Quasi-periodic solutions of the negative-order Jaulent–Miodek hierarchy

Abstract

A complete algorithm is developed to deduce quasi-periodic solutions for the negative-order KdV (nKdV) hierarchy by using the backward Neumann systems. From the nonlinearization of Lax pair, the nKdV hierarchy is reduced to a family of backward Neumann systems via separating temporal and spatial variables. The backward Neumann systems are shown to be integrable in the Liouville sense, whose involutive solutions yield the finite parametric solutions of nKdV hierarchy. The negative-order Novikov equation is given, which specifies a finite-dimensional invariant subspace of nKdV flows. By the Abel-Jacobi variable, the nKdV flows are integrated with Abel-Jacobi solutions on the Jacobi variety of a Riemann surface. Finally, the Riemann-Jacobi inversion of Abel-Jacobi solutions is studied, from which some quasi-periodic solutions to the nKdV hierarchy are obtained. 1 coupling of KdV and nKdV flows. From the configuration of KdV6, Kupershmidt depicted it as a nonholonomic deformation of bi-Hamiltonian system (i.e. the Kupershmidt deformation) [6], and Zhou further generalized the nonholonomic perturbation to the mixed hierarchy of soliton equations, which includes the Kupershmidt deformation as its special member [7]. Recently, the Hamiltonian structure, infinitely many conservation laws, N -soliton, N -kink and quasiperiodic wave solutions have been derived for the nKdV equation [8,9]. It is seen that soliton and kink solutions occur simultaneously in the nKdV equation. Also, the nKdV equation is found to be linked with the Camassa-Holm (CH) equation via a reciprocal transformation [10], such that the nKdV equation becomes a hot spot of integrable nonlinear evolution equations (INLEEs). Therefore, this paper is intend to design a constructing scheme for getting quasiperiodic solutions to the nKdV hierarchy characterized by the nKdV equation (1.1).Due to the negative powers of Olver's operator, the original form of nKdV equation is of a nonlocal INLEE, where the nonlocality is eliminated by introducing an additional spectral potential v or ρ. As known so far, most attention is paid to positive-order INLEEs (pINLEEs), while the negative-order INLEEs (nINLEEs) are not studied as comprehensively as the positiveorder ones. The main reason lies in the fact that the nonlocality of nINLEEs is out of the control of usual methods of analysis. Even in a long time, there was no a fascinating example of nINLEEs of physical interests in the community of mathematical physics. After the physical derivation of CH equation [11], the contented tranquility of such a subject has been completely disturbed ever since. It is found that the CH equation possesses not only the non-smooth soliton solutions (peakons) [11,12], but also the presence of wave breaking [13]. Soon afterwards, the interest in the study of non-smooth solitons leads to the advent of peakon systems of Degasperis-Procesi (DP), CH2, and some others with cubic nonlinearity [14][15][16][17]. Seemingly, the peakon systems are described as local INLEEs by the Helmholtz operator 1 ...

Exotic wave patterns in Riemann problem of the high‐order Jaulent–Miodek equation: Whitham modulation theory

“…The Lax pair 37 of the high‐order JM equation () is $$\begin{eqnarray} {} \psi _{xx}={\it A}\psi , \ \ \psi _{t}=-\frac{1}{2} {\it B}_x\psi +{\it B}\psi _x, \end{eqnarray}$$with $$\begin{eqnarray} {} {\it A}=-\lambda ^2+\lambda r+q, \ \ {\it B}=\lambda ^2+\frac{1}{2} r \lambda +\frac{1}{2} q+\frac{3}{8} r^2, \end{eqnarray}$$where λ is a free spectral parameter. Demanding that $$(\psi _{xx})_t=(\psi _t)_{xx}$$ for any λ, the high‐order JM equation () is reproduced.…”

“…Subsequently, Geng et al 35,36 expressed some quasi-periodic solutions for several (2+1)-dimensional nonlinear evolution equations by using the Riemann theta functions. Recently, Chen 37 presented the quasi-periodic solutions to the negativeorder JM hierarchy by using a family of backward Neumann-type systems. Tsuchida 38 proposed a systematic method to generate the modified JM equation 39 by an inverse Miura map.…”

“…Motivated by the work referred above, this paper investigates the Riemann problem of the second member in the JM hierarchy, i.e., the high‐order JM equation 30, 37. $$\begin{eqnarray} {} \begin{split} &q_{t}=-\frac{1}{4} q_{xxx}+\frac{3}{2} q q_{x}-\frac{9}{8}r_x r_{xx}-\frac{3}{8}r r_{xxx}+\frac{3}{2}q r r_x +\frac{3}{8} q_x r^2,\\ &r_{t}=-\frac{1}{4} r_{xxx}+\frac{3}{2} q_{x} r+\frac{3}{2}q r_{x}+\frac{15}{8}r_x r^2, \end{split} \end{eqnarray}$$which can be reduced to the KdV equation for $$r(x,t)=0$$.…”

Quasi‐periodic solutions to a hierarchy of integrable nonlinear differential–difference equations

“…[14][15][16][17][18][19] Using algebro-geometric method, quasi-periodic solutions of many integrable nonlinear evolution equations related to 2 × 2 matrix spectral problems are obtained on the basis of the theory of hyperelliptic curves. [20][21][22][23][24][25][26][27][28][29] The authors in previous studies 30,31 propose a unified framework for successfully constructing quasi-periodic solutions of the entire Boussinesq hierarchy associated with a third-order differential operator by using Burchnall-Chaundy polynomials and the theory of trigonal curves. In previous works, 32,33 a general method was developed to introduce trigonal curves by resorting to the characteristic polynomial of the Lax matrix.…”

“…This research began in the 1970s with the pioneering work of Novikov, Matveev, Dubrovin, Its, Lax, McKean, and others 14–19 . Using algebro‐geometric method, quasi‐periodic solutions of many integrable nonlinear evolution equations related to $$$ 2\times 2 $$$ matrix spectral problems are obtained on the basis of the theory of hyperelliptic curves 20–29 . The authors in previous studies 30,31 propose a unified framework for successfully constructing quasi‐periodic solutions of the entire Boussinesq hierarchy associated with a third‐order differential operator by using Burchnall–Chaundy polynomials and the theory of trigonal curves.…”