A new class of solutions for the multi-component extended Harry Dym equation

TL;DR

This paper introduces a novel transformation linking two integrable systems, leading to new wave-like solutions that extend the known multi-phase solutions of the multi-component extended Harry Dym equation.

Contribution

It presents a new point transformation between the multi-component Harry Dym and extended Harry Dym equations that generates a new class of wave-like solutions, expanding the solution space.

Findings

New wave-like solutions for the multi-component extended Harry Dym equation.

Transformation does not preserve multi-phase solution class.

Method applicable to other integrable systems with similar properties.

Abstract

We construct a point transformation between two integrable systems, the multi-component Harry Dym equation and the multi-component extended Harry Dym equation, that does not preserve the class of multi-phase solutions. As a consequence we obtain a new type of wave-like solutions, generalising the~multi-phase solutions of the multi-component extended Harry Dym equation. Our construction is easily transferable to other integrable systems with analogous properties.

Equations

$$\biggl{(}\sum_{m=0}^{M}\epsilon_{m}\lambda^{m}\biggr{)}\psi_{xx}=\biggl{(}\sum_{m=0}^{M}v_{m}\lambda^{m}\biggr{)}\psi$$

$$v_t=-\frac{1}{2}({\partial }_x^2-{\kappa }^2)(v^{-1/2}{)}_x$$

$$v=\frac{1}{c^2{r}^2},$$

$$\theta =\stackrel{r}{\int }\frac{\sqrt{\lambda }d\lambda }{\sqrt{4c^{-2}+s_1\lambda +s_0{\lambda }^2+{\kappa }^2{\lambda }^3}};$$

$$v=\frac{e^{2\kappa x}}{c^2{R}^2},$$

$$\frac{e^{\kappa x}-1}{\kappa }+ct=\vartheta =\stackrel{R}{\int }\frac{\sqrt{\lambda }d\lambda }{\sqrt{4c^{-2}+s_1\lambda +s_0{\lambda }^2}}.$$

$$4c^{-2}+s_1\lambda +s_0{\lambda }^2=s_0(\lambda -{\lambda }_1{)}^2,$$

$$\frac{e^{\kappa x}-1}{\kappa}+ct=\vartheta=\frac{1}{\sqrt{s_{0}}}\overset{R}{\int}\frac{\sqrt{\lambda}\,d\lambda}{\lambda-\lambda_{1}}=\frac{1}{\sqrt{s_{0}}}\biggl{(}2\sqrt{R}-2\sqrt{\smash[b]{\lambda_{1}}}\mathop{\rm arctanh}\sqrt{\frac{R}{\smash[b]{\lambda_{1}}}}\biggr{)}.$$

$$4c^{-2}+s_1\lambda +s_0{\lambda }^2+{\kappa }^2{\lambda }^3={\kappa }^2(\lambda -{\lambda }_2)(\lambda -{\lambda }_3{)}^2,$$

$$\theta$$

$${\psi }_{xx}=U\psi$$

$$U(x,t,\lambda )=\sigma +\frac{u^1}{\lambda }+\frac{u^2}{{\lambda }^2}+\frac{u^3}{{\lambda }^3}+...,$$

$${\psi }_t=a{\psi }_x-\frac{1}{2}{a}_x\psi .$$

$$U_t=\left(-\frac{1}{2}{\partial }_x^3+2U{\partial }_x+U_x\right)a,$$

$$u_t^k=u_x^{k+1}+a^1{u}_x^k+2u^k{a}_x^1,k=1,2,...,u^1+\xi =\frac{1}{2}{a}_{1,xx}-2\sigma a_1,$$

$$U(x,t,\lambda )=\sigma +\frac{u^1}{\lambda }+\frac{u^2}{{\lambda }^2}+...+\frac{u^M}{{\lambda }^M}$$

$$u_t^1=a_1{u}_x^1+2u^1{a}_{1,x},u^1+\xi =\frac{1}{2}{a}_{1,xx}-2\sigma a_1,$$

$$u_t^k=u_x^{k+1}+a^1{u}_x^k+2u^k{a}_x^1,k=1,2,...,M-1,u_t^M=a^1{u}_x^M+2u^M{a}_x^1,$$

$$u_t^k=u_x^{k+1}+a^1{u}_x^k+2u^k{a}_x^1,k=1,2,...,u^1+\xi =\frac{1}{2}{a}_{1,xx}.$$

$$u_t^1=a_1{u}_x^1+2u^1{a}_{1,x},u^1+\xi =\frac{1}{2}{a}_{1,xx},$$

$$u_t^k=u_x^{k+1}+a^1{u}_x^k+2u^k{a}_x^1,k=1,2,...,M-1,$$

$$u_t^M=a^1{u}_x^M+2u^M{a}_x^1,$$

$$u_t^1=-\frac{1}{2}(a_{-1}{)}_{xxx},$$

$$u_t^k=a_{-1}{u}_x^{k-1}+2u^{k-1}(a_{-1}{)}_x,k=2,\dots ,M,$$

$$a_{-1}=(u^M{)}^{-1/2}.$$

$${\psi }_{xx}=\left(\frac{u^1}{\lambda }+\frac{u^2}{{\lambda }^2}+...+\frac{u^M}{{\lambda }^M}\right)\psi$$

$${\psi }_{zz}=\left(\frac{u^1}{\lambda }+\frac{u^2}{{\lambda }^2}+...+\frac{u^M}{{\lambda }^M}\right)\psi .$$

$$z=\frac{e^{\kappa x}-1}{\kappa },$$

$${\phi }_{xx}=\left(\sigma +\frac{v^1}{\lambda }+\frac{v^2}{{\lambda }^2}+...+\frac{v^M}{{\lambda }^M}\right)\phi ,$$

$$v_t^1=-\frac{1}{2}({\partial }_x^3-{\kappa }^2{\partial }_x){\tilde{a}}_{-1},$$

$$v_t^k={\tilde{a}}_{-1}{v}_x^{k-1}+2v^{k-1}({\tilde{a}}_{-1}{)}_x,k=2,\dots ,M,$$

$${\tilde{a}}_{-1}=(v^M{)}^{-1/2}.$$

$${\phi }_t=\frac{1}{\lambda }\left({\tilde{a}}_{-1}{\phi }_x-\frac{1}{2}({\tilde{a}}_{-1}{)}_x\phi \right).$$

Full text

1 Introduction

In a number of papers [2, 3, 4], integrable systems associated with the energy-dependent linear Schrödinger equation

[TABLE]

were investigated in details. In these papers, two main classes were selected by the conditions $\epsilon_{M}=0$, $v_{M}=1$ (the so called “multi-component KdV systems”) and $\epsilon_{M}=0$, $v_{0}=1$ (the so called “multi-component extended Harry Dym systems” or “multi-component Camassa–Holm systems”). In the present paper we consider another class determined by a sole restriction $v_{M}=0$ (the so called “multi-component Harry Dym systems” (13) or “multi-component Hunter–Saxton equations”). We show that the multi-component Harry Dym equations are connected with the multi-component extended Harry Dym equations by the $\kappa$-transformation introduced below, see (15). The method presented here is applicable to any solutions. Without loss of generality and for simplicity we restrict our consideration to multi-phase solutions only. Applying the $\kappa$-transformation to the multi-phase solutions of multi-component Harry Dym systems we obtain a new class of solutions of multi-component extended Harry Dym systems, which we call the $\kappa$-deformed multi-phase solutions. This new class of solutions cannot be obtained as a reduction of multi-phase solutions. Whilst multi-phase solutions belong to the so called “solitonic sector” (i.e., the case of reflectionless potentials), the new class of solutions (the $\kappa$-deformed multi-phase solutions) has a rapidly increasing behaviour with respect to $x$.

Example 1

The extended Harry Dym equation ($\kappa$ is an arbitrary constant, $\kappa\neq 0$)

[TABLE]

possesses the one-phase solution ($c$ is an arbitrary constant)

[TABLE]

where $r$ is a function of $\theta=x+ct$ determined implicitly by ($s_{0},s_{1}$ are arbitrary constants)

[TABLE]

and simultaneously the $\kappa$-deformed one-phase solution

[TABLE]

where the function $R(\vartheta)$ is determined implicitly by

[TABLE]

These solutions are significantly different: the first solution is essentially one-dimensional (see Figure 1) and is obtained by a regular procedure (one can look for a travelling wave reduction determined by the ansatz $v(\theta)$, where $\theta=x+ct$), while the second solution is two-dimensional (see Figure 2) and is obtained (see below) by means of an invertible point transformation between the extended Harry Dym equation and its high-frequency limit ($\kappa=0$), which is the well-known Harry Dym equation.

Many integrable systems have such an invertible point transformation that connects them with their high-frequency limits. So, once a one-phase (or a multi-phase) solution is found, one can construct the so-called $\kappa$-deformed solution following our approach.

Moreover, in the particular case

[TABLE]

the second solution can be found in elementary functions, i.e.

[TABLE]

In another particular case

[TABLE]

the first solution can be found in elementary functions, i.e.

[TABLE]

Thus, both types of solutions have completely different behaviour.

In this paper we restrict our consideration to the simplest integrable system whose multi-phase solutions are associated with hyperelliptic Riemann surfaces: the multi-component extended Harry Dym equation.

2 A Special Class of Integrable Systems

The energy dependent linear Schrödinger equation

[TABLE]

was recently [8] investigated for the special case (with respect to the spectral parameter $\lambda$)

[TABLE]

where $\sigma$ is an arbitrary constant. If $\sigma\neq 0$, then the parameter $\sigma$ can be fixed to 1 by an appropriate scaling of independent variables and dependent functions without loss of generality.

Integrable systems associated with (1) can be obtained from the compatibility condition $(\psi_{t})_{xx}=(\psi_{xx})_{t}$, where

[TABLE]

The compatibility condition $(\psi_{t})_{xx}=(\psi_{xx})_{t}$ yields the relationship

[TABLE]

which leads to the dispersive integrable chain ($\xi$ is an integration constant)

[TABLE]

where $a=\lambda+a^{1}$ and the function $U$ is determined by (2). This dispersive integrable chain can be reduced to $M$-component integrable dispersive systems by simple reductions $u^{M+1}=0$ for any natural number $M$. This means that one should consider the linear problem (1), (3), where

[TABLE]

instead of (2). If $M=1$, one obtains the remarkable Camassa–Holm equation

[TABLE]

if $M>1$, the multi-component generalisation of the Camassa–Holm equation is

[TABLE]

where again $u^{1}+\xi=\frac{1}{2}a_{1,xx}-2\sigma a_{1}$.

Below we also investigate the special case $\sigma=0$ and discuss the relationship between integrable systems determined by both choices $\sigma=0$ and $\sigma\neq 0$. The corresponding dispersive integrable chain reduces to the form (cf. (4))

[TABLE]

So, the main difference between (4) and (7) is a difference between the constraints $u^{1}+\xi=\frac{1}{2}a_{1,xx}-2\sigma a_{1}$ and $u^{1}+\xi=\tfrac{1}{2}a_{1,xx}$. Again if $M=1$, one can obtain the Hunter–Saxton equation

[TABLE]

which is a high frequency limit of the Camassa–Holm equation (see detail in [7]). If $M>1$, the multi-component generalisation of the Hunter–Saxton equation is (cf. (6))

[TABLE]

where again $u^{1}+\xi=\frac{1}{2}a_{1,xx}$. If instead of the choice $a=\lambda+a_{1}$ we consider the dependence $a=a_{-1}/\lambda$, then

[TABLE]

If $M=1$, then this is well-known Harry Dym equation; if $M>1$, then this system (8) will be called the multi-component Harry Dym equation.

Below we show that integrable systems (8) associated with the energy dependent linear Schrödinger equation (see (1) and (5) in the limit $\sigma=0$)

[TABLE]

can be interpreted as a high frequency limit of the so called multi-component extended Harry Dym equation (see detail below).

Indeed, one can consider the linear spectral problem (9) written in the form

[TABLE]

Then we apply the point transformation

[TABLE]

where $\kappa$ is an arbitrary parameter. Then $\partial_{z}\rightarrow e^{-\kappa x}\partial_{x}$. If $\kappa\rightarrow 0$, then $z(x,\kappa)\rightarrow x$.

Under transformation (11) the linear spectral problem (10) becomes (see (1) and (2))

[TABLE]

where $\sigma=\kappa^{2}/4$, $v^{k}=u^{k}e^{2\kappa x}$ and $\psi=\varphi\exp(\kappa x/2)$. The high frequency limit $\kappa\rightarrow 0$ reduces the above linear problem to (9). We illustrate this property for the multi-component extended Harry Dym equation

[TABLE]

This system follows from the compatibility condition $(\varphi_{t})_{xx}=(\varphi_{xx})_{t}$, where the function $\varphi$ is a common solution of two linear equations, i.e. (12) and (cf. (3))

[TABLE]

The high frequency limit $\kappa\rightarrow 0$ leads to the system (8). Now we apply point transformation (11) to system (8) written in the form (here we simply replaced $x$ by $z$)

[TABLE]

Then we again obtain system (13), where $a_{-1}=\tilde{a}_{-1}e^{\kappa x},$ $v^{k}=u^{k}e^{2\kappa x}$. Thus integrable systems (13) and (14) are connected with each other by the point transformation

[TABLE]

and simultaneously system (8) $\equiv$ (14) is a high frequency limit of system (13).

3 Multi-Phase Solutions and a High Frequency Limit

To illustrate a difference between the general case $\kappa\neq 0$ and its high-frequency limit $\kappa=0$, in this section we consider multi-gap solutions of the multi-component extended Harry Dym equation

[TABLE]

where $\tilde{a}_{-1}=(v^{M})^{-1/2}$.

In this case linear problem (1), (3) reduces to the form

[TABLE]

where $\phi=\psi\psi^{+}$ (here $\psi$ and $\psi^{+}$ are two linearly independent solutions), $a=\tilde{a}_{-1}/\lambda$ and $S(\lambda)$ is a polynomial expression with constant coefficients.

As usual, finite-gap solutions connected with hyperelliptic Riemann surfaces can be constructed in several steps:

1. We seek polynomial solutions (with respect to the spectral parameter $\lambda$) for the function $\phi$ in the factorised form

[TABLE]

where $N$ is an arbitrary natural number.

2. Since function $\phi$ is a polynomial of the degree$\ N$, the dependence $S(\lambda)$ is a polynomial of the degree $2N+M$, i.e.

[TABLE]

where $s_{-1}=\kappa^{2},$ while other $s_{k}$ are “integration constants”.

3. Expanding $\phi$ by virtue of (18) in the first equation of (17) with respect to the spectral parameter $\lambda$, one can find expressions for field variables $v_{k}$. Indeed, substituting (18) into (17), one obtains

[TABLE]

where

[TABLE]

4. Following B.A. Dubrovin [5, 6], we consider the limit $\lambda\rightarrow r^{i}(x,t)$ of the two equations (17). This straightforward computation yields two autonomous systems

[TABLE]

where $a^{i}(\mathbf{r})=a(\lambda,\mathbf{r})|_{\lambda=r^{i}}$. In our case (see (20), $k=M$)

[TABLE]

5. A straightforward integration of (22) implies multi-phase solutions of (13) written in the implicit form222Explicit formulae for more wide class of integrable systems, whose multi-phase solutions are associated with hyperelliptic Riemann surfaces, were obtained in [1]

[TABLE]

[TABLE]

Remark: If $N=1$, then a one-phase solution is parameterised by a single function $r(\theta)$. Namely,

[TABLE]

where the function $r(\theta)$ is determined by the relationship (here $s_{M+1}=4c^{-2}$)

[TABLE]

3.1 Finite-Gap Solutions and $\kappa$-Transformation

Now we can compare finite-gap solutions for both cases $\kappa=0$ and $\kappa\neq 0$. So in the case $\kappa=0$ we have for system (14) multi-phase solutions (cf. (20) and (21))

[TABLE]

where

[TABLE]

The dependencies $R^{k}(z,t)$ are presented in implicit form (cf. (23))

[TABLE]

[TABLE]

where (cf. (19))

[TABLE]

Under transformation (11) system (14) becomes (13), while multi-phase solutions of system (14) take the form (we remind the reader that $v^{k}=u^{k}e^{2\kappa x}$)

[TABLE]

where the dependencies $R^{k}(x,t)$ are presented in implicit form (cf. (23), (24))

[TABLE]

[TABLE]

Thus, we found a new type of solutions of multi-component Harry Dym equation (13), which do not coincide with the corresponding multi-phase solutions (23).

3.2 The One-Phase Solution

In the particular case $N=1$, the multi-component extended Harry Dym equation (16) has the one-phase solution

[TABLE]

where the function $r(\theta)$ is determined by the relationship (here $s_{M+1}=4c^{-2}$)

[TABLE]

and the $\kappa$-deformed one-phase solution

[TABLE]

where the function $R(\vartheta)$ is determined by the relationship (here $s_{M+1}=4c^{-2}$)

[TABLE]

3.3 The Extended Harry Dym Equation

Here we consider the particular case $M=1$, i.e., the extended Harry Dym equation

[TABLE]

Its $N$-phase solutions are determined by333The simplest case $N=1$ is presented in the Introduction.

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

A new class of solutions ($\kappa$-deformed $N$-phase solutions) is determined by

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

The case $N=1$ for the extended Harry Dym equation was considered in the Introduction.

4 Conclusion

Using the multi-component extended Harry Dym equation as an example, we studied integrable systems connected with their high-frequency limits ($\kappa=0$) by an invertible point transformation and obtained a new class of their solutions. Applying transformation (11), the multi-phase solutions of the high-frequency limits could be recalculated into a new kind of solutions for the original systems. As a future perspective, one can apply the $\kappa$-transformation to, e.g., multi-peakon solutions of the Extended Harry Dym equation to obtain a new class of solutions for its high-frequency limit, well-known as the Hunter–Saxton equation (see again [7]), etc.

Acknowledgements

MM gratefully acknowledges the support from GAČR under project P201/12/G028. MVP’s work was partially supported by the grant of Presidium of RAS “Fundamental Problems of Nonlinear Dynamics” and by the RFBR grant 14-01-00012. MVP thanks V.E. Adler, L.V. Bogdanov, E.V. Ferapontov, V.G. Marikhin, A.I. Zenchuk for important discussions.