Fig 2. Sketch of the flow in the t-plane Fig 2. Sketch of the flow in the t-plane. The cross denotes the point vortex (it is not present in the first problem). The portion of the sketch below AB is the image of the flow into the bottom. From: Solitary gravity waves and free surface flows past a point vortex IMA J Appl Math. 2017;82(4):821-835. doi:10.1093/imamat/hxx015 IMA J Appl Math | © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Fig 1. Sketch of the flow in the z-plane. From: Solitary gravity waves and free surface flows past a point vortex IMA J Appl Math. 2017;82(4):821-835. doi:10.1093/imamat/hxx015 IMA J Appl Math | © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Fig 3. Plots of the solitary wave solutions computed in Section 3 (solid curves), compared against the KdV solution (dashed curves) given by (3.12) for various values of ω. The solutions correspond to: (i) ω=0.5, (ii) ω=0.4, (iii) ω=0.3, (iv) ω=0.2, (v) ω=0.15 and (vi) ω=0.1. From: Solitary gravity waves and free surface flows past a point vortex IMA J Appl Math. 2017;82(4):821-835. doi:10.1093/imamat/hxx015 IMA J Appl Math | © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Fig 10. Profiles of solutions from branch (a) in Fig Fig 10. Profiles of solutions from branch (a) in Fig. 8 (parameter values Γ=5 and γ=0.95). The maximum amplitude solution (i) has the values α=4.60, F=3.03. The other solutions have the values (ii) α=4.41, F=3.02, (iii) α=4.08, F=3.03 and (iv) α=2.68, F=10. From: Solitary gravity waves and free surface flows past a point vortex IMA J Appl Math. 2017;82(4):821-835. doi:10.1093/imamat/hxx015 IMA J Appl Math | © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Fig 8. A plot of the solution branches in the (1/F,α) plane Fig 8. A plot of the solution branches in the (1/F,α) plane. The solution branch (a) has parameter values γ=0.95, Γ=5, (b) γ=0.95, Γ=1 (c) γ=0.5, Γ=1, and (d) γ=0.5, Γ=0.1. The branch (e) is the solution branch of unperturbed gravity waves, computed in Section 3. The dashed line is the plot of αmax given by (4.13), at which the solution branches terminate, since the amplitude cannot exceed this critical value. From: Solitary gravity waves and free surface flows past a point vortex IMA J Appl Math. 2017;82(4):821-835. doi:10.1093/imamat/hxx015 IMA J Appl Math | © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Fig 9. Profiles of solutions from branch (b) in Fig Fig 9. Profiles of solutions from branch (b) in Fig. 8 (parameter values Γ=1 and γ=0.95). The maximum amplitude solution (i) has values α=1.52, F=1.75. The other solutions have the values (ii) α=1.49, F=1.74, (iii) α=1.41, F=1.73, (iv) α=1.17, F=1.77, (v) α=0.90, F=2.13 and (vi) α=0.71, F=10. From: Solitary gravity waves and free surface flows past a point vortex IMA J Appl Math. 2017;82(4):821-835. doi:10.1093/imamat/hxx015 IMA J Appl Math | © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Fig 11. Profiles for the parameter values Γ=−0. 1, γ=0. 5 and (i) α=0 Fig 11. Profiles for the parameter values Γ=−0.1, γ=0.5 and (i) α=0.45, (ii) α=0.35, (iii) α=0.25 and (iv) α=0.15. Notice that the maximum height achieved is no longer given by α. From: Solitary gravity waves and free surface flows past a point vortex IMA J Appl Math. 2017;82(4):821-835. doi:10.1093/imamat/hxx015 IMA J Appl Math | © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Fig 4. Plot of F versus ω for larger values of ω Fig 4. Plot of F versus ω for larger values of ω. The dotted and dashed line are computed by using the series (3.7) for N=100 and N=300 respectively. The solid line is computed using the series (3.16) with N=100. From: Solitary gravity waves and free surface flows past a point vortex IMA J Appl Math. 2017;82(4):821-835. doi:10.1093/imamat/hxx015 IMA J Appl Math | © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Fig 5. Plot of F versus ω computed with N=2000 Fig 5. Plot of F versus ω computed with N=2000. The first minimum is shown to occur at ω≈0.996. From: Solitary gravity waves and free surface flows past a point vortex IMA J Appl Math. 2017;82(4):821-835. doi:10.1093/imamat/hxx015 IMA J Appl Math | © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Fig 6. The solution of the problem formulated in Section 4 Fig 6. The solution of the problem formulated in Section 4.1 for a variety of parameter values. Plots (i) and (ii) are solutions of the form (4.4) with parameter values (i) Γ=1, γ=0.5 and (ii) Γ=−1, γ=0.5. Plots (iii) and (iv) are given by (4.9). Both are with Γ=−1. The values of γ are (iii) γ=0.6535≈γ* and (iv) γ=0.9. In plot (iii) we can see that the free surface touches itself at x=0. If we decrease γ further the free surface self intersects. We can see clearly in (iv) the location of the bifurcation points on the free surface. From: Solitary gravity waves and free surface flows past a point vortex IMA J Appl Math. 2017;82(4):821-835. doi:10.1093/imamat/hxx015 IMA J Appl Math | © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Fig 7. The top row consists of plots of the solution (4 Fig 7. The top row consists of plots of the solution (4.4) as γ→1, while the bottom row shows the solution (4.9) as γ→1. For the top row , Γ=1, (i) γ=0.999, (ii) γ=0.999999 and (iii) γ=1. For the bottom row, Γ=−1, (iv) γ=0.99, (v) γ=0.999 and (vi) γ=1. From: Solitary gravity waves and free surface flows past a point vortex IMA J Appl Math. 2017;82(4):821-835. doi:10.1093/imamat/hxx015 IMA J Appl Math | © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.