Algebro-Geometric Solutions of the Coupled Chaffee-Infante Reaction Diffusion Hierarchy

Yue, Chao; Xia, Tiecheng

doi:https://doi.org/10.1155/2021/6618932

Advances in Mathematical Physics

On this page

Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6618932 | https://doi.org/10.1155/2021/6618932

Algebro-Geometric Solutions of the Coupled Chaffee-Infante Reaction Diffusion Hierarchy

Chao Yue¹and Tiecheng Xia²

Academic Editor: Wen-Xiu Ma

Received17 Oct 2020

Accepted18 May 2021

Published22 Jul 2021

Abstract

The coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a matrix spectral problem is derived by using two sets of the Lenard recursion gradients. Based on the characteristic polynomial of the Lax matrix for the CCIRD hierarchy, we introduce a trigonal curve of arithmetic genus , from which the corresponding Baker-Akhiezer function and meromorphic functions on are constructed. Then, the CCIRD equations are decomposed into Dubrovin-type ordinary differential equations. Furthermore, the theory of the trigonal curve and the properties of the three kinds of Abel differentials are applied to obtain the explicit theta function representations of the Baker-Akhiezer function and the meromorphic functions. In particular, algebro-geometric solutions for the entire CCIRD hierarchy are obtained.

1. Introduction

It is significantly important to search for solutions of nonlinear partial differential equations of mathematical physics. There are many methods to find the exact solutions [1, 2] and approximate solutions [1–3] of various nonlinear partial differential equations. Reaction diffusion equations are effective and important mathematical models, which contribute to explaining processes of the transition, diffusion, and fluidity of matter. Constructing exact solutions of such equations has been widely used in mathematics, physics, chemistry, biology, and other fields. Therefore, it is necessary for us to study algebro-geometric constructions of the coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a matrix spectral problem. The third member in the hierarchy is which is called the CCIRD equation compared with Equation (0.4) in Ref. [4].

Algebro-geometric solution is closely associated with the inverse spectral theory [5, 6], and the solution of the KdV equation with an initial value problem was solved by the use of the method in Ref. [7]. Over the recent decades, integrable equations related to matrix spectral problems have been extensively researched. Several systematic methods have been developed to construct algebro-geometric solutions for integrable equations such as KdV, Kadomtsev-Petviashvili equation, modified KdV, sine-Gordon, Ablowitz-Kaup-Newell-Segur, the Camassa-Holm equations, and Ablowitz-Ladik lattice [8–27]. But the study of algebro-geometric solutions of the whole hierarchy of is still a challenging problem. Fortunately, in Ref. [28], a unified framework was proposed to yield algebro-geometric solutions of the whole Boussinesq hierarchy. Based on the work of that, a systematic method was proposed to define the trigonal curve and develop the framework to analyse soliton equations associated with the matrix spectral problems, from which the algebro-geometric solutions of some entire hierarchies are obtained [29–34]. In Ref. [29], algebro-geometric quasi-periodic solutions to the three-wave resonant interaction hierarchy related to the trigonal curve with three infinite points were obtained. Wang and Geng constructed algebro-geometric solutions of a new hierarchy of soliton equations associated with a matrix spectral problem [30] based on the methods used in [28, 29]. Later, Ma analysed the four-component AKNS soliton hierarchy, particularly asymptotics of the Baker-Akhiezer functions, in such a way that it proposes a general theory applicable to soliton hierarchies associated with matrix spectral problems [31]. As a continuous study of [31], Ma constructed algebro-geometric solutions of the four-component AKNS soliton hierarchy in terms of a general theory of trigonal curves [32]. However, as far as we know, algebro-geometric solutions to the CCIRD hierarchy have not been investigated. The most important result of this paper is to give the explicit algebro-geometric solutions to the CCIRD hierarchy related to matrix spectral problems by using the approaches used in [28–30], which complements the existing works in this area.

The outline of this paper is as follows. In Section 2, we obtain the CCIRD hierarchy related to a matrix spectral problem based on the Lenard recursion equations. In Section 3, a trigonal curve of arithmetic genus with three infinite points is introduced by the use of the characteristic polynomial of the Lax matrix for the stationary CCIRD equations, from which the stationary Baker-Akhiezer function and associated meromorphic functions are given on . Then, the stationary CCIRD equations are decomposed into the system of Dubrovin-type ordinary differential equations. In Section 4, we present the explicit theta function representations of the stationary Baker-Akhiezer function, of the meromorphic functions, and, in particular, of the potentials for the entire stationary CCIRD hierarchy. In Section 5, we extend all the Baker-Akhiezer functions, the meromorphic functions, the Dubrovin-type equations, and the theta function representations dealt with in Sections 3 and 4 to the time-dependent case.

2. The CCIRD Hierarchy

In the section, we shall derive the CCIRD hierarchy associated with a spectral problem: where the potential and is a spectral parameter. Next, we introduce the Lenard gradient sequences with two initial points and two operators are defined as

Then, the sequences and , can be uniquely determined and the first several members read as

In order to obtain the CCIRD hierarchy, we solve the stationary zero-curvature equation which is equivalent to where each entry is a Laurent expansion in : and satisfy . A direct calculation shows that (8)–(16) implies the Lenard equation

Substituting Equation (17) into Equation (18) and collecting the same powers of , we get the following recursion relations: where Since equation has the general solution then can be expressed as where and are arbitrary constants.

Consider the auxiliary problem: where each entry has the form and satisfies . Then, we introduce which is determined by and are arbitrary constants. It is easy to find that satisfies the Lenard equation

Then, from the compatibility condition of Equations (2) and (22), we have where the vector fields the constant vectors and the projective map . The third member in the hierarchy (22) is (for convenience, we take )

Taking in system (23), we have which is called the CCIRD equation compared with Equation (0.4) in Ref. [4].

3. The Stationary Baker-Akhiezer Function

In the section, we are devoted to detailed study of the stationary Baker-Akhiezer function and the associated meromorphic functions. Then, the system of Dubrovin-type differential equations is derived. Let us consider the stationary CCIRD hierarchy: which is equivalent to the stationary zero-curvature equation, and determined by (21). It is easy to verify that the matrix also satisfies the stationary zero-curvature equation. Then, the characteristic polynomial of Lax matrix is independent of variable with the expansion where and are polynomials with constant coefficients of :

It is easy to find that is a polynomial of degree with respect to as . Then, naturally leads to a trigonal curve with . For convenience, we denote the compactification of the curve by the same symbol . Hence, becomes a three-sheeted Riemann surface of arithmetic genus if it is nonsingular or smooth. Here, the meaning of nonsingular is that at each point holds. For , these curves are typically nonhyperelliptic. Point on is represented as pairs satisfying (29) along with the three different points at infinity, which can be computed from the curve by choosing . The complex structure on is defined in the usual way by introducing local coordinate near point which is neither branch nor singular point of except the three points at infinity with local coordinate and local coordinate near branch or singular point .

Next, we shall define the meromorphic functions and on as follows: with the stationary Baker-Akhiezer function defined by

By using (31)–(33), a direct calculation gives that where

Through straightforward calculations, we obtain some main interrelationships among polynomials some of which are summarized below:

By observing Equations (21) and (38), one infers that , , and are polynomials with respect to of degree . Therefore, we may write them in the following forms: where .

Define

In order to more distinctly put forward the properties of and we introduce the holomorphic map , changing sheets, which is defined by where satisfy , that is,

Furthermore, the positive divisors on of degree are defined as with

Further properties of and are summarized as follows:

The dynamics of the zeros , , and of , , and are described according to Dubrovin-type equations as follows.

Lemma 1. Assume the zeros , , and of , , and remain distinct, respectively, and let . Then, , , and satisfy the system of differential equations

Proof. Substituting into (40) and (43), we have Then, we get Then, inserting into the first equation of (45) and in view of (70) and (71), we arrive at On the other hand, differentiating (46) with respect to , we find Comparing (72) and (73), we derive (66). Similarly, we can prove that Equations (67) and (68) hold.☐

4. Algebro-Geometric Solutions of the Stationary CCIRD Hierarchy

In the section, we continue our study of the stationary CCIRD hierarchy and will obtain explicit Riemann theta function representations for the two meromorphic functions , , the Baker-Akhiezer function , and the algebro-geometric solutions and for the CCIRD hierarchy. By introducing the local coordinate near , we have the following lemma.

Lemma 2. Suppose that satisfies the th stationary CCIRD system . Moreover, let . Then,

Proof. Substituting the three sets of ansatz into Riccati-type Equations (53) and (54), and comparing the coefficients of the same powers of , we derive (74) and (75). Equation (76) then follows from inserting (74) and (75) into (64).
One infers from Equations (34), (35), (74), and (75) that the divisors and of and are as follows: which means that are zeros of and its poles, are zeros of and its poles.
A tedious calculation reveals that the asymptotic behaviors of and near are given as Equip the Riemann surface with an appropriate fixed homology basis , in such a way that the intersection matrix of cycles satisfies For the present, we introduce the holomorphic differentials on defined by By using the basis and , the matrices and can be constructed from and it is possible to show that matrices and are invertible. Now, we define the matrices and by One can see that matrix is symmetric, and it has a positive definite imaginary part. If we normalize into the new basis then we have
Then, the Laurent expansion of (83) near yields the following results: Let denote the normalized Abelian differential of the second kind satisfying and introduce then we have where are integral constants and is an appropriately chosen base point on . The -periods of the differential are denoted by Then, from (85) and (87), we have in which we used .
Furthermore, the normalized Abelian differential of the third kind is holomorphic on with simple poles at and with residues 1 and -1, respectively, that is, where are integration constants. Let be the period lattice . The complex torus is called the Jacobian variety of . An Abel map is defined by with the natural linear extension to the factor group Considering the nonspecial divisor , and we define where and
denotes the th symmetric power of .☐

Theorem 3. Assume that the curve is nonsingular, and let . Then,

Proof. We prove only the first linearity of the Abel map with respect to in (98). Assume that for then, one computes which yields by the use of the standard Lagrange interpolation argument that which implies the first representation of (98). The second and third equalities in (98) follow from the same calculation.
Denote by the Riemann theta function associated with equipped with a fixed homology basis. For convenience, the function is defined as where is the vector of Riemann constants. Then, we get In view of (98), we could rewrite them as where
Combined with the above results, the theta function representations of and the algebro-geometric solutions of the stationary CCIRD hierarchy are presented in the next theorem.☐

Theorem 4. Assume that the curve is nonsingular. Let and let . Suppose that or or is nonspecial. Then, Finally, the theta representations of and are of the form

Proof. Let be defined by the right-hand side of (106). We intend to prove that with given by (64). For that purpose, we first inspect the zeros and poles of . Since they can only come from zeros of and , one can compute by using (34) and (35) that Then, where . Consequently, all zeros and poles of and on are simple and coincident. It remains to identify the essential singularities of and at . Considering (64), (76), (87), and the expression for in (106), we deduce that and share the same singularities and zeros. The Riemann-Roch uniqueness results in the holomorphic function , where is a constant. By (76), (87), and the right-hand side of (106), we have Then, we conclude , with which the proof of (106) is completed. By using the asymptotic properties of near , we get (109). Equations (78), (79), and (93) immediately yield that and have the following forms: Taking into account the asymptotic expansions of and near , , we have which together with (109) show the expressions (107), (108), and (110).☐

5. Algebro-Geometric Solutions of the CCIRD Hierarchy

In this section, we extend the results of Sections 3 and 4 to the time-dependent CCIRD hierarchy. In particular, we obtain Riemann theta function representations for the time-dependent Baker-Akhiezer function, the meromorphic function, and algebro-geometric solutions of the CCIRD hierarchy.

Similar to (31)–(33), we consider the following time-dependent Baker-Akhiezer function:

The compatibility conditions of the first three equations in (116) show that

It is easy to find that satisfies (118) and (119). Then, the characteristic polynomial of Lax matrix for the CCIRD hierarchy is a constant independent of variables and with the expansion where and are defined as in (27) and (28). Then, the CCIRD curve is determined by

Closely related to are the following two meromorphic functions and on defined by which imply from (116) that where and are defined as in (36)–(38). Hence, (39)–(45) also hold in the present context. Similarly, we have

After defining as (46)–(48) by replacing with , one infers from (74), (75), (124), and (128) that the divisors and of and are as follows:

Differentiating (122) and (123) with respect to and using (116), we get

Further properties of and can be presented, similar to (55)–(63), replacing with with , etc. The four important ones of that are given as follows:

Lemma 5. Assume (116) and (117) and let . Then,

Proof. Differentiating (59) (by replacing with ) with respect to , we get Without loss of generality, we take the integration constant as zero, then get which implies equation (132). Differentiating (55) (by replacing with ) with respect to and using (55), (60), and (63), we can deduce Thus, we prove the expression (133). The last one can be proved in the same way.☐

We present some properties of as follows.

Lemma 6. Assume (116) and (122), , and let . Then, Similar to Lemma 1, the zeros , , and of , , and are described in terms of Dubrovin-type equations as follows.

Lemma 7. (i)Assume the zeros of remain distinct for , where is open and connected. Then, satisfy the system of differential equations(ii)Assume the zeros of remain distinct for , where is open and connected. Then, satisfy the system of differential equations(iii)Assume the zeros of remain distinct for , where is open and connected. Then, satisfy the system of differential equationsFor convenience, we introduce the notation where and the corresponding homogeneous cases with In view of (138), we denote the function by and is the associated homogeneous quantity replacing by the corresponding homogeneous polynomials that is, especially

Lemma 8. Assume denotes the local coordinate near Then,

Proof. We only prove (151) and accordingly obtain (152). From (149), it is easy to see that thus, (151) is right for . By using (116), (117), and (149), we have (i)When by investigating (153), one can assume that has the following expansionfor some coefficients . Suppose that where and are defined in (74) and (75). Substituting (155) and (156) into (154) and comparing the same powers of yield from which it can be inferred where and are integration constants. We find that the coefficients of the power series for near and the coefficients of the homogeneous polynomials are differential polynomials in , with no arbitrary integration constants in their construction, and the definition of , it follows that it also can have no arbitrary integration constants and must consist purely of differential polynomials in , and . Therefore, we have On the other hand, we find (ii)When by analysing (153), one can assume that has the following form:for some coefficients . Suppose that where and are defined in (74) and (75). Inserting (161) and (162) into (154) and comparing the same powers of imply from which it can be inferred where and are integration constants. Managed together, we find that . Then, On the other hand, we get (iii)When in terms of (153), one can assume that has the following expansion:for some coefficients . Suppose that where and are defined in (74) and (75). Substituting (167) and (168) into (154) and comparing the same powers of yield from which one can infer where is an integration constant. Similarly, one can conclude . Therefore, On the other hand, Thus, we complete the proof of (151). Similarly, we can prove that (152) is right.
From (116), one infers that Therefore, Let be the normalized differential of the second kind holomorphic on with a pole of order at , Furthermore, we define the normalized differential of the second kind by In addition, we define the vector of -periods of the differential of the second kind Integrating Equation (176) gives rise to where are constants.☐

Given these results, the theta function representations of and , particularly, the algebro-geometric solutions of CCIRD hierarchy, are shown as follows.

Theorem 9. Assume that the curve is nonsingular. Let and let . Suppose that or or is nonspecial. Then, Finally, the theta representations of and read

Proof. The proof of (179), (180), and (182)–(185) is similar to that of Theorem 4, so we only need to prove (181). Let be defined by the right-hand side of (181). We want to prove that with given by (138). Then, we compute by using (124) and (125) that Then, where . Consequently, all zeros and poles of and on are simple and coincident. Similar to Theorem 4, one can find that and have the same essential singularities at . Then, the Riemann-Roch uniqueness results in .
We can explicitly rewrite and as where with the aid of the following theorem.☐

Theorem 10. Assume that the curve is nonsingular and let . Then,

Proof. First, we consider a meromorphic differential From the representation (180), we have where denotes a holomorphic differential on , that is, for some . Because is single-valued on , all - and -periods of are integer multiples of , and hence, for some . Similarly, for some , in terms of By symmetry of , this is equivalent to Using this equality and the linear equivalence , and , that is, we can present the other two equalities

6. Conclusions

In this paper, firstly we obtain the CCIRD hierarchy related to a matrix spectral problem based on the Lenard recursion equations, and a trigonal curve of arithmetic genus with three infinite points is introduced by using the characteristic polynomial of Lax matrix for the stationary CCIRD equations, from which the stationary Baker-Akhiezer function and associated meromorphic functions are given on . Then, the stationary CCIRD equations are decomposed into the system of Dubrovin-type ordinary differential equations. Furthermore, we present the explicit theta function representations of the stationary Baker-Akhiezer function, of the meromorphic functions, and, in particular, of the potentials for the entire stationary CCIRD hierarchy. Finally, we extend all the Baker-Akhiezer function, the meromorphic functions, the Dubrovin-type equations, and the theta function representations dealt with in Sections 3 and 4 to the time-dependent case. The technology presented in this paper can be applied to other hierarchies related to matrix spectral problems, to get more various algebro-geometric solutions, which will enrich and supplement the known results.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Natural Science Foundation of China (Grant Nos. 11971475 and 11805114), the Natural Science Foundation of Shandong Province (Grant Nos. ZR2016AL04, ZR2016FL05, and ZR2017MF039), and the High-Level Training Project of Taishan Medical University (No. 2015GCC07).

References

A. Yusuf, F. Tchier, and M. Inc, “New interaction and combined multi-wave solutions for the Heisenberg ferromagnetic spin chain equation,” The European Physical Journal Plus, vol. 135, no. 5, 2020.
View at: Publisher Site | Google Scholar
T. A. Sulaiman, A. Yusuf, and A. Atangana, “New lump, lump-kink, breather waves and other interaction solutions to the (3+1)-dimensional soliton equation,” Communications in Theoretical Physics, vol. 72, no. 8, article 085004, 2020.
View at: Publisher Site | Google Scholar
F. Tchier, M. Inc, and A. Yusuf, “Symmetry analysis, exact solutions and numerical approximations for the space-time Carleman equation in nonlinear dynamical systems,” The European Physical Journal Plus, vol. 134, no. 6, p. 250, 2019.
View at: Publisher Site | Google Scholar
M. L. Wang and X. Bai, “Exact solutions for a generalized Burgers-Fisher equation,” Journal of Lanzhou University(Natural Sciences), vol. 35, no. 2, pp. 1–6, 1999.
View at: Google Scholar
E. D. Belokolos, A. I. Bobenko, V. Z. Enol'skii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer, Berlin, 1994.
V. Batchenko and F. Gesztesy, “On the spectrum of Schrödinger operators with quasi-periodic algebro-geometric KDV potentials,” Journal d'Analyse Mathématique, vol. 95, no. 1, pp. 333–387, 2005.
View at: Publisher Site | Google Scholar
P. D. Lax, “Periodic solutions of the KdV equation,” Communications on Pure and Applied Mathematics, vol. 28, no. 1, pp. 141–188, 1975.
View at: Publisher Site | Google Scholar
E. Date and S. Tanaka, “Periodic multi-soliton solutions of Korteweg-de Vries equation and Toda lattice,” Progress of Theoretical Physics Supplement, vol. 59, pp. 107–125, 1976.
View at: Publisher Site | Google Scholar
Y. C. Ma and M. J. Ablowitz, “The periodic cubic Schrödinger equation,” Applications of Mathematics, vol. 65, no. 2, pp. 113–158, 1981.
View at: Publisher Site | Google Scholar
C. Yue and T. Xia, “Algebro-geometric solutions for the complex Sharma-Tasso-Olver hierarchy,” Journal of Mathematical Physics, vol. 55, no. 8, pp. 083511–083535, 2014.
View at: Publisher Site | Google Scholar
Z. Qiao, “-matrix and algebraic-geometric solution for integrable symplectic map,” Chinese Science Bulletin, vol. 44, no. 2, pp. 114–118, 1999.
View at: Publisher Site | Google Scholar
Z. Qiao, “Generalized -matrix structure and algebro-geometric solution for integrable system,” Reviews in Mathematical Physics, vol. 13, no. 5, pp. 545–586, 2001.
View at: Publisher Site | Google Scholar
R. Zhou, “Dynamical -matrices for the constrained Harry-Dym flows,” Physics Letters A, vol. 220, no. 6, pp. 320–330, 1996.
View at: Publisher Site | Google Scholar
R. Zhou, “The finite-band solution of the Jaulent-Miodek equation,” Journal of Mathematical Physics, vol. 38, no. 5, pp. 2535–2546, 1997.
View at: Publisher Site | Google Scholar
H. Pan, T. Xia, and D. Chen, “A two-component generalization of Burgers' equation with quasi-periodic solution,” Reports on Mathematical Physics, vol. 74, no. 2, pp. 235–250, 2014.
View at: Publisher Site | Google Scholar
I. M. Krichever, “Algebraic-geometric construction of the Zaharov-Sabat equations and their periodic solutions,” Doklady Akademii Nauk SSSR, vol. 227, pp. 394–397, 1976.
View at: Google Scholar
I. M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry,” Functional Analysis and Its Applications, vol. 11, no. 1, pp. 12–26, 1977.
View at: Publisher Site | Google Scholar
B. A. Dubrovin, “Completely integrable Hamiltonian systems associated with matrix operators and Abelian varieties,” Functional Analysis and Its Applications, vol. 11, no. 4, pp. 265–277, 1978.
View at: Publisher Site | Google Scholar
B. A. Dubrovin, “Theta functions and nonlinear equations,” Russian Mathematical Surveys, vol. 36, no. 2, pp. 11–92, 1981.
View at: Publisher Site | Google Scholar
B. A. Dubrovin, “Matrix finite-gap operators,” Revue Scientifique et Technique, vol. 23, pp. 33–78, 1983.
View at: Google Scholar
F. Gesztesy and R. Ratnaseelan, “An alternative approach to algebro-geometric solutions of the AKNS hierarchy,” Reviews in Mathematical Physics, vol. 10, no. 3, pp. 345–391, 1998.
View at: Publisher Site | Google Scholar
F. Gesztesy and H. Holden, “Algebro-geometric solutions of the Camassa-Holm hierarchy,” Revista Matemática Iberoamericana, vol. 19, pp. 73–142, 2003.
View at: Google Scholar
F. Gesztesy and H. Holden, Soliton Equations and Their Algebro-Geometric Solutions. Vol. I:(1+1)- Dimensional Continuous Models, vol. 79 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2003.
F. Gesztesy, H. Holden, J. Michor, and G. Teschl, Soliton Equations and Their Algebro-Geometric Solutions, Vol. II: (1+1)-Dimensional Discrete Models, vol. 114 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2008.
J. S. Geronimo, F. Gesztesy, and H. Holden, “Algebro-geometric solutions of the Baxter–Szegő difference equation,” Communications in Mathematical Physics, vol. 258, no. 1, pp. 149–177, 2005.
View at: Publisher Site | Google Scholar
W. Bulla, F. Gesztesy, H. Holden, and G. Teschl, “Algebro-geometric quasi-periodic finite gap solutions of the Toda and Kac-van Moerbeke hierarchies,” Amer. Math. Soc., vol. 135, pp. 1–79, 1998.
View at: Google Scholar
E. Fan, “The positive and negative Camassa–Holm- hierarchies, zero curvature representations, bi-Hamiltonian structures, and algebro-geometric solutions,” Journal of Mathematical Physics, vol. 50, no. 1, pp. 013525–013547, 2009.
View at: Publisher Site | Google Scholar
R. Dickson, F. Gesztesy, and K. Unterkofler, “Algebro-geometric solutions of the Boussinesq hierarchy,” Reviews in Mathematical Physics, vol. 11, no. 7, pp. 823–879, 1999.
View at: Publisher Site | Google Scholar
G. He, X. Geng, and L. Wu, “Algebro-geometric quasi-periodic solutions to the three-wave resonant interaction hierarchy,” SIAM Journal on Mathematical Analysis, vol. 46, no. 2, pp. 1348–1384, 2014.
View at: Publisher Site | Google Scholar
H. Wang and X. G. Geng, “Algebro-geometric solutions to a new hierarchy of soliton equations,” Journal of Mathematical Physics, Analysis, Geometry, vol. 11, no. 4, pp. 359–398, 2015.
View at: Google Scholar
W. X. Ma, “Trigonal curves and algebro-geometric solutions to soliton hierarchies I,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 473, no. 2203, p. 20170232, 2017.
View at: Publisher Site | Google Scholar
W. X. Ma, “Trigonal curves and algebro-geometric solutions to soliton hierarchies II,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 473, no. 2203, p. 20170233, 2017.
View at: Publisher Site | Google Scholar
Y. Hou, P. Zhao, E. Fan, and Z. Qiao, “Algebro-geometric solutions for the Degasperis-Procesi hierarchy,” SIAM Journal on Mathematical Analysis, vol. 45, no. 3, pp. 1216–1266, 2013.
View at: Publisher Site | Google Scholar
Y. Hou, E. Fan, Z. Qiao, and Z. Wang, “Algebro-geometric solutions for the derivative Burgers hierarchy,” Journal of Nonlinear Science, vol. 25, no. 1, pp. 1–35, 2015.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 Chao Yue and Tiecheng Xia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies