Literature about solitons and Hirota derivatives

nonlinear systemnonlinear-analysispartial differential equationsreference-requestsoliton-theory

This summer I'll learn a mini-course about soliton theory ("Soliton equations and symmetric functions" in LHSM (Russian summer school in mathematics).
The web-page of this course is https://mccme.ru/dubna/2019/courses/rozhkovskaya.html
(warning: it's in Russian, so here's my translation of its program:

Examples of soliton equations, their solutions. Properties of solitons.
Definition and properties of Hirota derivatives.
KdV and KP equations in terms of Hirota derivatives.
Bilinear form of the KP hierarchy.
Symmetric functions: main definitions, properties of elementary, complete and power sum symmetric functions.
Interpretation of the bilinear form of KP hierarchy in terms of symmetric functions.
(If it'll be enough time) Some words about the action of fermions on the symmetric functions and solutions of KP hierarchy).

This program has intrigued me (despite my main interest is representation theory and related topics) and now I'm looking for a book in which the soliton theory will be outlined according to this program (maybe only without symmetric functions…).

I saw Kasman's book but he doesn't say too much about Hirota derivatives and his style seems simplistic to me…

I'll be grateful for anyone who'll give me some references. It'll be great if these books will contain an "algebraic approach" to the subject or some connections with representation theory… But any other reference will also be greeted with admiration!

Best Answer

I can't really tell if this is what you're looking for, but it's a couple of ideas at least:

R. Hirota, The Direct Method in Soliton Theory.
T. Miwa, M. Jimbo, E. Date, Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras.

Related Solutions

Non-linear waves and solitons: verifying solution for Nonlinear Schrödinger Equation

You have a lot of moving pieces, here.

Construct $U$ and $V$ with the given expressions, and $u_0(x,t)=e^{2it}.$ Note that $u_x=0$ and $\overline{u}_x=0.$ \begin{align*} U&=\left[\begin{matrix}0 &u\\ \overline{u} &0\end{matrix}\right]=\left[\begin{matrix}0 &e^{2it}\\ e^{-2it} &0\end{matrix}\right] \\ V&=\left[\begin{matrix}|u|^2 &iu_x \\ -i\,\overline{u}_x &-|u|^2\end{matrix}\right] =\left[\begin{matrix}1 &0 \\ 0 &-1\end{matrix}\right]. \end{align*}
Next we build the AKNS operators $X$ and $T$ as follows: \begin{align*} X&=-i\lambda\sigma_3+iU=-i\lambda\left[\begin{matrix}1 &0\\0&-1\end{matrix}\right]+i\left[\begin{matrix}0 &e^{2it}\\ e^{-2it} &0\end{matrix}\right]=\left[\begin{matrix}-i\lambda &ie^{2it}\\ ie^{-2it} &i\lambda\end{matrix}\right] \\ T&=2\lambda X+iV =2\lambda \left[\begin{matrix}-i\lambda &ie^{2it}\\ ie^{-2it} &i\lambda\end{matrix}\right]+i\left[\begin{matrix}1 &0 \\ 0 &-1\end{matrix}\right] =\left[\begin{matrix}i-2\lambda^2i &2\lambda i e^{2it} \\ 2\lambda i e^{-2it} &-i+2\lambda^2 i\end{matrix}\right]. \end{align*}
Moving on, we calculate $\Psi_0;$ but it has an annoying term $e^{it\sigma_3}$ in it. That is, we have to exponentiate a matrix. Fortunately, we do not have to go to the trouble of finding the eigenvalues and eigenvectors and exponentiating the result. Mathematica has the MatrixExp function that will do this for us. We get $$e^{it\sigma_3}=\left[\begin{matrix}e^{it} &0 \\ 0 &e^{-it}\end{matrix}\right]. $$ Hence, \begin{align*} \Psi_0&=\frac{1}{\sqrt{2\mu(\mu+\lambda)}}\left[\begin{matrix}e^{it} &0 \\ 0 &e^{-it}\end{matrix}\right]\left[\begin{matrix}e^{i\mu(x+2\lambda t)} &-(\mu+\lambda)e^{-i\mu(x+2\lambda t)}\\(\mu+\lambda)e^{i\mu(x+2\lambda t)}&e^{-i\mu(x+2\lambda t)}\end{matrix}\right] \\ &=\frac{1}{\sqrt{2\mu(\mu+\lambda)}}\left[\begin{matrix}e^{i[\mu(x+2\lambda t)+t]} &-(\mu+\lambda)e^{-i[\mu(x+2\lambda t)-t]}\\(\mu+\lambda)e^{i[\mu(x+2\lambda t)-t]}&e^{-i[\mu(x+2\lambda t)+t]}\end{matrix}\right]. \end{align*}
The only missing pieces to $(4)$ are $\Psi_x$ and $\Psi_t,$ which for our case, of course, are really $\dfrac{\partial}{\partial x}\Psi_0$ and $\dfrac{\partial}{\partial t}\Psi_0,$ respectively. I get \begin{align*} \dfrac{\partial}{\partial x}\Psi_0&= \frac{1}{\sqrt{2\mu(\mu+\lambda)}}\left[ \begin{matrix} i\mu e^{i[\mu(x+2\lambda t)+t]} &-(\mu+\lambda)(-i\mu)e^{-i[\mu(x+2\lambda t)-t]}\\ (\mu+\lambda)(i\mu)e^{i[\mu(x+2\lambda t)-t]} &-i\mu e^{-i[\mu(x+2\lambda t)+t]} \end{matrix}\right] \quad\text{and} \\ \dfrac{\partial}{\partial t}\Psi_0&= \frac{1}{\sqrt{2\mu(\mu+\lambda)}}\left[ \begin{matrix} (i(2\mu\lambda+1))e^{i[\mu(x+2\lambda t)+t]} &(\mu+\lambda)(i(2\mu\lambda-1))e^{-i[\mu(x+2\lambda t)-t]}\\ (\mu+\lambda)(i(2\mu\lambda-1))e^{i[\mu(x+2\lambda t)-t]} & -i(2\mu\lambda+1)e^{-i[\mu(x+2\lambda t)+t]}\end{matrix}\right]. \end{align*} To see that $\Psi_0$ satisfies \begin{align*} \Psi_x&=X\Psi \\ \Psi_t&=T\Psi \end{align*} should now be a matter of algebra.
Speaking of algebra, we can note that if $\Psi\mapsto\alpha\Psi,$ whether $\Psi$ is a solution or not remains unchanged, because $\partial_x(\alpha\Psi)=\alpha\,\partial_x\Psi,$ and $X(\alpha\Psi)=\alpha X\Psi$, and similarly for the $t$ equation. Because M.SE has a rather narrow window for typing up equations, this has a practical value in that we can ignore constants out in front of $\Psi$. That is, we need to check that \begin{align*} &\frac{1}{\sqrt{2\mu(\mu+\lambda)}}\left[\begin{matrix} i\mu e^{i[\mu(x+2\lambda t)+t]} &-(\mu+\lambda)(-i\mu)e^{-i[\mu(x+2\lambda t)-t]}\\ (\mu+\lambda)(i\mu)e^{i[\mu(x+2\lambda t)-t]} &-i\mu e^{-i[\mu(x+2\lambda t)+t]}\end{matrix}\right] \\ =&\left[\begin{matrix}-i\lambda &ie^{2it}\\ ie^{-2it} &i\lambda\end{matrix}\right]\frac{1}{\sqrt{2\mu(\mu+\lambda)}}\left[\begin{matrix}e^{i[\mu(x+2\lambda t)+t]} &-(\mu+\lambda)e^{-i[\mu(x+2\lambda t)-t]}\\(\mu+\lambda)e^{i[\mu(x+2\lambda t)-t]}&e^{-i[\mu(x+2\lambda t)+t]}\end{matrix}\right], \end{align*} or \begin{align*} &\left[ \begin{matrix} i\mu e^{i[\mu(x+2\lambda t)+t]} &-(\mu+\lambda)(-i\mu)e^{-i[\mu(x+2\lambda t)-t]}\\ (\mu+\lambda)(i\mu)e^{i[\mu(x+2\lambda t)-t]} &-i\mu e^{-i[\mu(x+2\lambda t)+t]} \end{matrix} \right] \\ =&\left[ \begin{matrix} -i\lambda &ie^{2it}\\ ie^{-2it} &i\lambda \end{matrix} \right] \left[ \begin{matrix} e^{i[\mu(x+2\lambda t)+t]} &-(\mu+\lambda)e^{-i[\mu(x+2\lambda t)-t]}\\(\mu+\lambda)e^{i[\mu(x+2\lambda t)-t]} &e^{-i[\mu(x+2\lambda t)+t]} \end{matrix} \right]. \end{align*} Performing the matrix multiplication on the RHS yields \begin{align*}&X\Psi_0= \\ &i\left[ \begin{matrix} -\lambda e^{i[\mu(x+2\lambda t)+t]}+(\mu+\lambda)e^{2it}e^{i[\mu(x+2\lambda t)-t]} &\lambda(\mu+\lambda)e^{-i[\mu(x+2\lambda t)-t]}+e^{2it}e^{-i[\mu(x+2\lambda t)+t]} \\ e^{-2it}e^{i[\mu(x+2\lambda t)+t]}+\lambda(\mu+\lambda)e^{i[\mu(x+2\lambda t)-t]} &-(\mu+\lambda)e^{-2it}e^{-i[\mu(x+2\lambda t)-t]}+\lambda e^{-i[\mu(x+2\lambda t)+t]} \end{matrix} \right] \end{align*} The $e^{2it}$ and $e^{-2it}$ have the effect of enabling the exponentials to be like terms, with the following spectacular simplification: \begin{align*}&X\Psi_0 \\ &=i\left[ \begin{matrix} -\lambda e^{i[\mu(x+2\lambda t)+t]}+(\mu+\lambda)e^{i[\mu(x+2\lambda t)+t]} &\lambda(\mu+\lambda)e^{-i[\mu(x+2\lambda t)-t]}+e^{-i[\mu(x+2\lambda t)-t]} \\ e^{i[\mu(x+2\lambda t)-t]}+\lambda(\mu+\lambda)e^{i[\mu(x+2\lambda t)-t]} &-(\mu+\lambda)e^{-i[\mu(x+2\lambda t)+t]}+\lambda e^{-i[\mu(x+2\lambda t)+t]} \end{matrix} \right] \\ &=i\left[ \begin{matrix} \mu e^{i[\mu(x+2\lambda t)+t]} &[\lambda(\mu+\lambda)+1]e^{-i[\mu(x+2\lambda t)-t]} \\ [\lambda(\mu+\lambda)+1]e^{i[\mu(x+2\lambda t)-t]} &-\mu e^{-i[\mu(x+2\lambda t)+t]} \end{matrix} \right] \\ &=i\left[ \begin{matrix} \mu e^{i[\mu(x+2\lambda t)+t]} &[\lambda\mu+\mu^2]e^{-i[\mu(x+2\lambda t)-t]} \\ [\lambda\mu+\mu^2]e^{i[\mu(x+2\lambda t)-t]} &-\mu e^{-i[\mu(x+2\lambda t)+t]} \end{matrix} \right] \\ &=i\mu\left[ \begin{matrix} e^{i[\mu(x+2\lambda t)+t]} &(\lambda+\mu)e^{-i[\mu(x+2\lambda t)-t]} \\ (\lambda+\mu)e^{i[\mu(x+2\lambda t)-t]} &-e^{-i[\mu(x+2\lambda t)+t]} \end{matrix} \right] \\ \end{align*} At this point, a bit of simplification of $\partial_x\Psi_0$ is in order. At last count, we had \begin{align*} &\phantom{=}\partial_x\Psi_0 \\ &=\left[ \begin{matrix} i\mu e^{i[\mu(x+2\lambda t)+t]} &-(\mu+\lambda)(-i\mu)e^{-i[\mu(x+2\lambda t)-t]}\\ (\mu+\lambda)(i\mu)e^{i[\mu(x+2\lambda t)-t]} &-i\mu e^{-i[\mu(x+2\lambda t)+t]} \end{matrix} \right] \\ &=\left[ \begin{matrix} i\mu e^{i[\mu(x+2\lambda t)+t]} &i\mu(\mu+\lambda)e^{-i[\mu(x+2\lambda t)-t]}\\ i\mu(\mu+\lambda)e^{i[\mu(x+2\lambda t)-t]} &-i\mu e^{-i[\mu(x+2\lambda t)+t]} \end{matrix} \right] \\ &=i\mu\left[ \begin{matrix} e^{i[\mu(x+2\lambda t)+t]} &(\mu+\lambda)e^{-i[\mu(x+2\lambda t)-t]}\\ (\mu+\lambda)e^{i[\mu(x+2\lambda t)-t]} &-e^{-i[\mu(x+2\lambda t)+t]} \end{matrix} \right]. \end{align*} Now you can see that these are, indeed, equal. The $\partial_t \Psi_0=T\Psi_0$ case should work out similarly.

Literature about the “Spaltungssatz”

The primary decomposition theorem in the book of Hoffmann and Kunze seems to be a fitting answer. Most of it follows by applying the Spaltungssatz of de Jong inductively. For reference I reproduce/reformulate its statement: (p. 220, Chapter 6, Theorem 12 & Corollary)

Let $\varphi : V \to V$ be a linear endomorphism. Decompose its minimal polynomial into irreducible factors (this can be done over any field). I.e. let $p_1, \dots, p_k$ be distinct irreducible monic polynomials and $n_1, \dots, n_k$ be positive integers such that $m_\varphi = \prod_{i=1}^k p_i^{n_i}$. Further, define $W_i := \ker (p_i(\varphi))$. Then the following statements hold:

The spaces $W_i$ are non-trivial and $\varphi$-invariant.
For each $i$, the restriction of $\varphi$ to $W_i$ has minimal polynomial $p_i^{n_i}$. Consequently, $\dim(W_i) \ge n_i \cdot \deg (p_i)$.
The spaces $W_i$ form a direct sum $V = \bigoplus_{i=1}^k W_i$. Call the projections associated with this direct sum $\pi_1, \dots, \pi_k$.
For each $i$ there is a polynomial $h_i$ such that $h_i(\varphi) =\pi_i$.
If another linear endomorphism $\psi : V\to V$ commutes with $\varphi$, then it commutes with each $\pi_i$ and each $W_i$ is $\psi$-invariant.
All $\varphi$-invariant subspaces can be written as direct sum of some of the $W_i$.

Parts of this can be expressed in matrix form. There exists a basis of $V$, such that the matrix of $\varphi$ wrt. this basis has block-diagonal form with non-empty blocks $A_1, \dots, A_k$ such that each $A_i$ has minimal polynomial $p_i^{n_i}$. The size of $A_i$ is at least $n_i \cdot \deg(p_i)$.

For the proof, Hoffmann and Kunze first construct the polynomials $h_i$ and derive the rest from there. The statements above which are not included in Hoffmann and Kunze are relatively easy corollaries of the others. All in all, this theorem produces a "best possible" block-diagonal form, without imposing restrictions on the underlying field. From another point of view, it characterises the relation between invariant subspaces and factors of the minimal polynomial.

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Non-linear waves and solitons: verifying solution for Nonlinear Schrödinger Equation

Literature about the “Spaltungssatz”

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