014), which shows the kind of answer expected but leaves the KdV and Toda questions open.

Problem Statement

For the KdV equation $u_t+6uu_x+u_{xxx}=0$ in Wronskian form — $\tau=W(\phi_1,\ldots,\phi_N)$, $u=2(\ln\tau)_{xx}$, with the entry vector satisfying the condition equations $-\phi_{xx}=A\phi$, $\phi_t=-4\phi_{xxx}$ for a constant $N\times N$ matrix $A$ considered up to similarity — determine necessary and sufficient conditions on a complex matrix $A$ under which the generated tau function is real up to a nonzero constant complex factor on the set where $\tau\neq0$, so that $u$ is a real solution of the KdV equation. The answer must specify the dependence on the free (initial) parameters of the entry vector, i.e. for which $A$ reality holds for all admissible data and for which it holds only on exceptional subsets, and it must reproduce the known instances: a real $A$ always generates real solutions; the conjugate-pair block canonical form (the complexiton case, Case 6 of the source) generates the real but singular complexiton solutions; and the variant of the source's complexiton construction in which the conjugate blocks $A^*,B^*$ are replaced by arbitrary complex blocks does not guarantee real solutions. Establish the parallel necessary and sufficient conditions for the Toda lattice $x_{n,tt}=e^{x_{n-1}-x_n}-e^{x_n-x_{n+1}}$ in Casoratian form — $\tau_n=\mathrm{Cas}(\phi_1,\ldots,\phi_N)=\det[\phi_j(n+i-1,t)]_{i,j=1}^N$, with the entry vector satisfying the difference condition equations $\phi(n+1,t)+\phi(n-1,t)=A\phi(n,t)$, $2\phi_t(n,t)=\phi(n+1,t)-\phi(n-1,t)$ for a constant $N\times N$ matrix $A$ considered up to similarity (Eqs. (4.20)-(4.21) of the source) — under which the Casoratian tau function is real up to a single nonzero constant complex factor independent of $n$ and $t$, so that $V_n=\tau_{n+1}\tau_{n-1}/\tau_n^2-1$ and hence the dependent variables of the lattice are real. As the direct completion of this classification, make explicit which real solution types (solitons, positons, negatons, rational solutions, complexitons, and their mixtures) arise, and in particular whether any complex $A$ yields real solutions of the KdV equation or the Toda lattice that are not reachable from real $A$ — thereby completing the reality part of the solution classification of the Wronskian/Casoratian technique for soliton equations with KdV-type bilinear forms posed in Zhang's notes (Remarks 3.7 and 4.5).

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