title: Rogue waves on the double periodic background in Hirota equation
authors: N. Sinthuja K. Manikandan M. Senthilvelan
text: 
  	
  	
  	
  	
  	Department of Nonlinear Dynamics, Bharathidasan University, Tiruchirappalli 620024, Tamil Nadu, India.
  	
  	
  	
  		We construct rogue wave solutions on the double periodic background for the Hirota equation through one fold Darboux transformation formula.  We consider two types of double periodic solutions as seed solutions.  We identify the squared eigenfunctions and eigenvalues that appear in the one fold Darboux transformation formula through an algebraic method with two eigenvalues.  We then construct the desired solution in two steps.  In the first step, we create double periodic waves as the background.  In the second step, we build rogue wave solution on the top of this double periodic solution. We present the localized structures for two different values of the arbitrary parameter each one with two different sets of eigenvalues.
  	Rogue waves on the double-periodic background in Hirota equation
  	N. Sinthuja et al.
  	Rogue waves on the double periodic background in Hirota equation
      N. Sinthuja 1, K. Manikandan 1 M. Senthilvelan 1, e-mail:velan@cnld.bdu.ac.in
      December 30, 2023
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  § INRODUCTION
  
  The nonlinear Schrödinger (NLS) equation describes the wave phenomena in the ocean <cit.> and the propagation of optical pulse in optical fiber <cit.>.  Considering the external factors such as depth of the sea, bottom friction and viscosity in the ocean and the femtosecond pulse propagation in fibers, several higher order NLS equations have been introduced in the literature <cit.>.  One such higher-order NLS equation is the Hirota equation.  In dimensionless form, the Hirota equation reads <cit.>,
  
      i ψ_t+α_2(ψ_xx+2|ψ|^2ψ)-i α_3(ψ_xxx+6|ψ|^2ψ_x)=0,
   
  where the function ψ =ψ(x,t) denotes the complex wave envelope,  α_2 and α_3 are arbitrary parameters.  The subscripts t and x stand for partial derivatives with respect to temporal and spatial variables, respectively. When α_2=1,α_3=0, Eq. (<ref>) reduces to the standard NLS equation.  During the past two decades, several works have been devoted to construct and analyze the characteristics of localized solutions in the Hirota equation <cit.>.  Multisolitons, breathers and rogue wave (RW) solutions have been constructed for the Hirota equation using different techniques, see for example Refs. <cit.>.  In these studies, RW solutions have been constructed only on the constant wave background.  
  
  In nature, RWs can arise from a variety of backgrounds, including  constant background, multi-soliton background and periodic wave background <cit.>. Recently, interest has been shown to construct RW solutions on the periodic background.  Initially, the occurrence of RWs on periodic background has been brought out only through numerical schemes.  For example, computations of RWs on the periodic background were performed in <cit.>, where the authors have computed the solutions of Zakharov-Shabat spectral problem only numerically.  Similarly, RWs on the double-periodic background were constructed in <cit.> but only via a numerical scheme. The resulting wave patterns exhibit periodicity in both space and time.  Differing from these, very recently, Chen and his collaborators have reported an algorithm to construct RW solutions on the periodic wave background for the focusing NLS equation <cit.>.  The same authors have also derived RW solutions on the double-periodic background for the focusing NLS equation using an effective algebraic method <cit.>.   Subsequently, Peng et al have studied the characteristics of RWs on the periodic background for the Hirota equation <cit.>. However, the authors have reported RWs only for a limiting case.  The studies on RWs on the periodic background can be found potential applications both in experiment and theory in their relative fields <cit.>.  
  
  
  In this work, we construct RW solutions on the double-periodic background for the Hirota equation.  We utilize one fold Darboux transformation formula and the method of nonlinearization of Lax pairs to derive these solutions.  To begin, we assume a constraint between the potential and squared eigenfunctions of the Lax system for the Hirota equation.  Using this constraint we identify Hamiltonians associated with the spatial and temporal parts of the Lax pair.  We notice that the Hamiltonian derived from spatial part of the Lax pair of Hirota equation matches with the one constructed for the NLS equation.  This is because the spatial part of the spectral problem of NLS equation and the Hirota equation are one and the same.  However, the Hamiltonian associated with the time part of the Hirota equation differs from NLS counterpart.  Since the spatial part is same, we obtain the same sets of differential constraints which connect the potential with eigenvalues that are given in Ref. <cit.>.  We construct periodic eigenfunctions by solving the differential constraints in terms of known double periodic solutions of the Hirota equation.  Substituting these periodic eigenfunctions and the identified set of eigenvalues in the one-fold Darboux transformation formula, we create the double-periodic wave background.  We then construct another linearly independent solution of the same Lax equations for the same set of eigenvalues which in turn creates the RW solution on top of the double-periodic wave  background.  Since the time evolution part in the Lax pair of Hirota equation differs from the NLS equation the second independent solution which we construct for the Hirota equation differs from the NLS equation.  We present surface plots of |r̂| for two different values of α_3 each one with two different sets of eigenvalues.  We find that the amplitude of RWs increases in the order of eigenvalues, such as λ_2 < λ_1 < λ_3 in the (x-t) plane.  When we vary the value of the arbitrary parameter α_3, we notice that the amplitude of RWs remains the same whereas the double-periodic background waves are localized in different positions in the (x-t) plane.   
  
  We organize our presentation as follows.  In Sec. 2, to begin, we present the Lax pair and one-fold Darboux transformation formula for the Hirota equation and two types of double periodic solutions of the Hirota equation.  We also briefly recall the method of constructing RW solution on the double periodic background.  In Sec. 3, we apply the algebraic method with two eigenvalues to Hirota equation and construct the RWs on the desired background.  In Sec. 4, we summarize our work.
  
  
  § LAX PAIR AND DARBOUX TRANSFORMATION
  
  The complete integrability of the nonlinear system (<ref>) is guaranteed by the presence of Lax pair.  Once the Lax pair is known, one can efficiently exploit the Darboux transformation technique to construct various localized solutions <cit.> for the concerned equation.   With the help of Darboux transformation method, we can construct a new solution ψ(x,t)=r̂(x,t) by providing a suitable seed solution ψ(x,t)=r(x,t).  The solution ψ(x,t)=r(x,t) to the Hirota Eq. (<ref>) comes out from the compatibility condition (φ_xt=φ_tx) of the pair of linear equations on φ. The Lax pair for the Hirota equation is given by <cit.>
  
      φ_x=U(λ,r) φ,     U(λ,r)=[   λ   r; -r̅  -λ ],
  
  and
  
      φ_t = V(λ,r) φ,
         V(λ,r)=λ^3 [  4α_3     0;     0 -4α_3 ]+λ^2 [   2 iα_2   4α_3 r; -4α_3 r̅ -2 i α_2 ]
          + λ[        2 α_3 |r|^2 2α_3 r_x+2 i α_2 r; 2α_3 r̅_x-2iα_2 r̅      -2  α_3 |r|^2 ]
          + [     iα_2 |r|^2-α_3(r r̅_x-r̅ r_x)    iα_2 r_x+ α_3( r_xx+2|r|^2 r ); iα_2 r̅_x- α_3( r̅_xx+2|r|^2 r̅ )    -iα_2 |r|^2+α_3(r r̅_x-r̅ r_x) ],
  
  where r is the potential and r̅ denotes the complex conjugate of r. One can easily verify that the compatibility condition U_t-V_x+[U,V]=0 gives rise to Eq. (<ref>).   
  
  
  
  The one-fold Darboux transformation of the Eq. (<ref>) reads <cit.>
  
      r̂(x,t)=r(x,t)+2(λ_1+λ̅_1)f_1 g̅_1/|f_1|^2+|g_1|^2,
  
  where φ=(f_1(x,t), g_1(x,t))^T - a non-zero solution of the Lax pair (<ref>) and (<ref>) for a fixed eigenvalue λ=λ_1 and r(x,t) and r̂(x,t) represents the seed and first iterated solutions of Eq. (<ref>), respectively.  With the chosen seed solution, the linear Eqs. (<ref>) and (<ref>) can be solved in a number of ways.  For example, it is trivial to integrate the Eqs. (<ref>) and (<ref>) with the plane wave background (r(x,t)=e^i(k x-ω t)) and generate the first-order RW solution (r̂(x,t)).  However, it is highly nontrivial to integrate the Eqs. (<ref>) and (<ref>) with double periodic solutions.  So one has to adopt a suitable methodology to integrate them with the genus-2 solutions as background.
  
  
  Very recently double periodic solutions for the Hirota equation with free real parameters have been constructed in Ref. <cit.>.  In our present study, we consider two types of double-periodic solutions which are given by the following rational functions of Jacobian elliptic functions sn, cn, and dn in <cit.>:
  
  
  
      (i)     r(x,t) = k √(1+k)sn(M t,k)-i C(x+v t)cn(M t,k)/√(1+k)-C(x+v t)dn(M t,k) e^i b t,
  
  where the function C(x+vt) is given by
  
      C(x+vt) = cn(√(1+k)x,κ)/dn(√(1+k)x,κ),   κ=√(1-k/1+k).
  
  
  
  
  
      (ii)     r(x,t)=ksn(M t/k,k)-i S(x+v t)dn(M t/k, k)/k(1-S(x+v t)cn(M t/k,k))e^ibt,
  
  where the function S(x+vt) is given by
  
      S(x+vt)=√(k/1+k)cn(√(2/k)x,κ),   κ=√(1-k/2),
  
  
  and k ∈ (0,1), M=2α_2, v=4α_3, b=2α_2, α_2=1 and α_3 is the arbitrary constant.  It follows from (<ref>) and (<ref>) that |r(x,t)|=|r(x+L,t)|=|r(x,t+T)|,    (x,t) ∈ℝ^2, with the fundamental periods L=4K(κ)/√(1+k) and T=2K(k) for (<ref>) and L=2√(2)K(κ) and T=4K(k) for (<ref>), where K(k) represents the complete elliptic integral of the first kind with the elliptic modulus parameter κ.  In Fig. <ref>, we display the qualitative profiles of double-periodic wave coming out from the solutions (<ref>) and (<ref>) for the modulus k=0.9 and α_2=1 with two different values of α_3.  In Figs. <ref>(a)-<ref>(b), double-periodic waves of (<ref>) are shown for α_3=1/6 (or) 0.1667 and α_3=0.85,  respectively.  Figures <ref>(c) and <ref>(d) represent the double-periodic profiles of (<ref>) for α_3=0.1667 and α_3=0.5, respectively.  From these plots, we observe that double-periodic wave is more localized for low values of α_3 when compared to the higher values of α_3 which can be visualized from Figs. <ref>(a) and <ref>(c).  We also notice that the orientation of double-periodic wave changes when we vary the value of the arbitrary parameter α_3.
  
  We intend to construct RWs on the double periodic profiles given in (<ref>) and (<ref>).  However, these solutions are travelling wave solutions.  We invoke the method of nonlinearization of Lax pairs <cit.> in order to identify the eigenvalues for which these solutions appear.  In this procedure one usually identifies the Hamiltonian associated with the spatial and temporal parts of the Lax pair by introducing a constraint between the potential r(x,t) and the eigenfunctions. In our case, the constraint reads r = ⟨f, f⟩: = f_1^2+g̅_1^2+f_2^2+g̅_2^2,  where the functions (f_1, g_1, f_2, g_2) are nothing but the solutions of the Lax pair Eqs. (<ref>) and (<ref>) and bar denotes the complex conjugate of the respective function. 
  
  
  After introducing this constraint we can proceed to construct a set of ordinary differential equations (ODEs) with r(x,t) as the dependent variable.  The order and number of these ODEs normally extends depending upon the number of unknowns to be determined.  In our case, we have four unknowns, namely (f_1, g_1, f_2, g_2) and so we sequentially go upto fourth order ODE. The last differential equation turns out to be the fourth-order Lax-Novikov equation in the hierarchy of stationary NLS equation.  The fourth-order complex-valued Lax-Novikov equation is integrable with two complex-valued constants of motion <cit.>. From these integrals of motion, we identify a set of admissible eigenvalues λ_1 and λ_2 which are symmetric about the imaginary axis.  Once the eigenvalues are fixed, substituting the double periodic solution and the obtained squared eigenfunctions in the one-fold Darboux transformation formula (<ref>), we can create double periodic solution as the background for the considered equation.  The resulting solution is periodic both in space and time. We then proceed to construct RWs on top of this double periodic waves.  To  achieve this we consider a second linearly independent solution for the Lax system (<ref>) and (<ref>) with the same set of eigenvalues and eigenfunctions with an unknown function. We determine this unknown function by substituting the considered second independent solution back in the Lax system and integrating the underlying equations. Substituting back the second solution in the one-fold Darboux transformation formula (<ref>), we obtain the RW solution on top of the double-periodic wave background.        
  
  
  § DEMONSTRATION ON HIROTA EQUATION
  
  In this section, we demonstrate the method of constructing RW solutions on the double periodic background for the Hirota equation.   In Ref. <cit.>, the authors have constructed the rogue wave solutions on the double periodic background for the focusing NLS equation.  We adopt the same procedure and derive the RW solution on the double periodic background to Eq. (<ref>). While implementing this procedure, we observe that Hirota equation also yields the same results upto section III in Ref. <cit.> with α_2=1.  While performing the procedure of nonlinearization of Lax pair for the Eqs. (<ref>) and (<ref>) with the constraint r = f_1^2+g̅_1^2+f_2^2+g̅_2^2 in the time part, we come across certain differences.  For instance, substituting the assumed constraint into Eqs. (<ref>) and (<ref>), we obtain the following finite-dimensional Hamiltonians, namely
  
      df_j/dx=∂ H_0/∂ g_j,    dg_j/dx=-∂ H_0/∂ f_j,     j=1,2,
  
  where
  
      H_0=⟨Λf,g⟩+1/2⟨f,f⟩⟨g,g⟩,
  
  and 
  
      df_j/dt=∂ H_1/∂ g_j,    dg_j/dt=-∂ H_1/∂ f_j,     j=1,2.
  
  where
  
      H_1=   1/4[i⟨f,g⟩+2/3⟨f,f⟩^2⟨g,g⟩^2+i1/2⟨f,f⟩⟨Λg,g⟩
         +i1/2⟨Λf,f⟩⟨g,g⟩  +4/3⟨Λf,g⟩⟨f,f⟩⟨g,g⟩ +2 i⟨Λ^2 f,g⟩
         +2/3⟨Λ^2 g,g⟩⟨f,f⟩+2/3⟨Λ^2 f,f⟩⟨g,g⟩  +8/3⟨Λ^3 f,g⟩
         -i/3⟨f,f⟩⟨f,g^3⟩ -i/3⟨g,g⟩⟨f^3,g⟩-1/6⟨f,f⟩^2⟨g^2,g^2⟩
         -1/6⟨g,g⟩⟨f^2,f^2⟩ -4/9⟨f,f⟩⟨Λf,g^3⟩-4/9⟨g,g⟩⟨Λf^3,g⟩].
  
  In Eqs. (<ref>) and (<ref>), the notations f, g and Λ denote f=(f_1,f_2,g̅_1,g̅_2)^T, g=(g_1,g_2,-f̅_1,-f̅_2)^T and 
   Λ=diag(λ_1,λ_2,-λ̅_1,-λ̅_2), respectively.  One may notice that the Hamiltonian (<ref>) which comes from the time evolution part of Lax pair of the Hirota equation differs from the NLS equation <cit.>.
  
  
  
  
  
  
  
  
  The squared eigenfunctions f_1^2 and g_1^2 which appear in the  Eq. (<ref>) can be obtained by substituting the expressions (<ref>) and (<ref>) and their derivatives in the following equations:  
  
      f_1^2=    λ_1/4(λ_1+λ̅_1)(λ_1-λ_2)(λ_1+λ̅_2)
            ×[r”+2|r|^2 r+4(b+λ_1^2)r+2λ_1 r'], 
      
      g_1^2=    λ_1/4(λ_1+λ̅_1)(λ_1-λ_2)(λ_1+λ̅_2)
            ×[r̅”+2|r|^2r̅+4(b+λ_1^2)r̅-2λ_1 r̅'], 
      
      f_1g_1=     -λ_1/4(λ_1+λ̅_1)(λ_1-λ_2)(λ_1+λ̅_2)
            ×[r'r̅-rr̅'+2λ_1 (2b+2λ_1^2+|r|^2)].
  
  (since the details of getting (<ref>) have already been given in Ref. <cit.> we do not reproduce them here).
  
  
  Similarly we can determine the expressions of squared eigenfunctions, namely f_2^2, g_2^2 and f_2g_2 from the above equations by replacing λ_1 by λ_2 and λ_2 by λ_1.  We consider two different sets of three admissible pairs of eigenvalues among ±λ_1, ±λ_2 and ±λ_3 which were found in Ref. <cit.> for the family of double-periodic wave solutions given in Eqs. (<ref>) and (<ref>).  Each pair of eigenvalues can be taken in place of λ_1 and λ_2.  In the first set, all the eigenvalues are considered to be real, that is λ_1=±√(τ_1), λ_2=±√(τ_2), and λ_3 =±√(τ_3) where τ_1 varies from 0 to 1, τ_2=τ_3-τ_1 and τ_3=1.  In the second set, it is considered that one of the eigenvalues λ_1=±√(τ_1) is positive and the other two eigenvalues are complex-conjugate, namely λ_2=±√(β + i γ) and λ_3=±√(β - i γ), where τ_1=2β and β varies from 0 to 0.5.   Now, substituting the resultant expressions coming out from (<ref>) and considering three admissible pairs of eigenvalues given above in (<ref>), we can set up the double-periodic wave pattern as background for the Hirota Eq. (<ref>).
  
  
  
  
  
  
  To generate RWs on this double periodic background, we construct a second linearly independent solution for the Lax pair Eqs. (<ref>) and (<ref>) with the same eigenvalues. The second linearly independent solution which we intend to construct should be non-periodic and grow linearly in x and t.  We consider the second linearly independent solution φ=(f̂_1,ĝ_1)^T for the Lax pair Eqs. (<ref>) and (<ref>) in the form 
  
      f̂_1 = f_1 η_1-2g̅_1/|f_1|^2+|g_1|^2,   ĝ_1 =  g_1 η_1+2f̅_1/|f_1|^2+|g_1|^2,
  
  where η_1 is the unknown function to be determined.  The Wronskian between the two solutions (<ref>) and (<ref>) is nonzero, 
  
      f_1ĝ_1-f̂_1g_1=   f_1(g_1 η_1+2f̅_1/|f_1|^2+|g_1|^2)
         -g_1(f_1 η_1-2g̅_1/|f_1|^2+|g_1|^2)=2,
  
  which in turn proves that the second solution is linearly independent from the first one.  
  
  
  
  Substituting (<ref>) into (<ref>) and utilizing (<ref>) for φ =(f̂_1,ĝ_1)^T, we obtain the following first-order partial differential equation for the unknown function η_1, that is 
  
      ∂η_1/∂ x = W_1:=     -4(λ_1+λ̅_1)f̅_1g̅_1/(|f̂_1|^2+|ĝ_1|^2)^2.
  
  Similarly, inserting (<ref>) into (<ref>) and utilizing (<ref>) for φ =(f̂_1,ĝ_1)^T, we obtain another first-order partial differential equation for η_1 in time derivative, namely
  
      ∂η_1/∂ t = W_2: =   2 (λ_1+λ̅_1)S_1f̅_1^2/(|f_1|^2+|g_1|^2)^2
      +2 (λ_1+λ̅_1)S_2g̅_1^2/(|f_1|^2+|g_1|^2)^2
         
      -4 (λ_1+λ̅_1)S_3f̅_1g̅_1/(|f_1|^2+|g_1|^2)^2,
  
  where S_1=2 r_xα_3+(i+4α_3(λ_1- λ̅_1))r, S_2=-2r̅_xα_3+(i+4α_3(λ_1- λ̅_1))r̅ and S_3=2|r|^2α_3+4α_3λ_1^2+(i-4α_3λ̅_1)(λ_1-λ̅_1).  
  The first-order equations (<ref>) and (<ref>) are compatible with each other (W_1t=W_2x).  Upon integrating the expressions (<ref>) and (<ref>), we obtain  
  
      η_1(x,t)=    ∫_x_0^x W_1(x',t)dx'+ ∫_t_0^t W_2(x_0,t')dt',
  
  where (x_0,t_0) is arbitrarily fixed. It is very difficult to integrate Eq. (<ref>) and obtain an explicit expression for η_1(x,t) analytically.  So we evaluate the expression (<ref>) numerically by Newton-Raphson method.
  
  Substituting the double periodic solutions r(x,t) (given in (<ref>) and (<ref>)) and the eigenfunctions of the second solution φ =(f̂_1,ĝ_1)^T of the linear equations (<ref>)-(<ref>) with λ=λ_1 in the one-fold Darboux transformation formula (<ref>), we obtain a new form of solution (RW solution on the double periodic background) to the Hirota Eq. (<ref>) in the form 
  
      r̂(x,t)=     r(x,t)+2(λ_1+λ̅_1)f̂_1ĝ̅_1/|f̂_1|^2+|ĝ_1|^2,
  
  where f̂_1 and ĝ_1 are given in (<ref>). We note here that this second solution differs from the one reported for the NLS equation.
  
  
  In Figs. <ref>(a)-(c), we show the surface plots of |r̂| of RWs on the double periodic background using the solution (<ref>) with k=0.8 and α_3=1/6 (or) 0.1667 for three real eigenvalues, namely λ_1=√(τ_1), λ_2=√(τ_2) and λ_3=√(τ_3) with τ_1=0.8, τ_3=1 and τ_2=τ_3-τ_1.  In this figure, in all the cases, RW attains their maximum amplitude at their origin, that is (x_0, t_0)=(0,0).  The amplitude of RWs is different in each case and is found to be |r̂| ≈ 4.13, 3.236 and 4.342 for the eigenvalues λ_1, λ_2 and λ_3, respectively.  The outcome is shown in Figs. <ref>(a), <ref>(b) and <ref>(c).  When we increase the value of the arbitrary parameter α_3 to 0.85, the amplitude of RWs remains the same as in the case of α_3=0.1667.  But we observe that the double-periodic background waves are localized in different positions in the (x-t) plane which can be visualized from the Figs. <ref>(a)-(c).  For k=0.95 and α_3=0.1667, the surface plots of |r̂| of RWs on the double periodic background using the solution (<ref>) with three real eigenvalues, namely λ_1=√(τ_1), λ_2=√(τ_2) and λ_3=√(τ_3) with τ_1=0.655, τ_3=1 and τ_2=τ_3-τ_1 are displayed in Figs. <ref>(a)-(c).  For this set of eigenvalues also, we observe that the amplitude of RWs exhibits a maximum value at their origins.  The amplitudes of RWs for the eigenvalues λ_1, λ_2 and λ_3 are |r̂| ≈ 4.015, 3.571 and 4.397 which are shown in Figs. <ref>(a), <ref>(b) and <ref>(c), respectively.  From these observations, we infer that the amplitude of RWs increases in the order of eigenvalues as λ_2 < λ_1 < λ_3 in the x-t plane.   The amplitude of RWs for the eigenvalue λ_3 is higher than that of the other two eigenvalues (λ_1 and λ_2).   
  
  
  Figure <ref> represents the surface plots of |r̂| of RWs on the double periodic background using the solution (<ref>) with k=0.9 for (a) λ_1=√(τ_1), τ_1=2β and (b) λ_2=√(β + i γ), β=0.45 and γ=0.217945. From Figs. <ref>(a) and <ref>(b), one can visualize the amplitude of RWs attains the maximum value |r̂| ≈ 4.35 and 3.841 for the real eigenvalue λ_1 and the complex eigenvalue λ_2, respectively.  The surface plots of |r̂| for the same set of eigenvalues but for the choice of the arbitrary parameter, α_3=0.5 are presented in Figs. <ref>(a) and <ref>(b).  Here we notice that the localization of double-periodic background waves occurs in different orientations whereas the amplitudes of RWs remain the same as in the case of α_3=0.1667. The RWs on the double periodic background which are shown in Figs. <ref>(a)-(b) are drawn for the same set of values as given in Fig. <ref> with k=0.95 and α_3=0.1667.  The maximum amplitudes of RWs for the eigenvalues λ_1 and λ_2 are |r̂| ≈ 4.025 and 3.811.  This is shown in Figs. <ref>(a) and <ref>(b).  We  conclude that the amplitude of RWs for the case of real eigenvalue (λ_1) is higher than that of the complex eigenvalue (λ_2).  
  
  
  § SUMMARY
  
  In this work, we have constructed RW solutions on the double-periodic background for the Hirota equation in an algebraic way.  One of the efficient tools to derive RW solution for the given nonlinear partial differential equation is the Darboux transformation method.  Through this method, one can construct the desired solution by appropriately choosing the seed solution.  However, it is very difficult to integrate the underlying equations arising from Lax pair when we consider double periodic solution as seed solution.  To circumvent this difficulty, we considered another procedure through which the necessary expressions for the squared eigenfunctions and the eigenvalues that appear in the one fold Darboux transformation formula can be identified.  The procedure which we brought-in to identify the eigenfunctions and eigenvalues is the method of nonlinearization of Lax pairs introduced long ago in a different context in the nonlinear dynamics literature.  With the help of this method and the spectral theory of Lax pairs, we have captured the eigenvalues and the squared eigenfunctions which appear in the one-fold Darboux transformation formula.  With the obtained expressions, we have created the double periodic waves as background.  In the second step, we have created the RW on top of this background.  We have presented the localized structures for two different values of the arbitrary parameter each one with two sets of eigenvalues.  Our studies reveal that the amplitudes of RWs increase in the order of eigenvalues as λ_2 < λ_1 < λ_3 in the (x-t) plane.  Our results also show that when we vary the value of the arbitrary parameter α_3 the amplitudes of RWs remain the same and localization of the double-periodic background waves occur in different orientations in the (x-t) plane. The application of this approach to few other evolutionary equations is under progress.
  
  
  
  § ACKNOWLEDGMENTS
  
  NS thanks the University for providing University Research Fellowship.  KM wishes to thank the Council of Scientific and Industrial Research, Government of India, for providing the Research Associateship under the Grant No. 03/1397/17/EMR-II. The work of MS forms part of a research project sponsored by National Board for Higher Mathematics, Government of India, under the Grant No. 02011/20/2018NBHM(R.P)/R&D 24II/15064.  The authors thank Prof. Dmitry E. Pelinovsky for helping to evaluate the numerical part of this work.   
  
  90 
  
  dudley
  J.M. Dudley, G. Genty, A. Mussot, A. Chabchoub and F. Dias, Nat. Rev. Phys. 1, 675-689 (2019).
  
  fujim
  W. Fujimoto, T. Waseda and A. Webb, Ocean. Dyn. 69, 101-121 (2019).
  
  degas
  A. Degasperis, S. Lombardo, M. Sommacal, J. Nonlinear Sci. 28, 1251-1291 (2018). 
  
  resi
  S. Residori, M. Onorato, U. Bortolozzo and F. T. Arecchi, Contemporary Physics 58, 53-69 (2017).
  
  ysong
  Y. Song, Z. Wang, C. Wang, K. Panajotov and H. Zhang, Advanced photonics 2, 024001 (2020).
  
  copie
  F. Copie, S. Randoux and P. Suret, Reviews in physics 5, 100037 (2020).
  
  onorato
  M. Onorato and P. Suret, Nat. Hazards 84, 541-548 (2016).
  	
  grimshaw
  R. Grimshaw and C. Yuan, Nat. Hazards 84, 493 (2016).
  
  akmdiev
  N. Akhmediev et al., J. Opt. 18, 063001 (2016).
  
  schen
  S. Chen and Z. Yan, Phys. Lett. A 383, 125906 (2019).
  
  vithya
  A. Vithya and M. S. Manirajan, Journal of Ocean Engineering and Science 5, 205-213 (2020).
  
  zlan
  Z. Lan, Appl. Math. Lett. 102, 106132 (2020).
  
  yyang
  Y. Yang, Z. Yan and B.A. Malomed, Chaos 25, 103112 (2015).
  
  zhangx
  X. Zhang and Y. Chen, Nonlinear Dyn. 93, 2169-2184 (2018).
  
  crabb
  M. Crabb and N. Akhmediev, Phys. Rev. E 99, 052217 (2019).
  
  anki:soto
  A. Ankiewicz, J.M. Soto-Crespo and N. Akhmediev, Phys. Rev. E 81, 0466012 (2010).
  
  wang
  X. Wang, Y. O. Li and Y. Chen, Wave motion 51, 1149-1160 (2014).
  
  chow:anki
  A. Chowdury, A. Ankiewicz and N. Akhmediev, Proc. R. Soc. A 471, 20150130 (2015).
  
  mlix
  M. Li, X. Zhang, T. Xu and L. Li, J. Phys. Soc. Jpn. 89,  054004 (2020).
  
  jcen
  J. Cen and A. Fring, Physica D: Nonlinear Phenomena 397,  17-24 (2019).
  
  chen6
  S. Chen and Z. Yan, App. Math. Lett 95, 65-71 (2019).
  
  lli
  L. Li, Z. Wu, L. Wang and J.S. He, Ann. Phy. 334, 198-211 (2013).
  
  tao
  Y. S. Tao and J. S. He, Phys. Rev. E 85, 026601 (2012).
  
  agaf
  D. S. Agafontsev and V. E. Zakharov, Nonlinearity 28,  2791-2821 (2015).
  
  mu
  G. Mu, Z. Qin and R. Grimshaw, SIAM J. Appl. Math. 75,  1-20 (2015).
  
  agaf1
  D. S. Agafontsev, and V. E. Zakharov, Nonlinearity 29, 3551-3578 (2016).
  
  mani
  K. Manikandan, P. Muruganandam, M. Senthilvelan and M. Lakshmanan, Phys Rev E.  90,  062905 (2014).
  
  mani2
  K. Manikandan, P. Muruganandam, M. Senthilvelan and M. Lakshmanan, Phys Rev E.  92 093202 (2016).
  
  kedziora
  D. J. Kedziora, A. Ankiewicz and N. Akhmediev, Euro. Phys. J. Spec. Topics 223, 43-62 (2014).
  
  calini
  A. Calini and C. M. Schober, Wave motion 71, 5-17 (2017). 
  
  chen2
  J. Chen and D. E. Pelinovsky, Proc. R. Soc. A 474, 20170814 (2018). 
  
  chen4
  J. Chen, D. E. Pelinovsky, and R. E. White, Phys. Rev. E 100, 052219 (2019). 
  
  peng
  W. Q. Peng, S. F. Tian,  X. B. Wang and T. T. Zhang, Wave motion 93, 102454 (2020).
  
  rand
  S. Randoux, P. Suret, A. Chabchoub, B. Kibler, and G. El, Phys. Rev. E 98, 022219 (2018).
  
  kim
  O. Kimmoun et al., Sci. Rep. 6, 28516 (2007).
  
  zhou
  R. G. Zhou, J. Math. Phys 48, 013510 (2007).
  
  zhou1
  R. G. Zhou, Stud. Appl. Math 123, 311 (2009).
  
  chen1
  J. Chen and D. E. Pelinovsky, Nonlinearity 31, 1955 (2018).
  
  chen3
  J. Chen and D. E. Pelinovsky, J. Nonlinear Sci. 51, 424 (2019). 
  
  
  
  
  