Geodesic
Numerical computations of integrals over paths on Riemann surfaces of genus $N$
Teoretičeskaâ i matematičeskaâ fizika, Tome 101 (1994) no. 2, pp. 179-188 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper is a continuation of work by Forest and Lee [1,2]. In [1,2] it was proved that the function theory of periodic soliton solutions occurs on the Riemann surfaces $\Re$ of genus $N$, where the integrals over path on $\Re$ play the most fundamental role. In this paper a numerical method is developed to evaluate these integrals. Precisely, the aim is to develop a computational code for integrals of the form $$ \displaystyle \int _{\gamma }\,f(z)\frac {dz}{R(z)},\qquad \text {or}\qquad \displaystyle \int _{\gamma }\, f(z)R(z)\,dz,$$ where $f(z)$ is any single-valued analytic function on the complex plane $\mathbf C$, and $R(z)$ is two-valued function on $\mathbf C$ of the form $$ R^2(z)=\displaystyle \prod ^{2N+\delta }_{k=1}\,(z-z_0(k)),\qquad \delta =0\quad \text {or}\quad 1,$$ where $\bigl \{z_0(k),1\le k\le 2N+\delta \bigr \}$ are distinct complex numbers which play the role of the branch points of the Riemann surface $\Re =\bigl \{(z,R(z))\bigr \}$ of genus $N-1+\delta$. The integral path $\gamma$ is continuous on $\Re$. The numerical code is developed in “Mathematica” [3]. 
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     title = {Numerical computations of integrals over paths on {Riemann} surfaces of genus $N$},
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J.-E. Lee. Numerical computations of integrals over paths on Riemann surfaces of genus $N$. Teoretičeskaâ i matematičeskaâ fizika, Tome 101 (1994) no. 2, pp. 179-188. http://geodesic.mathdoc.fr/item/TMF_1994_101_2_a1/

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[2] Lee J. E., “The Multi-Phase Averaging Techiques and the Modulation Theory of the Focusing and Defocusing Nonlinear Schrödinger Equations”, Commun. in PDE, 15(9) (1990), 1293–1311 | DOI | MR | Zbl

[3] Wolfram S., Mathematica (A System for Doing Mathematics by Computer), 2-nd ed., Addison-Wesley, 1991 ; Mathematica software, Mathematica v. 2.1, 1992, distributed by Wolfram Research Inc.

[4] Overman E. A., McLaughlin D. W., Bishop A. R., “Coherence and chaos in the driver, damped sine-Gordon equation: measurement of the solution spectrum”, Physica D, 19 (1985), 1–41 | DOI | MR

[5] Bishop A. R., Forest M. G., McLaughlin D. W., Overman E. A., “A quasi-periodic route to chaos in a near-integrable PDE”, Physica D, 23 (1986), 293–328 | DOI | MR | Zbl

[6] Ercolani N. M., Forest M. G., McLaughlin D. W., “Geometry of modulational instability (III Homoclinic orbits for the periodic sine-Gordon equation)”, Physica D, 43 (1990), 349–384 | DOI | MR | Zbl

[7] Forest M. G., Pagano S., Parmentier R. D., Christiansen P. L., Soerensen M. P., Sheu S. P., “Numerical evidence for global bifurcations leading to switching phenomena in long Josephson junctions”, Wave Motion, 12 (1990) | DOI | Zbl

[8] Springer S. G., Introduction to Riemann surface, Chelsea, N. Y., 1981 | MR | Zbl